8498:
7271:
8493:{\displaystyle {\begin{aligned}E_{\mathbf {k} }e^{i\mathbf {k} \cdot \mathbf {x} }u_{\mathbf {k} }(\mathbf {x} )&={\frac {-\hbar ^{2}}{2m}}\nabla \cdot \left(i\mathbf {k} e^{i\mathbf {k} \cdot \mathbf {x} }u_{\mathbf {k} }(\mathbf {x} )+e^{i\mathbf {k} \cdot \mathbf {x} }\nabla u_{\mathbf {k} }(\mathbf {x} )\right)+U(\mathbf {x} )e^{i\mathbf {k} \cdot \mathbf {x} }u_{\mathbf {k} }(\mathbf {x} )\\E_{\mathbf {k} }e^{i\mathbf {k} \cdot \mathbf {x} }u_{\mathbf {k} }(\mathbf {x} )&={\frac {-\hbar ^{2}}{2m}}\left(i\mathbf {k} \cdot \left(i\mathbf {k} e^{i\mathbf {k} \cdot \mathbf {x} }u_{\mathbf {k} }(\mathbf {x} )+e^{i\mathbf {k} \cdot \mathbf {x} }\nabla u_{\mathbf {k} }(\mathbf {x} )\right)+i\mathbf {k} \cdot e^{i\mathbf {k} \cdot \mathbf {x} }\nabla u_{\mathbf {k} }(\mathbf {x} )+e^{i\mathbf {k} \cdot \mathbf {x} }\nabla ^{2}u_{\mathbf {k} }(\mathbf {x} )\right)+U(\mathbf {x} )e^{i\mathbf {k} \cdot \mathbf {x} }u_{\mathbf {k} }(\mathbf {x} )\\E_{\mathbf {k} }e^{i\mathbf {k} \cdot \mathbf {x} }u_{\mathbf {k} }(\mathbf {x} )&={\frac {\hbar ^{2}}{2m}}\left(\mathbf {k} ^{2}e^{i\mathbf {k} \cdot \mathbf {x} }u_{\mathbf {k} }(\mathbf {x} )-2i\mathbf {k} \cdot e^{i\mathbf {k} \cdot \mathbf {x} }\nabla u_{\mathbf {k} }(\mathbf {x} )-e^{i\mathbf {k} \cdot \mathbf {x} }\nabla ^{2}u_{\mathbf {k} }(\mathbf {x} )\right)+U(\mathbf {x} )e^{i\mathbf {k} \cdot \mathbf {x} }u_{\mathbf {k} }(\mathbf {x} )\\E_{\mathbf {k} }u_{\mathbf {k} }(\mathbf {x} )&={\frac {\hbar ^{2}}{2m}}\left(-i\nabla +\mathbf {k} \right)^{2}u_{\mathbf {k} }(\mathbf {x} )+U(\mathbf {x} )u_{\mathbf {k} }(\mathbf {x} )\end{aligned}}}
11948:
10706:
2590:
11509:
10267:
2189:
11049:
11943:{\displaystyle {\frac {\partial ^{2}\varepsilon _{n}(\mathbf {k} )}{\partial k_{i}\partial k_{j}}}={\frac {\hbar ^{2}}{m}}\delta _{ij}+\left({\frac {\hbar ^{2}}{m}}\right)^{2}\sum _{n'\neq n}{\frac {\langle n\mathbf {k} |-i\nabla _{i}|n'\mathbf {k} \rangle \langle n'\mathbf {k} |-i\nabla _{j}|n\mathbf {k} \rangle +\langle n\mathbf {k} |-i\nabla _{j}|n'\mathbf {k} \rangle \langle n'\mathbf {k} |-i\nabla _{i}|n\mathbf {k} \rangle }{\varepsilon _{n}(\mathbf {k} )-\varepsilon _{n'}(\mathbf {k} )}}}
10701:{\displaystyle {\frac {\partial ^{2}\varepsilon _{n}(\mathbf {k} )}{\partial k_{i}\partial k_{j}}}={\frac {\hbar ^{2}}{m}}\delta _{ij}+\left({\frac {\hbar ^{2}}{m}}\right)^{2}\sum _{n'\neq n}{\frac {\langle n\mathbf {k} |-i\nabla _{i}|n'\mathbf {k} \rangle \langle n'\mathbf {k} |-i\nabla _{j}|n\mathbf {k} \rangle +\langle n\mathbf {k} |-i\nabla _{j}|n'\mathbf {k} \rangle \langle n'\mathbf {k} |-i\nabla _{i}|n\mathbf {k} \rangle }{\varepsilon _{n}(\mathbf {k} )-\varepsilon _{n'}(\mathbf {k} )}}}
11450:
13162:
13590:
2585:{\displaystyle {\begin{aligned}u(\mathbf {r} +\mathbf {a} _{j})&=e^{-i\mathbf {k} \cdot (\mathbf {r} +\mathbf {a} _{j})}\psi (\mathbf {r} +\mathbf {a} _{j})\\&={\big (}e^{-i\mathbf {k} \cdot \mathbf {r} }e^{-i\mathbf {k} \cdot \mathbf {a} _{j}}{\big )}{\big (}e^{2\pi i\theta _{j}}\psi (\mathbf {r} ){\big )}\\&=e^{-i\mathbf {k} \cdot \mathbf {r} }e^{-2\pi i\theta _{j}}e^{2\pi i\theta _{j}}\psi (\mathbf {r} )\\&=u(\mathbf {r} ).\end{aligned}}}
27:
10718:
13614:
3130:
42:
11142:
716:
8826:
13626:
3828:
9227:
13602:
2899:
4264:
11044:{\displaystyle {\frac {1}{2}}\sum _{ij}{\frac {\partial ^{2}\varepsilon _{n}}{\partial k_{i}\partial k_{j}}}q_{i}q_{j}={\frac {\hbar ^{2}}{2m}}q^{2}+\sum _{n'\neq n}{\frac {|\int d\mathbf {r} \,u_{n\mathbf {k} }^{*}{\frac {\hbar ^{2}}{m}}\mathbf {q} \cdot (-i\nabla +\mathbf {k} )u_{n'\mathbf {k} }|^{2}}{\varepsilon _{n\mathbf {k} }-\varepsilon _{n'\mathbf {k} }}}}
3623:
8983:
9913:
9720:
8644:
11445:{\displaystyle {\frac {1}{2}}\sum _{ij}{\frac {\partial ^{2}\varepsilon _{n}}{\partial k_{i}\partial k_{j}}}q_{i}q_{j}={\frac {\hbar ^{2}}{2m}}q^{2}+\sum _{n'\neq n}{\frac {|\langle n\mathbf {k} |{\frac {\hbar ^{2}}{m}}\mathbf {q} \cdot (-i\nabla )|n'\mathbf {k} \rangle |^{2}}{\varepsilon _{n\mathbf {k} }-\varepsilon _{n'\mathbf {k} }}}}
7264:
6643:
3277:
10122:
4041:
9475:
10250:
4504:
6890:
686:
The most common example of Bloch's theorem is describing electrons in a crystal, especially in characterizing the crystal's electronic properties, such as electronic band structure. However, a Bloch-wave description applies more generally to any wave-like phenomenon in a periodic medium. For example,
3125:{\displaystyle {\begin{aligned}{\hat {\mathbf {T} }}_{\mathbf {n} }\psi (\mathbf {r} )&=\psi (\mathbf {r} +\mathbf {T} _{\mathbf {n} })\\&=\psi (\mathbf {r} +n_{1}\mathbf {a} _{1}+n_{2}\mathbf {a} _{2}+n_{3}\mathbf {a} _{3})\\&=\psi (\mathbf {r} +\mathbf {A} \mathbf {n} )\end{aligned}}}
9730:
4872:
6427:
9480:
1414:
The defining property of a crystal is translational symmetry, which means that if the crystal is shifted an appropriate amount, it winds up with all its atoms in the same places. (A finite-size crystal cannot have perfect translational symmetry, but it is a useful approximation.)
7061:
1384:
8821:{\displaystyle {\frac {\partial \varepsilon _{n}}{\partial \mathbf {k} }}={\frac {\hbar ^{2}}{m}}\int d\mathbf {r} \,\psi _{n\mathbf {k} }^{*}(-i\nabla )\psi _{n\mathbf {k} }={\frac {\hbar }{m}}\langle {\hat {\mathbf {p} }}\rangle =\hbar \langle {\hat {\mathbf {v} }}\rangle }
12121:
6523:
3135:
3823:{\displaystyle {\hat {\mathbf {T} }}_{\mathbf {n} _{1}}{\hat {\mathbf {T} }}_{\mathbf {n} _{2}}\psi (\mathbf {x} )=\psi (\mathbf {x} +\mathbf {A} \mathbf {n} _{1}+\mathbf {A} \mathbf {n} _{2})={\hat {\mathbf {T} }}_{\mathbf {n} _{1}+\mathbf {n} _{2}}\psi (\mathbf {x} )}
9222:{\displaystyle \varepsilon _{n}(\mathbf {k} +\mathbf {q} )=\varepsilon _{n}(\mathbf {k} )+\sum _{i}{\frac {\partial \varepsilon _{n}}{\partial k_{i}}}q_{i}+{\frac {1}{2}}\sum _{ij}{\frac {\partial ^{2}\varepsilon _{n}}{\partial k_{i}\partial k_{j}}}q_{i}q_{j}+O(q^{3})}
9991:
2876:
operator. This basis is what we are looking for. The wave functions in this basis are energy eigenstates (because they are eigenstates of the
Hamiltonian), and they are also Bloch states (because they are eigenstates of the translation operators; see Lemma above).
9324:
10127:
5377:
4593:
3951:
3578:
4378:
1226:
6687:
11138:
4672:
8961:
5664:
5259:
719:
A Bloch wave function (bottom) can be broken up into the product of a periodic function (top) and a plane-wave (center). The left side and right side represent the same Bloch state broken up in two different ways, involving the wave vector
4259:{\displaystyle 1=\int _{V}|\psi (\mathbf {x} )|^{2}d\mathbf {x} =\int _{V}\left|{\hat {\mathbf {T} }}_{\mathbf {n} }\psi (\mathbf {x} )\right|^{2}d\mathbf {x} =|\lambda _{\mathbf {n} }|^{2}\int _{V}|\psi (\mathbf {x} )|^{2}d\mathbf {x} }
5770:
2037:
12232:
6964:
2184:
6518:
2668:
1130:
834:
8568:
154:
4788:
12253:
The concept of the Bloch state was developed by Felix Bloch in 1928 to describe the conduction of electrons in crystalline solids. The same underlying mathematics, however, was also discovered independently several times: by
3428:
1534:
5577:
4942:
7276:
1879:
1289:
4008:
3342:
9977:
8879:
4982:
operators gives three commuting cyclic subgroups (given they can be generated by only one element) which are infinite, 1-dimensional and abelian. All irreducible representations of abelian groups are one dimensional.
4783:
6251:
6036:
5918:
641:
9908:{\displaystyle \sum _{i}{\frac {\partial \varepsilon _{n}}{\partial k_{i}}}q_{i}=\sum _{i}\int d\mathbf {r} \,u_{n\mathbf {k} }^{*}{\frac {\hbar ^{2}}{m}}(-i\nabla +\mathbf {k} )_{i}q_{i}u_{n\mathbf {k} }}
5270:
9273:
12037:
3864:
4682:
Apart from the group theory technicalities this proof is interesting because it becomes clear how to generalize the Bloch theorem for groups that are not only translations. This is typically done for
3500:
5487:
9319:
973:. Therefore, the first Brillouin zone is often used to depict all of the Bloch states without redundancy, for example in a band structure, and it is used for the same reason in many calculations.
9715:{\displaystyle E_{n}=E_{n}^{0}+\int d\mathbf {r} \,\psi _{n}^{*}{\hat {V}}\psi _{n}+\sum _{n'\neq n}{\frac {|\int d\mathbf {r} \,\psi _{n}^{*}{\hat {V}}\psi _{n}|^{2}}{E_{n}^{0}-E_{n'}^{0}}}+...}
3618:
1146:
7259:{\displaystyle E_{\mathbf {k} }\left(e^{i\mathbf {k} \cdot \mathbf {x} }u_{\mathbf {k} }(\mathbf {x} )\right)=\left\left(e^{i\mathbf {k} \cdot \mathbf {x} }u_{\mathbf {k} }(\mathbf {x} )\right)}
2904:
2194:
12358:
6133:
4980:
2874:
4316:
7026:
2744:
1720:
6638:{\displaystyle {\hat {P}}_{\varepsilon |{\boldsymbol {\tau }}}\psi (\mathbf {r} )=\psi (\mathbf {r} )e^{i\mathbf {k} \cdot {\boldsymbol {\tau }}}=\psi (\mathbf {r} +{\boldsymbol {\tau }})}
4509:
3272:{\displaystyle \mathbf {A} ={\begin{bmatrix}\mathbf {a} _{1}&\mathbf {a} _{2}&\mathbf {a} _{3}\end{bmatrix}},\quad \mathbf {n} ={\begin{pmatrix}n_{1}\\n_{2}\\n_{3}\end{pmatrix}}}
12018:
5582:
5133:
11985:
4698:. In this proof it is also possible to notice how it is key that the extra point group is driven by a symmetry in the effective potential but it shall commute with the Hamiltonian.
10117:{\displaystyle {\frac {\partial \varepsilon _{n}}{\partial \mathbf {k} }}={\frac {\hbar ^{2}}{m}}\int d\mathbf {r} \,u_{n\mathbf {k} }^{*}(-i\nabla +\mathbf {k} )u_{n\mathbf {k} }}
6071:
553:
499:
381:
5674:
5095:
1953:
3859:
11054:
8622:
9470:{\displaystyle {\hat {H}}_{\mathbf {k} +\mathbf {q} }={\hat {H}}_{\mathbf {k} }+{\frac {\hbar ^{2}}{m}}\mathbf {q} \cdot (-i\nabla +\mathbf {k} )+{\frac {\hbar ^{2}}{2m}}q^{2}}
6895:
4373:
4036:
2111:
10245:{\displaystyle {\frac {\partial \varepsilon _{n}}{\partial \mathbf {k} }}={\frac {\hbar ^{2}}{m}}\int d\mathbf {r} \,\psi _{n\mathbf {k} }^{*}(-i\nabla )\psi _{n\mathbf {k} }}
6432:
8597:
7050:
6988:
1061:
765:
8506:
5053:
4598:
942:(see figure at right). Therefore, wave vectors that differ by a reciprocal lattice vector are equivalent, in the sense that they characterize the same set of Bloch states.
12146:
5403:
5125:
1284:
1262:
663:
579:
521:
427:
252:
179:
8884:
4694:
and it is used for computing the band structure, spectrum and specific heats of crystals given a specific crystal group symmetry like FCC or BCC and eventually an extra
13031:"A tutorial survey on waves propagating in periodic media: Electronic, photonic and phononic crystals. Perception of the Bloch theorem in both real and Fourier domains"
4499:{\displaystyle \mathbf {{\hat {T}}_{n}} \psi (\mathbf {x} )=\psi (\mathbf {x} +\mathbf {n} \cdot \mathbf {a} )=e^{ik\mathbf {n} \cdot \mathbf {a} }\psi (\mathbf {x} ),}
3351:
1447:
1009:
5020:
13076:
6885:{\displaystyle {\hat {H}}_{\mathbf {k} }u_{\mathbf {k} }(\mathbf {r} )=\leftu_{\mathbf {k} }(\mathbf {r} )=\varepsilon _{\mathbf {k} }u_{\mathbf {k} }(\mathbf {r} )}
5498:
4877:
11504:
11477:
6243:
6216:
6189:
6162:
5799:
5430:
1056:
908:
860:
199:
4345:
1811:
12165:
3958:
3495:
3282:
1399:
Being Bloch's theorem a statement about lattice periodicity, in this proof all the symmetries are encoded as translation symmetries of the wave function itself.
9918:
6672:(as the invariants of the irreducible representations) can be treated as the fundamental building blocks instead of the irreducible representations themselves.
4720:
467:
447:
401:
300:
276:
223:
2599:
88:
12612:
5923:
5804:
3497:
and the two operators shall have a common set of eigenfunctions. Therefore, we start to look at the eigen-functions of the translation operator:
8838:
8624:. More precisely the crystal momentum is not a momentum but it stands for the momentum in the same way as the electromagnetic momentum in the
13069:
4867:{\displaystyle {\hat {\boldsymbol {\tau }}}={\hat {\boldsymbol {\tau }}}_{1}{\hat {\boldsymbol {\tau }}}_{2}{\hat {\boldsymbol {\tau }}}_{3}}
6422:{\displaystyle \chi _{k_{1}}(n_{1},a_{1})\chi _{k_{2}}(n_{2},a_{2})\chi _{k_{3}}(n_{3},a_{3})=e^{i\mathbf {k} \cdot {\boldsymbol {\tau }}}}
4998:
which in this case is a one dimensional matrix. All these subgroups, given they are cyclic, they have characters which are appropriate
12595:
12285:
1379:{\displaystyle \psi _{\mathbf {k} }(\mathbf {x} +\mathbf {a} )=e^{i\mathbf {k} \cdot \mathbf {a} }\psi _{\mathbf {k} }(\mathbf {x} ).}
13606:
12558:
4269:
584:
13062:
12116:{\displaystyle \mathbf {M} (\mathbf {k} )\mathbf {a} =\mp e\left(\mathbf {E} +\mathbf {v} (\mathbf {k} )\times \mathbf {B} \right)}
4706:
1723:
6649:
In the generalized version of the Bloch theorem, the
Fourier transform, i.e. the wave function expansion, gets generalized from a
9232:
842:
is periodic with the same periodicity as the crystal lattice. The actual quantum state of the electron is entirely determined by
13652:
12238:
4987:
20:
12237:
As an intuitive interpretation, both of the previous two equations resemble formally and are in a semi-classical analogy with
12993:
12972:
5435:
9278:
1008:
For a detailed example in which the consequences of Bloch's theorem are worked out in a specific situation, see the article
10256:
3583:
64:
12466:
12814:
Translated by A. T. Fuller from
Edouard Davaux's French translation (1907) of the original Russian dissertation (1892).
6654:
6076:
5097:. Note that this is straightforward in the finite cyclic group case but in the countable infinite case of the infinite
3345:
45:
Solid line: A schematic of the real part of a typical Bloch state in one dimension. The dotted line is from the factor
5372:{\displaystyle V\left(\mathbf {r} +\sum _{i}N_{i}\mathbf {a} _{i}\right)=V(\mathbf {r} +\mathbf {L} )=V(\mathbf {r} )}
4949:
2809:
13662:
13594:
12838:
12475:
12451:
Bloch, F. (1929). Über die quantenmechanik der elektronen in kristallgittern. Zeitschrift für physik, 52(7), 555-600.
6993:
4588:{\displaystyle \psi _{\mathbf {k} }(\mathbf {x} )=e^{i\mathbf {k} \cdot \mathbf {x} }u_{\mathbf {k} }(\mathbf {x} )}
2680:
1656:
12382:
Mathematically Bloch's theorem is interpreted in terms of unitary characters of a lattice group, and is applied to
3946:{\displaystyle \lambda _{\mathbf {n} _{1}}\lambda _{\mathbf {n} _{2}}=\lambda _{\mathbf {n} _{1}+\mathbf {n} _{2}}}
3573:{\displaystyle {\hat {\mathbf {T} }}_{\mathbf {n} }\psi (\mathbf {x} )=\lambda _{\mathbf {n} }\psi (\mathbf {x} )}
12828:
12421:
12267:
704:
11990:
12500:
13630:
13575:
13186:
11956:
1221:{\displaystyle u_{\mathbf {k} }(\mathbf {x} )=u_{\mathbf {k} }(\mathbf {x} +\mathbf {n} \cdot \mathbf {a} ).}
12737:"On the part of the motion of the lunar perigee which is a function of the mean motions of the sun and moon"
13121:
13012:
12416:
11133:{\displaystyle \psi _{n\mathbf {k} }=|n\mathbf {k} \rangle =e^{i\mathbf {k} \mathbf {x} }u_{n\mathbf {k} }}
6041:
526:
472:
354:
12906:
Kotani M; Sunada T. (2000). "Albanese maps and an off diagonal long time asymptotic for the heat kernel".
5058:
4321:
12401:
5264:
3833:
8602:
3433:
1242:
For any wave function that satisfies the Schrödinger equation and for a translation of a lattice vector
13657:
13366:
13275:
12411:
12279:
6650:
4350:
4013:
670:
4667:{\displaystyle u_{\mathbf {k} }(\mathbf {x} )=u_{\mathbf {k} }(\mathbf {x} +\mathbf {A} \mathbf {n} )}
13315:
13290:
12406:
8573:
1642:
1548:
345:
279:
7031:
6969:
4990:
are the same. The character is the representation over the complex numbers of the group or also the
2886:
In this proof all the symmetries are encoded as commutation properties of the translation operators
19:
This article is about a theorem in quantum mechanics. For the theorem used in complex analysis, see
13485:
13239:
13229:
13085:
8956:{\displaystyle {\frac {\partial ^{2}\varepsilon _{n}(\mathbf {k} )}{\partial k_{i}\partial k_{j}}}}
8629:
5025:
3430:
Given the
Hamiltonian is invariant for translations it shall commute with the translation operator
56:
12129:
6990:
is a parameter of the
Hamiltonian and therefore we arrive at a "continuous family" of eigenvalues
5386:
5104:
1267:
1245:
961:
is equivalent to one (and only one) vector in the first
Brillouin zone. Therefore, if we restrict
646:
562:
504:
410:
235:
162:
13331:
12372:
5659:{\displaystyle \psi \left(\mathbf {r} +\sum _{i}N_{i}\mathbf {a} _{i}\right)=\psi (\mathbf {r} )}
5254:{\displaystyle \chi _{k_{1}}({\hat {\boldsymbol {\tau }}}_{1}(n_{1},a_{1}))=e^{ik_{1}n_{1}a_{1}}}
983:
12582:
The vibrational spectrum and specific heat of a face centered cubic crystal, Robert B. Leighton
6681:
5492:
1444:. If the crystal is shifted by any of these three vectors, or a combination of them of the form
9915:
where the integrations are over a primitive cell or the entire crystal, given if the integral
5130:
Given the character is a root of unity, for each subgroup the character can be then written as
1028:
2798:
are integers. Because the crystal has translational symmetry, this operator commutes with the
13382:
13361:
13295:
5765:{\displaystyle {\hat {P}}_{\varepsilon |\tau _{i}+L_{i}}={\hat {P}}_{\varepsilon |\tau _{i}}}
5005:
2802:. Moreover, every such translation operator commutes with every other. Therefore, there is a
2032:{\displaystyle \psi (\mathbf {r} +\mathbf {a} _{j})=e^{2\pi i\theta _{j}}\psi (\mathbf {r} )}
12227:{\displaystyle \hbar {\dot {k}}=-e\left(\mathbf {E} +\mathbf {v} \times \mathbf {B} \right)}
6245:. In this context, the wave vector serves as a quantum number for the translation operator.
13351:
13126:
12915:
12698:
12583:
11482:
11455:
6966:
Given this is defined in a finite volume we expect an infinite family of eigenvalues; here
6221:
6194:
6167:
6140:
5777:
5408:
4995:
2799:
1041:
946:
893:
845:
184:
12547:
6959:{\displaystyle u_{\mathbf {k} }(\mathbf {r} )=u_{\mathbf {k} }(\mathbf {r} +\mathbf {R} )}
2179:{\displaystyle u(\mathbf {r} )=e^{-i\mathbf {k} \cdot \mathbf {r} }\psi (\mathbf {r} )\,.}
8:
13305:
13285:
13270:
13219:
12732:
12543:
12426:
12255:
6513:{\displaystyle {\hat {P}}_{R}\psi _{j}=\sum _{\alpha }\psi _{\alpha }\chi _{\alpha j}(R)}
12919:
12702:
1543:
are three integers, then the atoms end up in the same set of locations as they started.
13452:
13442:
13191:
12931:
12888:
12805:
12714:
12396:
12263:
2803:
2663:{\displaystyle \psi (\mathbf {r} )=e^{i\mathbf {k} \cdot \mathbf {r} }u(\mathbf {r} ),}
1125:{\displaystyle \psi (\mathbf {r} )=e^{i\mathbf {k} \cdot \mathbf {r} }u(\mathbf {r} ),}
939:
829:{\displaystyle \psi (\mathbf {r} )=e^{i\mathbf {k} \cdot \mathbf {r} }u(\mathbf {r} ),}
755:
556:
452:
432:
386:
285:
261:
208:
8563:{\displaystyle {\hat {\mathbf {p} }}_{\text{eff}}=-i\hbar \nabla +\hbar \mathbf {k} ,}
994:
of an electron can be calculated based on how the energy of a Bloch state varies with
149:{\displaystyle \psi (\mathbf {r} )=e^{i\mathbf {k} \cdot \mathbf {r} }u(\mathbf {r} )}
13613:
13560:
13525:
13432:
13356:
12999:
12989:
12968:
12962:
12935:
12834:
12718:
12496:
12471:
12383:
1808:
which is an eigenstate of all the translation operators. As a special case of this,
1409:
226:
72:
5774:
From here we can see that also the character shall be invariant by a translation of
13530:
13422:
13402:
13397:
13392:
13387:
13244:
13224:
13181:
13146:
13116:
13046:
13042:
12923:
12880:
12868:
12785:
12748:
12706:
12431:
12376:
8625:
6669:
6658:
4710:
987:
700:
696:
692:
255:
13029:
J. Gazalet; S. Dupont; J.C. Kastelik; Q. Rolland & B. Djafari-Rouhani (2013).
3423:{\displaystyle {\hat {H}}={\frac {{\hat {\mathbf {p} }}^{2}}{2m}}+U(\mathbf {x} )}
1529:{\displaystyle n_{1}\mathbf {a} _{1}+n_{2}\mathbf {a} _{2}+n_{3}\mathbf {a} _{3},}
13555:
13510:
13176:
13093:
12958:
12371:
is a periodic potential. Specific periodic one-dimensional equations include the
4695:
2674:
Finally, we are ready for the main proof of Bloch's theorem which is as follows.
407:, which is present because there are many different wave functions with the same
13030:
13618:
13565:
13447:
13336:
12824:
12769:
12461:
12271:
12259:
12029:
12021:
6653:
which is applicable only for cyclic groups, and therefore translations, into a
5572:{\displaystyle {\hat {H}}=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V(\mathbf {r} )}
4999:
4991:
4937:{\displaystyle {\hat {\boldsymbol {\tau }}}_{i}=n_{i}{\hat {\mathbf {a} }}_{i}}
4038:
if we use the normalization condition over a single primitive cell of volume V
1774:
are integers). The following fact is helpful for the proof of Bloch's theorem:
1038:
each of these wave functions is a Bloch state, meaning that this wave function
991:
666:
326:
322:
318:
303:
34:
13646:
13490:
13472:
13457:
13437:
13341:
13310:
13141:
13003:
12954:
12611:
Group
Representations and Harmonic Analysis from Euler to Langlands, Part II
12242:
4714:
1874:{\displaystyle \psi (\mathbf {r} +\mathbf {a} _{j})=C_{j}\psi (\mathbf {r} )}
1789:
202:
8503:
This shows how the effective momentum can be seen as composed of two parts,
13427:
13249:
13151:
13054:
12149:
5920:
and from the last equation we get for each dimension a periodic condition:
5098:
4003:{\displaystyle \lambda _{\mathbf {n} }=e^{s\mathbf {n} \cdot \mathbf {a} }}
3337:{\displaystyle U(\mathbf {x} +\mathbf {T} _{\mathbf {n} })=U(\mathbf {x} )}
1143:
has the same periodicity as the atomic structure of the crystal, such that
13028:
13013:"Periodic Potentials and Bloch's Theorem – lectures in "Semiconductors I""
12927:
9972:{\displaystyle \int d\mathbf {r} \,u_{n\mathbf {k} }^{*}u_{n\mathbf {k} }}
8874:{\displaystyle {\frac {\partial \varepsilon _{n}}{\partial \mathbf {k} }}}
4778:{\displaystyle {\boldsymbol {\tau }}=\sum _{i=1}^{3}n_{i}\mathbf {a} _{i}}
3830:
If we substitute here the eigenvalue equation and dividing both sides for
13547:
13265:
13234:
13214:
13161:
12686:
12153:
6662:
4691:
4687:
4683:
955:
with the property that no two of them are equivalent, yet every possible
404:
230:
76:
12774:"Sur les équations différentielles linéaires à coefficients périodiques"
12278:). The general form of a one-dimensional periodic potential equation is
13462:
13300:
13136:
12892:
12790:
12773:
12753:
12710:
12266:(1892). As a result, a variety of nomenclatures are common: applied to
1896:
688:
79:, who discovered the theorem in 1929. Mathematically, they are written
68:
30:
26:
12871:(1987). "Homology and closed geodesics in a compact Riemann surface".
12689:(1928). "Über die Quantenmechanik der Elektronen in Kristallgittern".
13520:
13346:
13131:
12983:
12884:
12759:
This work was initially published and distributed privately in 1877.
12736:
12160:
First order semi-classical equation of motion for electron in a band
6031:{\displaystyle k_{1}n_{1}a_{1}=k_{1}(n_{1}a_{1}+L_{1})-2\pi m_{1}}
1410:
Preliminaries: Crystal symmetries, lattice, and reciprocal lattice
1232:
A second and equivalent way to state the theorem is the following
758:
vector. In all plots, blue is real part and red is imaginary part.
13570:
13537:
13515:
13495:
5913:{\displaystyle e^{ik_{1}n_{1}a_{1}}=e^{ik_{1}(n_{1}a_{1}+L_{1})}}
4986:
Given they are one dimensional the matrix representation and the
2108:
are the reciprocal lattice vectors (see above). Finally, define
967:
to the first
Brillouin zone, then every Bloch state has a unique
336:
The description of electrons in terms of Bloch functions, termed
330:
41:
13505:
13500:
13106:
7052:
and thus at the basic concept of an electronic band structure.
715:
636:{\displaystyle \psi _{n\mathbf {k} }=\psi _{n(\mathbf {k+K} )}}
12804:
8963:
given they are the coefficients of the following expansion in
6164:
identify the irreducible representation in the same manner as
13480:
13101:
6218:
is a macroscopic periodic length of the crystal in direction
12985:
Group theory: application to the physics of condensed matter
12495:(2nd ed.). Cambridge University Press. pp. 17–20.
5668:
And for each dimension a translation operator with a period
13024:. Texts in Mathematics. Edinburgh: Scottish Academic Press.
12020:
and we can use it to write a semi-classical equation for a
9268:{\displaystyle \varepsilon _{n}(\mathbf {k} +\mathbf {q} )}
1792:
of all of the translation operators (simultaneously), then
1010:
Particle in a one-dimensional lattice (periodic potential)
13111:
9321:
We can consider the following perturbation problem in q:
5482:{\textstyle \mathbf {L} =\sum _{i}N_{i}\mathbf {a} _{i}}
12548:"Applications of Group Theory to the Physics of Solids"
6429:
and the generic formula for the wave function becomes:
5101:(i.e. the translation group here) there is a limit for
4717:. Translations can be written in terms of unit vectors
1575:(with units of inverse length), with the property that
13022:
The
Spectral Theory of Periodic Differential Equations
9314:{\displaystyle {\hat {H}}_{\mathbf {k} +\mathbf {q} }}
5438:
3220:
3152:
703:. It is generally treated in the various forms of the
12288:
12168:
12132:
12040:
12028:
Second order semi-classical equation of motion for a
11993:
11959:
11512:
11485:
11458:
11145:
11057:
10721:
10270:
10130:
9994:
9921:
9733:
9483:
9327:
9281:
9235:
8986:
8887:
8841:
8647:
8605:
8576:
8509:
7274:
7064:
7034:
6996:
6972:
6898:
6690:
6526:
6435:
6254:
6224:
6197:
6170:
6143:
6079:
6044:
5926:
5807:
5780:
5677:
5585:
5501:
5411:
5389:
5273:
5136:
5107:
5061:
5028:
5008:
4952:
4880:
4791:
4723:
4601:
4512:
4381:
4353:
4324:
4272:
4044:
4016:
3961:
3867:
3836:
3626:
3586:
3503:
3436:
3354:
3285:
3138:
2902:
2812:
2683:
2602:
2192:
2114:
1956:
1814:
1659:
1450:
1292:
1270:
1248:
1149:
1064:
1044:
1035:
each of these wave functions is an energy eigenstate,
1031:
of wave functions with the following two properties:
896:
848:
768:
649:
587:
565:
529:
507:
475:
455:
435:
413:
389:
357:
288:
264:
238:
211:
187:
165:
91:
12866:
9477:
Perturbation theory of the second order states that
3613:{\displaystyle {\hat {\mathbf {T} }}_{\mathbf {n} }}
1648:
12778:
12731:
3279:We use the hypothesis of a mean periodic potential
12981:
12352:
12226:
12140:
12115:
12012:
11979:
11942:
11498:
11471:
11444:
11132:
11043:
10700:
10244:
10116:
9971:
9907:
9714:
9469:
9313:
9267:
9221:
8955:
8873:
8820:
8616:
8591:
8562:
8492:
7258:
7044:
7020:
6982:
6958:
6884:
6637:
6512:
6421:
6237:
6210:
6183:
6156:
6127:
6065:
6030:
5912:
5793:
5764:
5658:
5571:
5481:
5424:
5397:
5371:
5253:
5119:
5089:
5047:
5014:
4974:
4936:
4866:
4777:
4666:
4587:
4498:
4367:
4339:
4310:
4258:
4030:
4002:
3945:
3853:
3822:
3612:
3572:
3489:
3422:
3336:
3271:
3124:
2868:
2738:
2662:
2584:
2178:
2031:
1873:
1714:
1528:
1378:
1278:
1256:
1220:
1124:
1050:
902:
854:
828:
657:
635:
573:
547:
515:
493:
461:
441:
421:
395:
375:
294:
270:
246:
217:
193:
173:
148:
12905:
12855:Floquet theory for partial differential equations
12538:
12536:
12353:{\displaystyle {\frac {d^{2}y}{dt^{2}}}+f(t)y=0,}
6128:{\displaystyle k_{1}={\frac {2\pi m_{1}}{L_{1}}}}
4390:
2865:
351:These eigenstates are written with subscripts as
75:. The theorem is named after the Swiss physicist
13644:
12248:
11953:The quantity on the right multiplied by a factor
10124:and we can reinsert the complete wave functions
4975:{\displaystyle {\hat {\boldsymbol {\tau }}}_{i}}
2869:{\displaystyle {\hat {T}}_{n_{1},n_{2},n_{3}}\!}
676:
4311:{\displaystyle 1=|\lambda _{\mathbf {n} }|^{2}}
2806:of the Hamiltonian operator and every possible
1912:in a different form, by choosing three numbers
1546:Another helpful ingredient in the proof is the
1027:For electrons in a perfect crystal, there is a
16:Fundamental theorem in condensed matter physics
13019:
12953:
12817:
12810:The General Problem of the Stability of Motion
12768:
12673:
12661:
12649:
12637:
12622:
12533:
12527:
12515:
9979:is normalized across the cell or the crystal.
7021:{\displaystyle \varepsilon _{n}(\mathbf {k} )}
6675:
5579:induces a periodicity with the wave function:
5383:is a macroscopic periodicity in the direction
2750:that shifts every wave function by the amount
2739:{\displaystyle {\hat {T}}_{n_{1},n_{2},n_{3}}}
1726:that shifts every wave function by the amount
1715:{\displaystyle {\hat {T}}_{n_{1},n_{2},n_{3}}}
229:with the same periodicity as the crystal, the
13070:
12823:
4785:We can think of these as commuting operators
2670:that proves that the state is a Bloch state.
2441:
2394:
2387:
2321:
13084:
11882:
11832:
11829:
11779:
11773:
11723:
11720:
11670:
11384:
11310:
11089:
10640:
10590:
10587:
10537:
10531:
10481:
10478:
10428:
8815:
8798:
8789:
8772:
1394:
67:in a periodic potential can be expressed as
12685:
12542:
13160:
13077:
13063:
12013:{\displaystyle \mathbf {M} (\mathbf {k} )}
5491:This substituting in the time independent
2596:has the periodicity of the lattice. Since
699:, and a periodic acoustic medium leads to
12789:
12752:
10897:
10191:
10055:
9933:
9812:
9607:
9526:
8708:
8635:For the effective velocity we can derive
6059:
4361:
4024:
2172:
2046:are three numbers which do not depend on
1000:; for more details see crystal momentum.
429:(each has a different periodic component
12967:. New York: Holt, Rinehart and Winston.
6248:We can generalize this for 3 dimensions
4506:which is true for a Bloch wave i.e. for
762:Suppose an electron is in a Bloch state
714:
40:
25:
11980:{\displaystyle {\frac {1}{\hbar ^{2}}}}
6628:
6604:
6550:
6520:i.e. specializing it for a translation
6413:
5405:that can also be seen as a multiple of
5161:
4957:
4885:
4849:
4830:
4811:
4795:
4725:
13645:
12484:
12460:
7028:dependent on the continuous parameter
1418:A three-dimensional crystal has three
13058:
12633:
12631:
12490:
6684:to the Bloch wave function we obtain
6668:Also here is possible to see how the
6066:{\displaystyle m_{1}\in \mathbb {Z} }
4677:
1905:. It is helpful to write the numbers
548:{\displaystyle \psi _{n\mathbf {k} }}
494:{\displaystyle \psi _{n\mathbf {k} }}
376:{\displaystyle \psi _{n\mathbf {k} }}
37:of a Bloch state in a silicon lattice
13601:
12564:from the original on 1 November 2019
12152:. This equation is analogous to the
8975:is considered small with respect to
6645:and we have proven Bloch’s theorem.
5495:with a simple effective Hamiltonian
5127:where the character remains finite.
5090:{\displaystyle \chi (\gamma )^{n}=1}
1804:Assume that we have a wave function
874:directly. This is important because
309:Functions of this form are known as
51:. The light circles represent atoms.
13625:
13010:
12593:
12467:Introduction to Solid State Physics
6661:are given from the specific finite
3854:{\displaystyle \psi (\mathbf {x} )}
2896:We define the translation operator
1264:, there exists at least one vector
1003:
21:Bloch's theorem (complex variables)
13:
12946:
12667:
12655:
12643:
12628:
12616:
12493:Principles of the theory of solids
11860:
11802:
11751:
11693:
11562:
11549:
11517:
11360:
11207:
11194:
11173:
10952:
10783:
10770:
10749:
10618:
10560:
10509:
10451:
10320:
10307:
10275:
10221:
10149:
10134:
10085:
10013:
9998:
9859:
9762:
9747:
9418:
9161:
9148:
9127:
9071:
9056:
8937:
8924:
8892:
8860:
8845:
8738:
8666:
8651:
8617:{\displaystyle \hbar \mathbf {k} }
8586:
8543:
8401:
8221:
8168:
7874:
7821:
7753:
7462:
7371:
7169:
6777:
5543:
5265:Born–von Karman boundary condition
5114:
5002:. In fact they have one generator
3346:independent electron approximation
2881:
449:). Within a band (i.e., for fixed
14:
13674:
12521:
12509:
8795:
8639:mean velocity of a Bloch electron
8583:
8549:
8540:
6680:If we apply the time-independent
4368:{\displaystyle k\in \mathbb {R} }
4031:{\displaystyle s\in \mathbb {C} }
1649:Lemma about translation operators
949:is a restricted set of values of
13624:
13612:
13600:
13589:
13588:
12215:
12207:
12199:
12134:
12104:
12093:
12085:
12077:
12058:
12050:
12042:
12003:
11995:
11987:is called effective mass tensor
11930:
11901:
11878:
11844:
11825:
11786:
11769:
11735:
11716:
11677:
11540:
11433:
11410:
11380:
11344:
11317:
11124:
11109:
11104:
11085:
11067:
11032:
11009:
10980:
10959:
10936:
10907:
10893:
10688:
10659:
10636:
10602:
10583:
10544:
10527:
10493:
10474:
10435:
10298:
10236:
10201:
10187:
10153:
10108:
10092:
10065:
10051:
10017:
9963:
9943:
9929:
9899:
9866:
9822:
9808:
9603:
9522:
9425:
9402:
9375:
9351:
9343:
9305:
9297:
9258:
9250:
9033:
9009:
9001:
8915:
8864:
8805:
8779:
8753:
8718:
8704:
8670:
8610:
8553:
8515:
8479:
8469:
8456:
8439:
8429:
8408:
8353:
8343:
8331:
8314:
8304:
8292:
8284:
8268:
8246:
8236:
8214:
8206:
8187:
8177:
8162:
8154:
8138:
8121:
8111:
8099:
8091:
8072:
8029:
8019:
8007:
7999:
7984:
7967:
7957:
7945:
7937:
7921:
7899:
7889:
7867:
7859:
7840:
7830:
7815:
7807:
7791:
7772:
7762:
7747:
7739:
7720:
7710:
7698:
7690:
7677:
7661:
7611:
7601:
7589:
7581:
7566:
7549:
7539:
7527:
7519:
7503:
7481:
7471:
7456:
7448:
7429:
7419:
7407:
7399:
7386:
7330:
7320:
7308:
7300:
7285:
7244:
7234:
7222:
7214:
7188:
7121:
7111:
7099:
7091:
7071:
7037:
7011:
6975:
6949:
6941:
6931:
6915:
6905:
6875:
6865:
6853:
6837:
6827:
6809:
6784:
6728:
6718:
6706:
6620:
6596:
6580:
6563:
6405:
5649:
5624:
5595:
5562:
5469:
5440:
5391:
5362:
5345:
5337:
5312:
5283:
4918:
4765:
4657:
4652:
4644:
4634:
4618:
4608:
4578:
4568:
4556:
4548:
4529:
4519:
4486:
4473:
4465:
4443:
4435:
4427:
4410:
4397:
4387:
4290:
4252:
4229:
4189:
4171:
4149:
4136:
4124:
4096:
4073:
3994:
3986:
3968:
3931:
3916:
3894:
3875:
3844:
3813:
3794:
3779:
3766:
3745:
3739:
3725:
3719:
3711:
3694:
3675:
3662:
3645:
3632:
3604:
3592:
3563:
3550:
3534:
3521:
3509:
3472:
3460:
3413:
3377:
3327:
3308:
3302:
3293:
3208:
3185:
3171:
3157:
3140:
3111:
3106:
3098:
3068:
3043:
3018:
2999:
2973:
2967:
2958:
2937:
2924:
2912:
2650:
2637:
2629:
2610:
2568:
2544:
2476:
2468:
2432:
2373:
2364:
2346:
2338:
2296:
2287:
2265:
2256:
2245:
2213:
2204:
2165:
2152:
2144:
2122:
2022:
1973:
1964:
1864:
1831:
1822:
1513:
1488:
1463:
1366:
1356:
1344:
1336:
1317:
1309:
1299:
1272:
1250:
1208:
1200:
1192:
1182:
1166:
1156:
1112:
1099:
1091:
1072:
816:
803:
795:
776:
681:
651:
624:
621:
618:
597:
567:
555:is unique only up to a constant
539:
509:
485:
415:
403:is a discrete index, called the
367:
240:
167:
139:
126:
118:
99:
12899:
12860:
12847:
12798:
12762:
12725:
12679:
12422:Symmetries in quantum mechanics
12268:ordinary differential equations
12241:for an electron in an external
8592:{\displaystyle -i\hbar \nabla }
6657:of the wave function where the
3206:
1403:Proof Using lattice periodicity
705:dynamical theory of diffraction
665:can be restricted to the first
13047:10.1016/j.wavemoti.2012.12.010
12806:Alexander Mihailovich Lyapunov
12605:
12587:
12576:
12454:
12445:
12335:
12329:
12097:
12089:
12054:
12046:
12007:
11999:
11934:
11926:
11905:
11897:
11870:
11849:
11812:
11791:
11761:
11740:
11703:
11682:
11544:
11536:
11389:
11367:
11363:
11351:
11322:
11306:
11077:
10988:
10963:
10943:
10882:
10692:
10684:
10663:
10655:
10628:
10607:
10570:
10549:
10519:
10498:
10461:
10440:
10302:
10294:
10224:
10212:
10096:
10076:
9871:
9850:
9722:To compute to linear order in
9647:
9629:
9592:
9548:
9429:
9409:
9367:
9335:
9289:
9262:
9246:
9216:
9203:
9037:
9029:
9013:
8997:
8919:
8911:
8809:
8783:
8741:
8729:
8519:
8483:
8475:
8460:
8452:
8443:
8435:
8357:
8349:
8318:
8310:
8272:
8264:
8250:
8242:
8191:
8183:
8125:
8117:
8033:
8025:
7971:
7963:
7925:
7917:
7903:
7895:
7844:
7836:
7776:
7768:
7724:
7716:
7615:
7607:
7553:
7545:
7507:
7499:
7485:
7477:
7433:
7425:
7334:
7326:
7248:
7240:
7192:
7184:
7125:
7117:
7045:{\displaystyle {\mathbf {k} }}
7015:
7007:
6983:{\displaystyle {\mathbf {k} }}
6953:
6937:
6919:
6911:
6879:
6871:
6841:
6833:
6813:
6805:
6732:
6724:
6698:
6632:
6616:
6584:
6576:
6567:
6559:
6545:
6534:
6507:
6501:
6443:
6390:
6364:
6344:
6318:
6298:
6272:
6006:
5970:
5905:
5869:
5746:
5735:
5696:
5685:
5653:
5645:
5566:
5558:
5508:
5366:
5358:
5349:
5333:
5205:
5202:
5176:
5164:
5154:
5111:
5072:
5065:
5055:, and therefore the character
4960:
4922:
4888:
4852:
4833:
4814:
4798:
4661:
4640:
4622:
4614:
4582:
4574:
4533:
4525:
4490:
4482:
4447:
4423:
4414:
4406:
4298:
4280:
4238:
4233:
4225:
4218:
4197:
4179:
4153:
4145:
4128:
4082:
4077:
4069:
4062:
3848:
3840:
3817:
3809:
3770:
3755:
3707:
3698:
3690:
3666:
3636:
3596:
3567:
3559:
3538:
3530:
3513:
3478:
3464:
3446:
3437:
3417:
3409:
3381:
3361:
3331:
3323:
3314:
3289:
3115:
3094:
3078:
2995:
2979:
2954:
2941:
2933:
2916:
2820:
2691:
2654:
2646:
2614:
2606:
2572:
2564:
2548:
2540:
2436:
2428:
2306:
2283:
2275:
2252:
2223:
2200:
2169:
2161:
2126:
2118:
2026:
2018:
1983:
1960:
1868:
1860:
1841:
1818:
1667:
1370:
1362:
1321:
1305:
1212:
1188:
1170:
1162:
1116:
1108:
1076:
1068:
910:can be written as above using
820:
812:
780:
772:
710:
628:
614:
143:
135:
103:
95:
1:
13653:Eponymous theorems of physics
13187:Spontaneous symmetry breaking
12833:. Courier Dover. p. 11.
12812:. London: Taylor and Francis.
12438:
12249:History and related equations
5048:{\displaystyle \gamma ^{n}=1}
4686:which are a combination of a
677:Applications and consequences
643:. Therefore, the wave vector
344:), underlies the concept of
63:states that solutions to the
12601:. University of Puget Sound.
12417:Periodic boundary conditions
12141:{\displaystyle \mathbf {a} }
8835:We evaluate the derivatives
5398:{\displaystyle \mathbf {a} }
5120:{\displaystyle n\to \infty }
1279:{\displaystyle \mathbf {k} }
1257:{\displaystyle \mathbf {a} }
1015:
658:{\displaystyle \mathbf {k} }
574:{\displaystyle \mathbf {K} }
523:, as does its energy. Also,
516:{\displaystyle \mathbf {k} }
422:{\displaystyle \mathbf {k} }
247:{\displaystyle \mathbf {k} }
174:{\displaystyle \mathbf {r} }
7:
12982:Dresselhaus, M. S. (2010).
12857:, RUSS MATH SURV., 37, 1–60
12389:
6676:Velocity and effective mass
4702:Proof with character theory
1058:can be written in the form
986:, it equals the electron's
10:
13679:
13367:Spin gapless semiconductor
13276:Nearly free electron model
12674:Ashcroft & Mermin 1976
12662:Ashcroft & Mermin 1976
12650:Ashcroft & Mermin 1976
12638:Ashcroft & Mermin 1976
12623:Ashcroft & Mermin 1976
12594:Roy, Ricky (May 2, 2010).
12528:Ashcroft & Mermin 1976
12516:Ashcroft & Mermin 1976
12412:Nearly free electron model
6651:discrete Fourier transform
1552:. These are three vectors
1549:reciprocal lattice vectors
671:without loss of generality
669:of the reciprocal lattice
346:electronic band structures
317:, and serve as a suitable
18:
13584:
13546:
13471:
13415:
13375:
13324:
13316:Density functional theory
13291:electronic band structure
13258:
13207:
13200:
13169:
13158:
13092:
13015:. The University of Kiel.
12407:Electronic band structure
6892:with boundary conditions
4946:The commutativity of the
3620:is an additive operator
1899:) which do not depend on
1643:reciprocal lattice vector
1420:primitive lattice vectors
1395:Using lattice periodicity
940:reciprocal lattice vector
890:unique. Specifically, if
738:(right). The difference (
501:varies continuously with
13663:Condensed matter physics
13486:Bogoliubov quasiparticle
13230:Quantum spin Hall effect
13122:Bose–Einstein condensate
13086:Condensed matter physics
12640:, p. 765 Appendix E
12276:Lyapunov–Floquet theorem
8630:canonical transformation
1389:
57:condensed matter physics
13020:M.S.P. Eastham (1973).
12596:"Representation Theory"
12402:Bloch wave – MoM method
8599:and a crystal momentum
5015:{\displaystyle \gamma }
2804:simultaneous eigenbasis
1895:are three numbers (the
1630:. (For the formula for
990:. Related to this, the
984:reduced Planck constant
256:crystal momentum vector
12691:Zeitschrift für Physik
12354:
12228:
12156:type of approximation
12142:
12117:
12014:
11981:
11944:
11500:
11473:
11446:
11134:
11045:
10715:The second order term
10702:
10262:effective mass theorem
10246:
10118:
9973:
9909:
9716:
9471:
9315:
9269:
9223:
8957:
8875:
8822:
8618:
8593:
8564:
8494:
7260:
7046:
7022:
6984:
6960:
6886:
6639:
6514:
6423:
6239:
6212:
6185:
6158:
6129:
6067:
6032:
5914:
5795:
5766:
5660:
5573:
5483:
5426:
5399:
5373:
5255:
5121:
5091:
5049:
5016:
4976:
4938:
4868:
4779:
4752:
4668:
4589:
4500:
4369:
4341:
4312:
4260:
4032:
4004:
3947:
3855:
3824:
3614:
3574:
3491:
3424:
3338:
3273:
3126:
2870:
2740:
2664:
2586:
2180:
2033:
1875:
1716:
1530:
1380:
1280:
1258:
1222:
1126:
1052:
904:
856:
830:
759:
659:
637:
575:
549:
517:
495:
463:
443:
423:
397:
377:
296:
272:
248:
219:
195:
175:
150:
52:
38:
13362:Topological insulator
13296:Anderson localization
12928:10.1007/s002200050033
12827:; Winkler, S (2004).
12491:Ziman, J. M. (1972).
12355:
12274:(or occasionally the
12229:
12143:
12118:
12015:
11982:
11945:
11501:
11499:{\displaystyle q_{j}}
11474:
11472:{\displaystyle q_{i}}
11447:
11135:
11046:
10703:
10247:
10119:
9982:We can simplify over
9974:
9910:
9717:
9472:
9316:
9270:
9224:
8958:
8876:
8823:
8619:
8594:
8565:
8495:
7261:
7047:
7023:
6985:
6961:
6887:
6640:
6515:
6424:
6240:
6238:{\displaystyle a_{1}}
6213:
6211:{\displaystyle L_{1}}
6186:
6184:{\displaystyle m_{1}}
6159:
6157:{\displaystyle k_{1}}
6130:
6068:
6033:
5915:
5796:
5794:{\displaystyle L_{i}}
5767:
5661:
5574:
5484:
5427:
5425:{\displaystyle a_{i}}
5400:
5374:
5256:
5122:
5092:
5050:
5017:
4977:
4939:
4869:
4780:
4732:
4669:
4590:
4501:
4370:
4342:
4313:
4261:
4033:
4005:
3948:
3856:
3825:
3615:
3575:
3492:
3425:
3339:
3274:
3127:
2890:Proof using operators
2871:
2741:
2665:
2587:
2181:
2034:
1876:
1717:
1531:
1381:
1281:
1259:
1223:
1127:
1053:
1051:{\displaystyle \psi }
982:is multiplied by the
905:
903:{\displaystyle \psi }
857:
855:{\displaystyle \psi }
831:
718:
660:
638:
576:
550:
518:
496:
464:
444:
424:
398:
378:
297:
273:
249:
220:
196:
194:{\displaystyle \psi }
176:
151:
44:
29:
13240:Aharonov–Bohm effect
13127:Fermionic condensate
12853:Kuchment, P.(1982),
12286:
12166:
12130:
12038:
11991:
11957:
11510:
11506:we have the theorem
11483:
11456:
11143:
11055:
10719:
10268:
10128:
9992:
9919:
9731:
9481:
9325:
9279:
9233:
8984:
8885:
8839:
8645:
8603:
8574:
8570:a standard momentum
8507:
7272:
7062:
7032:
6994:
6970:
6896:
6688:
6682:Schrödinger equation
6524:
6433:
6252:
6222:
6195:
6168:
6141:
6077:
6042:
5924:
5805:
5778:
5675:
5583:
5499:
5493:Schrödinger equation
5436:
5409:
5387:
5271:
5263:If we introduce the
5134:
5105:
5059:
5026:
5022:which shall obey to
5006:
4950:
4878:
4789:
4721:
4599:
4510:
4379:
4351:
4340:{\displaystyle s=ik}
4322:
4270:
4042:
4014:
3959:
3865:
3834:
3624:
3584:
3501:
3434:
3352:
3348:with an Hamiltonian
3283:
3136:
2900:
2810:
2800:Hamiltonian operator
2748:translation operator
2681:
2600:
2190:
2112:
1954:
1812:
1724:translation operator
1657:
1448:
1290:
1268:
1246:
1147:
1062:
1042:
947:first Brillouin zone
894:
846:
766:
647:
585:
563:
527:
505:
473:
453:
433:
411:
387:
355:
286:
262:
236:
209:
185:
163:
89:
65:Schrödinger equation
13631:Physics WikiProject
13306:tight binding model
13286:Fermi liquid theory
13271:Free electron model
13220:Quantum Hall effect
13201:Electrons in solids
12988:. Springer-Verlag.
12964:Solid State Physics
12920:2000CMaPh.209..633K
12733:George William Hill
12703:1929ZPhy...52..555B
12470:. New York: Wiley.
12427:Tight-binding model
12373:Kronig–Penney model
12256:George William Hill
12239:Newton's second law
10917:
10211:
10075:
9953:
9832:
9696:
9673:
9622:
9541:
9511:
9275:are eigenvalues of
8728:
8628:, and as part of a
6655:character expansion
1784:If a wave function
1782: —
1240: —
1025: —
13192:Critical phenomena
12791:10.24033/asens.220
12754:10.1007/BF02417081
12711:10.1007/BF01339455
12544:Dresselhaus, M. S.
12397:Bloch oscillations
12377:Mathieu's equation
12350:
12264:Alexander Lyapunov
12224:
12138:
12113:
12010:
11977:
11940:
11666:
11496:
11469:
11442:
11301:
11168:
11130:
11041:
10898:
10877:
10744:
10713:
10698:
10424:
10242:
10192:
10114:
10056:
9969:
9934:
9905:
9813:
9800:
9743:
9712:
9677:
9659:
9608:
9587:
9527:
9497:
9467:
9311:
9265:
9219:
9122:
9052:
8953:
8871:
8833:
8818:
8709:
8614:
8589:
8560:
8490:
8488:
7256:
7057:
7042:
7018:
6980:
6956:
6882:
6635:
6510:
6477:
6419:
6235:
6208:
6181:
6154:
6125:
6073:is an integer and
6063:
6028:
5910:
5791:
5762:
5656:
5611:
5569:
5479:
5456:
5422:
5395:
5369:
5299:
5267:on the potential:
5251:
5117:
5087:
5045:
5012:
4972:
4934:
4864:
4775:
4703:
4678:Using group theory
4664:
4585:
4496:
4365:
4337:
4308:
4256:
4028:
4000:
3955:This is true for
3943:
3851:
3820:
3610:
3570:
3490:{\displaystyle =0}
3487:
3420:
3334:
3269:
3263:
3197:
3122:
3120:
2891:
2866:
2736:
2660:
2582:
2580:
2176:
2029:
1871:
1802:
1796:is a Bloch state.
1780:
1712:
1526:
1404:
1376:
1276:
1254:
1238:
1218:
1122:
1048:
1023:
900:
852:
826:
760:
756:reciprocal lattice
655:
633:
571:
557:reciprocal lattice
545:
513:
491:
459:
439:
419:
393:
373:
331:crystalline solids
292:
268:
244:
215:
191:
171:
146:
73:periodic functions
53:
39:
13658:Quantum mechanics
13640:
13639:
13526:Exciton-polariton
13411:
13410:
13383:Thermoelectricity
12995:978-3-642-06945-1
12974:978-0-03-083993-1
12384:spectral geometry
12321:
12181:
11975:
11938:
11646:
11634:
11596:
11576:
11440:
11341:
11281:
11266:
11221:
11156:
11154:
11039:
10933:
10857:
10842:
10797:
10732:
10730:
10711:
10696:
10404:
10392:
10354:
10334:
10178:
10158:
10042:
10022:
9848:
9791:
9776:
9734:
9698:
9632:
9567:
9551:
9455:
9399:
9370:
9338:
9292:
9175:
9110:
9108:
9085:
9043:
8951:
8869:
8831:
8812:
8786:
8770:
8695:
8675:
8632:of the momentum.
8528:
8522:
8387:
8063:
7650:
7369:
7166:
7055:
6763:
6701:
6537:
6468:
6446:
6123:
5738:
5688:
5602:
5540:
5511:
5447:
5290:
5167:
4963:
4925:
4891:
4855:
4836:
4817:
4801:
4701:
4393:
4131:
3773:
3669:
3639:
3599:
3516:
3467:
3449:
3401:
3384:
3364:
2919:
2889:
2823:
2694:
2592:This proves that
1800:
1778:
1670:
1402:
1236:
1021:
920:be written using
701:phononic crystals
697:photonic crystals
462:{\displaystyle n}
442:{\displaystyle u}
396:{\displaystyle n}
295:{\displaystyle i}
271:{\displaystyle e}
227:periodic function
218:{\displaystyle u}
13670:
13628:
13627:
13616:
13604:
13603:
13592:
13591:
13531:Phonon polariton
13423:Amorphous magnet
13403:Electrostriction
13398:Flexoelectricity
13393:Ferroelectricity
13388:Piezoelectricity
13245:Josephson effect
13225:Spin Hall effect
13205:
13204:
13182:Phase transition
13164:
13147:Luttinger liquid
13094:States of matter
13079:
13072:
13065:
13056:
13055:
13050:
13025:
13016:
13007:
12978:
12959:Mermin, N. David
12940:
12939:
12908:Comm. Math. Phys
12903:
12897:
12896:
12864:
12858:
12851:
12845:
12844:
12821:
12815:
12813:
12802:
12796:
12795:
12793:
12766:
12760:
12758:
12756:
12729:
12723:
12722:
12697:(7–8): 555–600.
12683:
12677:
12671:
12665:
12659:
12653:
12647:
12641:
12635:
12626:
12620:
12614:
12609:
12603:
12602:
12600:
12591:
12585:
12580:
12574:
12573:
12571:
12569:
12563:
12552:
12540:
12531:
12525:
12519:
12513:
12507:
12506:
12488:
12482:
12481:
12458:
12452:
12449:
12432:Wannier function
12370:
12359:
12357:
12356:
12351:
12322:
12320:
12319:
12318:
12305:
12301:
12300:
12290:
12233:
12231:
12230:
12225:
12223:
12219:
12218:
12210:
12202:
12183:
12182:
12174:
12147:
12145:
12144:
12139:
12137:
12122:
12120:
12119:
12114:
12112:
12108:
12107:
12096:
12088:
12080:
12061:
12053:
12045:
12019:
12017:
12016:
12011:
12006:
11998:
11986:
11984:
11983:
11978:
11976:
11974:
11973:
11961:
11949:
11947:
11946:
11941:
11939:
11937:
11933:
11925:
11924:
11923:
11904:
11896:
11895:
11885:
11881:
11873:
11868:
11867:
11852:
11847:
11842:
11828:
11823:
11815:
11810:
11809:
11794:
11789:
11772:
11764:
11759:
11758:
11743:
11738:
11733:
11719:
11714:
11706:
11701:
11700:
11685:
11680:
11668:
11665:
11658:
11645:
11644:
11639:
11635:
11630:
11629:
11620:
11610:
11609:
11597:
11592:
11591:
11582:
11577:
11575:
11574:
11573:
11561:
11560:
11547:
11543:
11535:
11534:
11525:
11524:
11514:
11505:
11503:
11502:
11497:
11495:
11494:
11478:
11476:
11475:
11470:
11468:
11467:
11451:
11449:
11448:
11443:
11441:
11439:
11438:
11437:
11436:
11431:
11415:
11414:
11413:
11399:
11398:
11397:
11392:
11383:
11378:
11370:
11347:
11342:
11337:
11336:
11327:
11325:
11320:
11309:
11303:
11300:
11293:
11277:
11276:
11267:
11265:
11257:
11256:
11247:
11242:
11241:
11232:
11231:
11222:
11220:
11219:
11218:
11206:
11205:
11192:
11191:
11190:
11181:
11180:
11170:
11167:
11155:
11147:
11139:
11137:
11136:
11131:
11129:
11128:
11127:
11114:
11113:
11112:
11107:
11088:
11080:
11072:
11071:
11070:
11050:
11048:
11047:
11042:
11040:
11038:
11037:
11036:
11035:
11030:
11014:
11013:
11012:
10998:
10997:
10996:
10991:
10985:
10984:
10983:
10978:
10962:
10939:
10934:
10929:
10928:
10919:
10916:
10911:
10910:
10896:
10885:
10879:
10876:
10869:
10853:
10852:
10843:
10841:
10833:
10832:
10823:
10818:
10817:
10808:
10807:
10798:
10796:
10795:
10794:
10782:
10781:
10768:
10767:
10766:
10757:
10756:
10746:
10743:
10731:
10723:
10707:
10705:
10704:
10699:
10697:
10695:
10691:
10683:
10682:
10681:
10662:
10654:
10653:
10643:
10639:
10631:
10626:
10625:
10610:
10605:
10600:
10586:
10581:
10573:
10568:
10567:
10552:
10547:
10530:
10522:
10517:
10516:
10501:
10496:
10491:
10477:
10472:
10464:
10459:
10458:
10443:
10438:
10426:
10423:
10416:
10403:
10402:
10397:
10393:
10388:
10387:
10378:
10368:
10367:
10355:
10350:
10349:
10340:
10335:
10333:
10332:
10331:
10319:
10318:
10305:
10301:
10293:
10292:
10283:
10282:
10272:
10251:
10249:
10248:
10243:
10241:
10240:
10239:
10210:
10205:
10204:
10190:
10179:
10174:
10173:
10164:
10159:
10157:
10156:
10147:
10146:
10145:
10132:
10123:
10121:
10120:
10115:
10113:
10112:
10111:
10095:
10074:
10069:
10068:
10054:
10043:
10038:
10037:
10028:
10023:
10021:
10020:
10011:
10010:
10009:
9996:
9987:
9978:
9976:
9975:
9970:
9968:
9967:
9966:
9952:
9947:
9946:
9932:
9914:
9912:
9911:
9906:
9904:
9903:
9902:
9889:
9888:
9879:
9878:
9869:
9849:
9844:
9843:
9834:
9831:
9826:
9825:
9811:
9799:
9787:
9786:
9777:
9775:
9774:
9773:
9760:
9759:
9758:
9745:
9742:
9727:
9721:
9719:
9718:
9713:
9699:
9697:
9695:
9690:
9689:
9672:
9667:
9657:
9656:
9655:
9650:
9644:
9643:
9634:
9633:
9625:
9621:
9616:
9606:
9595:
9589:
9586:
9579:
9563:
9562:
9553:
9552:
9544:
9540:
9535:
9525:
9510:
9505:
9493:
9492:
9476:
9474:
9473:
9468:
9466:
9465:
9456:
9454:
9446:
9445:
9436:
9428:
9405:
9400:
9395:
9394:
9385:
9380:
9379:
9378:
9372:
9371:
9363:
9356:
9355:
9354:
9346:
9340:
9339:
9331:
9320:
9318:
9317:
9312:
9310:
9309:
9308:
9300:
9294:
9293:
9285:
9274:
9272:
9271:
9266:
9261:
9253:
9245:
9244:
9228:
9226:
9225:
9220:
9215:
9214:
9196:
9195:
9186:
9185:
9176:
9174:
9173:
9172:
9160:
9159:
9146:
9145:
9144:
9135:
9134:
9124:
9121:
9109:
9101:
9096:
9095:
9086:
9084:
9083:
9082:
9069:
9068:
9067:
9054:
9051:
9036:
9028:
9027:
9012:
9004:
8996:
8995:
8980:
8974:
8968:
8962:
8960:
8959:
8954:
8952:
8950:
8949:
8948:
8936:
8935:
8922:
8918:
8910:
8909:
8900:
8899:
8889:
8880:
8878:
8877:
8872:
8870:
8868:
8867:
8858:
8857:
8856:
8843:
8827:
8825:
8824:
8819:
8814:
8813:
8808:
8803:
8788:
8787:
8782:
8777:
8771:
8763:
8758:
8757:
8756:
8727:
8722:
8721:
8707:
8696:
8691:
8690:
8681:
8676:
8674:
8673:
8664:
8663:
8662:
8649:
8626:minimal coupling
8623:
8621:
8620:
8615:
8613:
8598:
8596:
8595:
8590:
8569:
8567:
8566:
8561:
8556:
8530:
8529:
8526:
8524:
8523:
8518:
8513:
8499:
8497:
8496:
8491:
8489:
8482:
8474:
8473:
8472:
8459:
8442:
8434:
8433:
8432:
8422:
8421:
8416:
8412:
8411:
8388:
8386:
8378:
8377:
8368:
8356:
8348:
8347:
8346:
8336:
8335:
8334:
8317:
8309:
8308:
8307:
8297:
8296:
8295:
8287:
8271:
8257:
8253:
8249:
8241:
8240:
8239:
8229:
8228:
8219:
8218:
8217:
8209:
8190:
8182:
8181:
8180:
8167:
8166:
8165:
8157:
8141:
8124:
8116:
8115:
8114:
8104:
8103:
8102:
8094:
8081:
8080:
8075:
8064:
8062:
8054:
8053:
8044:
8032:
8024:
8023:
8022:
8012:
8011:
8010:
8002:
7989:
7988:
7987:
7970:
7962:
7961:
7960:
7950:
7949:
7948:
7940:
7924:
7910:
7906:
7902:
7894:
7893:
7892:
7882:
7881:
7872:
7871:
7870:
7862:
7843:
7835:
7834:
7833:
7820:
7819:
7818:
7810:
7794:
7783:
7779:
7775:
7767:
7766:
7765:
7752:
7751:
7750:
7742:
7723:
7715:
7714:
7713:
7703:
7702:
7701:
7693:
7680:
7664:
7651:
7649:
7641:
7640:
7639:
7626:
7614:
7606:
7605:
7604:
7594:
7593:
7592:
7584:
7571:
7570:
7569:
7552:
7544:
7543:
7542:
7532:
7531:
7530:
7522:
7506:
7492:
7488:
7484:
7476:
7475:
7474:
7461:
7460:
7459:
7451:
7432:
7424:
7423:
7422:
7412:
7411:
7410:
7402:
7389:
7370:
7368:
7360:
7359:
7358:
7345:
7333:
7325:
7324:
7323:
7313:
7312:
7311:
7303:
7290:
7289:
7288:
7265:
7263:
7262:
7257:
7255:
7251:
7247:
7239:
7238:
7237:
7227:
7226:
7225:
7217:
7199:
7195:
7191:
7177:
7176:
7167:
7165:
7157:
7156:
7155:
7142:
7132:
7128:
7124:
7116:
7115:
7114:
7104:
7103:
7102:
7094:
7076:
7075:
7074:
7051:
7049:
7048:
7043:
7041:
7040:
7027:
7025:
7024:
7019:
7014:
7006:
7005:
6989:
6987:
6986:
6981:
6979:
6978:
6965:
6963:
6962:
6957:
6952:
6944:
6936:
6935:
6934:
6918:
6910:
6909:
6908:
6891:
6889:
6888:
6883:
6878:
6870:
6869:
6868:
6858:
6857:
6856:
6840:
6832:
6831:
6830:
6820:
6816:
6812:
6798:
6797:
6792:
6788:
6787:
6764:
6762:
6754:
6753:
6744:
6731:
6723:
6722:
6721:
6711:
6710:
6709:
6703:
6702:
6694:
6644:
6642:
6641:
6636:
6631:
6623:
6609:
6608:
6607:
6599:
6583:
6566:
6555:
6554:
6553:
6548:
6539:
6538:
6530:
6519:
6517:
6516:
6511:
6500:
6499:
6487:
6486:
6476:
6464:
6463:
6454:
6453:
6448:
6447:
6439:
6428:
6426:
6425:
6420:
6418:
6417:
6416:
6408:
6389:
6388:
6376:
6375:
6363:
6362:
6361:
6360:
6343:
6342:
6330:
6329:
6317:
6316:
6315:
6314:
6297:
6296:
6284:
6283:
6271:
6270:
6269:
6268:
6244:
6242:
6241:
6236:
6234:
6233:
6217:
6215:
6214:
6209:
6207:
6206:
6190:
6188:
6187:
6182:
6180:
6179:
6163:
6161:
6160:
6155:
6153:
6152:
6137:The wave vector
6134:
6132:
6131:
6126:
6124:
6122:
6121:
6112:
6111:
6110:
6094:
6089:
6088:
6072:
6070:
6069:
6064:
6062:
6054:
6053:
6037:
6035:
6034:
6029:
6027:
6026:
6005:
6004:
5992:
5991:
5982:
5981:
5969:
5968:
5956:
5955:
5946:
5945:
5936:
5935:
5919:
5917:
5916:
5911:
5909:
5908:
5904:
5903:
5891:
5890:
5881:
5880:
5868:
5867:
5847:
5846:
5845:
5844:
5835:
5834:
5825:
5824:
5800:
5798:
5797:
5792:
5790:
5789:
5771:
5769:
5768:
5763:
5761:
5760:
5759:
5758:
5749:
5740:
5739:
5731:
5724:
5723:
5722:
5721:
5709:
5708:
5699:
5690:
5689:
5681:
5665:
5663:
5662:
5657:
5652:
5638:
5634:
5633:
5632:
5627:
5621:
5620:
5610:
5598:
5578:
5576:
5575:
5570:
5565:
5551:
5550:
5541:
5539:
5531:
5530:
5521:
5513:
5512:
5504:
5488:
5486:
5485:
5480:
5478:
5477:
5472:
5466:
5465:
5455:
5443:
5431:
5429:
5428:
5423:
5421:
5420:
5404:
5402:
5401:
5396:
5394:
5378:
5376:
5375:
5370:
5365:
5348:
5340:
5326:
5322:
5321:
5320:
5315:
5309:
5308:
5298:
5286:
5260:
5258:
5257:
5252:
5250:
5249:
5248:
5247:
5238:
5237:
5228:
5227:
5201:
5200:
5188:
5187:
5175:
5174:
5169:
5168:
5160:
5153:
5152:
5151:
5150:
5126:
5124:
5123:
5118:
5096:
5094:
5093:
5088:
5080:
5079:
5054:
5052:
5051:
5046:
5038:
5037:
5021:
5019:
5018:
5013:
4981:
4979:
4978:
4973:
4971:
4970:
4965:
4964:
4956:
4943:
4941:
4940:
4935:
4933:
4932:
4927:
4926:
4921:
4916:
4912:
4911:
4899:
4898:
4893:
4892:
4884:
4873:
4871:
4870:
4865:
4863:
4862:
4857:
4856:
4848:
4844:
4843:
4838:
4837:
4829:
4825:
4824:
4819:
4818:
4810:
4803:
4802:
4794:
4784:
4782:
4781:
4776:
4774:
4773:
4768:
4762:
4761:
4751:
4746:
4728:
4673:
4671:
4670:
4665:
4660:
4655:
4647:
4639:
4638:
4637:
4621:
4613:
4612:
4611:
4594:
4592:
4591:
4586:
4581:
4573:
4572:
4571:
4561:
4560:
4559:
4551:
4532:
4524:
4523:
4522:
4505:
4503:
4502:
4497:
4489:
4478:
4477:
4476:
4468:
4446:
4438:
4430:
4413:
4402:
4401:
4400:
4395:
4394:
4386:
4374:
4372:
4371:
4366:
4364:
4346:
4344:
4343:
4338:
4317:
4315:
4314:
4309:
4307:
4306:
4301:
4295:
4294:
4293:
4283:
4265:
4263:
4262:
4257:
4255:
4247:
4246:
4241:
4232:
4221:
4216:
4215:
4206:
4205:
4200:
4194:
4193:
4192:
4182:
4174:
4166:
4165:
4160:
4156:
4152:
4141:
4140:
4139:
4133:
4132:
4127:
4122:
4112:
4111:
4099:
4091:
4090:
4085:
4076:
4065:
4060:
4059:
4037:
4035:
4034:
4029:
4027:
4009:
4007:
4006:
4001:
3999:
3998:
3997:
3989:
3973:
3972:
3971:
3952:
3950:
3949:
3944:
3942:
3941:
3940:
3939:
3934:
3925:
3924:
3919:
3905:
3904:
3903:
3902:
3897:
3886:
3885:
3884:
3883:
3878:
3860:
3858:
3857:
3852:
3847:
3829:
3827:
3826:
3821:
3816:
3805:
3804:
3803:
3802:
3797:
3788:
3787:
3782:
3775:
3774:
3769:
3764:
3754:
3753:
3748:
3742:
3734:
3733:
3728:
3722:
3714:
3697:
3686:
3685:
3684:
3683:
3678:
3671:
3670:
3665:
3660:
3656:
3655:
3654:
3653:
3648:
3641:
3640:
3635:
3630:
3619:
3617:
3616:
3611:
3609:
3608:
3607:
3601:
3600:
3595:
3590:
3579:
3577:
3576:
3571:
3566:
3555:
3554:
3553:
3537:
3526:
3525:
3524:
3518:
3517:
3512:
3507:
3496:
3494:
3493:
3488:
3477:
3476:
3475:
3469:
3468:
3463:
3458:
3451:
3450:
3442:
3429:
3427:
3426:
3421:
3416:
3402:
3400:
3392:
3391:
3386:
3385:
3380:
3375:
3371:
3366:
3365:
3357:
3343:
3341:
3340:
3335:
3330:
3313:
3312:
3311:
3305:
3296:
3278:
3276:
3275:
3270:
3268:
3267:
3260:
3259:
3246:
3245:
3232:
3231:
3211:
3202:
3201:
3194:
3193:
3188:
3180:
3179:
3174:
3166:
3165:
3160:
3143:
3131:
3129:
3128:
3123:
3121:
3114:
3109:
3101:
3084:
3077:
3076:
3071:
3065:
3064:
3052:
3051:
3046:
3040:
3039:
3027:
3026:
3021:
3015:
3014:
3002:
2985:
2978:
2977:
2976:
2970:
2961:
2940:
2929:
2928:
2927:
2921:
2920:
2915:
2910:
2875:
2873:
2872:
2867:
2864:
2863:
2862:
2861:
2849:
2848:
2836:
2835:
2825:
2824:
2816:
2797:
2790:
2745:
2743:
2742:
2737:
2735:
2734:
2733:
2732:
2720:
2719:
2707:
2706:
2696:
2695:
2687:
2669:
2667:
2666:
2661:
2653:
2642:
2641:
2640:
2632:
2613:
2595:
2591:
2589:
2588:
2583:
2581:
2571:
2554:
2547:
2536:
2535:
2534:
2533:
2510:
2509:
2508:
2507:
2481:
2480:
2479:
2471:
2449:
2445:
2444:
2435:
2424:
2423:
2422:
2421:
2398:
2397:
2391:
2390:
2384:
2383:
2382:
2381:
2376:
2367:
2351:
2350:
2349:
2341:
2325:
2324:
2312:
2305:
2304:
2299:
2290:
2279:
2278:
2274:
2273:
2268:
2259:
2248:
2222:
2221:
2216:
2207:
2185:
2183:
2182:
2177:
2168:
2157:
2156:
2155:
2147:
2125:
2107:
2096:
2051:
2045:
2038:
2036:
2035:
2030:
2025:
2014:
2013:
2012:
2011:
1982:
1981:
1976:
1967:
1949:
1934:
1911:
1904:
1894:
1887:
1880:
1878:
1877:
1872:
1867:
1856:
1855:
1840:
1839:
1834:
1825:
1807:
1795:
1787:
1783:
1773:
1766:
1721:
1719:
1718:
1713:
1711:
1710:
1709:
1708:
1696:
1695:
1683:
1682:
1672:
1671:
1663:
1640:
1629:
1619:
1598:
1574:
1542:
1535:
1533:
1532:
1527:
1522:
1521:
1516:
1510:
1509:
1497:
1496:
1491:
1485:
1484:
1472:
1471:
1466:
1460:
1459:
1443:
1385:
1383:
1382:
1377:
1369:
1361:
1360:
1359:
1349:
1348:
1347:
1339:
1320:
1312:
1304:
1303:
1302:
1285:
1283:
1282:
1277:
1275:
1263:
1261:
1260:
1255:
1253:
1241:
1227:
1225:
1224:
1219:
1211:
1203:
1195:
1187:
1186:
1185:
1169:
1161:
1160:
1159:
1142:
1131:
1129:
1128:
1123:
1115:
1104:
1103:
1102:
1094:
1075:
1057:
1055:
1054:
1049:
1026:
1004:Detailed example
999:
988:crystal momentum
981:
972:
966:
960:
954:
937:
931:
915:
909:
907:
906:
901:
885:
879:
873:
867:
861:
859:
858:
853:
841:
835:
833:
832:
827:
819:
808:
807:
806:
798:
779:
753:
737:
728:
693:electromagnetism
664:
662:
661:
656:
654:
642:
640:
639:
634:
632:
631:
627:
602:
601:
600:
580:
578:
577:
572:
570:
554:
552:
551:
546:
544:
543:
542:
522:
520:
519:
514:
512:
500:
498:
497:
492:
490:
489:
488:
468:
466:
465:
460:
448:
446:
445:
440:
428:
426:
425:
420:
418:
402:
400:
399:
394:
382:
380:
379:
374:
372:
371:
370:
329:of electrons in
301:
299:
298:
293:
277:
275:
274:
269:
253:
251:
250:
245:
243:
224:
222:
221:
216:
200:
198:
197:
192:
180:
178:
177:
172:
170:
155:
153:
152:
147:
142:
131:
130:
129:
121:
102:
50:
13678:
13677:
13673:
13672:
13671:
13669:
13668:
13667:
13643:
13642:
13641:
13636:
13580:
13561:Granular matter
13556:Amorphous solid
13542:
13467:
13453:Antiferromagnet
13443:Superparamagnet
13416:Magnetic phases
13407:
13371:
13320:
13281:Bloch's theorem
13254:
13196:
13177:Order parameter
13170:Phase phenomena
13165:
13156:
13088:
13083:
13053:
12996:
12975:
12949:
12947:Further reading
12944:
12943:
12904:
12900:
12885:10.2307/2374542
12865:
12861:
12852:
12848:
12841:
12830:Hill's Equation
12822:
12818:
12803:
12799:
12767:
12763:
12730:
12726:
12684:
12680:
12672:
12668:
12660:
12656:
12648:
12644:
12636:
12629:
12621:
12617:
12610:
12606:
12598:
12592:
12588:
12581:
12577:
12567:
12565:
12561:
12550:
12541:
12534:
12526:
12522:
12514:
12510:
12503:
12489:
12485:
12478:
12462:Kittel, Charles
12459:
12455:
12450:
12446:
12441:
12436:
12392:
12361:
12314:
12310:
12306:
12296:
12292:
12291:
12289:
12287:
12284:
12283:
12280:Hill's equation
12270:, it is called
12251:
12235:
12214:
12206:
12198:
12197:
12193:
12173:
12172:
12167:
12164:
12163:
12154:de Broglie wave
12133:
12131:
12128:
12127:
12124:
12103:
12092:
12084:
12076:
12075:
12071:
12057:
12049:
12041:
12039:
12036:
12035:
12002:
11994:
11992:
11989:
11988:
11969:
11965:
11960:
11958:
11955:
11954:
11951:
11929:
11916:
11915:
11911:
11900:
11891:
11887:
11886:
11877:
11869:
11863:
11859:
11848:
11843:
11835:
11824:
11816:
11811:
11805:
11801:
11790:
11785:
11768:
11760:
11754:
11750:
11739:
11734:
11726:
11715:
11707:
11702:
11696:
11692:
11681:
11676:
11669:
11667:
11651:
11650:
11640:
11625:
11621:
11619:
11615:
11614:
11602:
11598:
11587:
11583:
11581:
11569:
11565:
11556:
11552:
11548:
11539:
11530:
11526:
11520:
11516:
11515:
11513:
11511:
11508:
11507:
11490:
11486:
11484:
11481:
11480:
11463:
11459:
11457:
11454:
11453:
11432:
11424:
11423:
11419:
11409:
11405:
11401:
11400:
11393:
11388:
11387:
11379:
11371:
11366:
11343:
11332:
11328:
11326:
11321:
11316:
11305:
11304:
11302:
11286:
11285:
11272:
11268:
11258:
11252:
11248:
11246:
11237:
11233:
11227:
11223:
11214:
11210:
11201:
11197:
11193:
11186:
11182:
11176:
11172:
11171:
11169:
11160:
11146:
11144:
11141:
11140:
11123:
11119:
11115:
11108:
11103:
11099:
11095:
11084:
11076:
11066:
11062:
11058:
11056:
11053:
11052:
11031:
11023:
11022:
11018:
11008:
11004:
11000:
10999:
10992:
10987:
10986:
10979:
10971:
10970:
10966:
10958:
10935:
10924:
10920:
10918:
10912:
10906:
10902:
10892:
10881:
10880:
10878:
10862:
10861:
10848:
10844:
10834:
10828:
10824:
10822:
10813:
10809:
10803:
10799:
10790:
10786:
10777:
10773:
10769:
10762:
10758:
10752:
10748:
10747:
10745:
10736:
10722:
10720:
10717:
10716:
10709:
10687:
10674:
10673:
10669:
10658:
10649:
10645:
10644:
10635:
10627:
10621:
10617:
10606:
10601:
10593:
10582:
10574:
10569:
10563:
10559:
10548:
10543:
10526:
10518:
10512:
10508:
10497:
10492:
10484:
10473:
10465:
10460:
10454:
10450:
10439:
10434:
10427:
10425:
10409:
10408:
10398:
10383:
10379:
10377:
10373:
10372:
10360:
10356:
10345:
10341:
10339:
10327:
10323:
10314:
10310:
10306:
10297:
10288:
10284:
10278:
10274:
10273:
10271:
10269:
10266:
10265:
10253:
10235:
10231:
10227:
10206:
10200:
10196:
10186:
10169:
10165:
10163:
10152:
10148:
10141:
10137:
10133:
10131:
10129:
10126:
10125:
10107:
10103:
10099:
10091:
10070:
10064:
10060:
10050:
10033:
10029:
10027:
10016:
10012:
10005:
10001:
9997:
9995:
9993:
9990:
9989:
9983:
9962:
9958:
9954:
9948:
9942:
9938:
9928:
9920:
9917:
9916:
9898:
9894:
9890:
9884:
9880:
9874:
9870:
9865:
9839:
9835:
9833:
9827:
9821:
9817:
9807:
9795:
9782:
9778:
9769:
9765:
9761:
9754:
9750:
9746:
9744:
9738:
9732:
9729:
9728:
9723:
9691:
9682:
9681:
9668:
9663:
9658:
9651:
9646:
9645:
9639:
9635:
9624:
9623:
9617:
9612:
9602:
9591:
9590:
9588:
9572:
9571:
9558:
9554:
9543:
9542:
9536:
9531:
9521:
9506:
9501:
9488:
9484:
9482:
9479:
9478:
9461:
9457:
9447:
9441:
9437:
9435:
9424:
9401:
9390:
9386:
9384:
9374:
9373:
9362:
9361:
9360:
9350:
9342:
9341:
9330:
9329:
9328:
9326:
9323:
9322:
9304:
9296:
9295:
9284:
9283:
9282:
9280:
9277:
9276:
9257:
9249:
9240:
9236:
9234:
9231:
9230:
9210:
9206:
9191:
9187:
9181:
9177:
9168:
9164:
9155:
9151:
9147:
9140:
9136:
9130:
9126:
9125:
9123:
9114:
9100:
9091:
9087:
9078:
9074:
9070:
9063:
9059:
9055:
9053:
9047:
9032:
9023:
9019:
9008:
9000:
8991:
8987:
8985:
8982:
8981:
8976:
8970:
8964:
8944:
8940:
8931:
8927:
8923:
8914:
8905:
8901:
8895:
8891:
8890:
8888:
8886:
8883:
8882:
8863:
8859:
8852:
8848:
8844:
8842:
8840:
8837:
8836:
8829:
8804:
8802:
8801:
8778:
8776:
8775:
8762:
8752:
8748:
8744:
8723:
8717:
8713:
8703:
8686:
8682:
8680:
8669:
8665:
8658:
8654:
8650:
8648:
8646:
8643:
8642:
8609:
8604:
8601:
8600:
8575:
8572:
8571:
8552:
8525:
8514:
8512:
8511:
8510:
8508:
8505:
8504:
8501:
8487:
8486:
8478:
8468:
8467:
8463:
8455:
8438:
8428:
8427:
8423:
8417:
8407:
8394:
8390:
8389:
8379:
8373:
8369:
8367:
8360:
8352:
8342:
8341:
8337:
8330:
8329:
8325:
8322:
8321:
8313:
8303:
8302:
8298:
8291:
8283:
8279:
8275:
8267:
8245:
8235:
8234:
8230:
8224:
8220:
8213:
8205:
8201:
8197:
8186:
8176:
8175:
8171:
8161:
8153:
8149:
8145:
8137:
8120:
8110:
8109:
8105:
8098:
8090:
8086:
8082:
8076:
8071:
8070:
8069:
8065:
8055:
8049:
8045:
8043:
8036:
8028:
8018:
8017:
8013:
8006:
7998:
7994:
7990:
7983:
7982:
7978:
7975:
7974:
7966:
7956:
7955:
7951:
7944:
7936:
7932:
7928:
7920:
7898:
7888:
7887:
7883:
7877:
7873:
7866:
7858:
7854:
7850:
7839:
7829:
7828:
7824:
7814:
7806:
7802:
7798:
7790:
7771:
7761:
7760:
7756:
7746:
7738:
7734:
7730:
7719:
7709:
7708:
7704:
7697:
7689:
7685:
7681:
7676:
7672:
7668:
7660:
7656:
7652:
7642:
7635:
7631:
7627:
7625:
7618:
7610:
7600:
7599:
7595:
7588:
7580:
7576:
7572:
7565:
7564:
7560:
7557:
7556:
7548:
7538:
7537:
7533:
7526:
7518:
7514:
7510:
7502:
7480:
7470:
7469:
7465:
7455:
7447:
7443:
7439:
7428:
7418:
7417:
7413:
7406:
7398:
7394:
7390:
7385:
7381:
7377:
7361:
7354:
7350:
7346:
7344:
7337:
7329:
7319:
7318:
7314:
7307:
7299:
7295:
7291:
7284:
7283:
7279:
7275:
7273:
7270:
7269:
7268:We remain with
7243:
7233:
7232:
7228:
7221:
7213:
7209:
7205:
7204:
7200:
7187:
7172:
7168:
7158:
7151:
7147:
7143:
7141:
7140:
7136:
7120:
7110:
7109:
7105:
7098:
7090:
7086:
7082:
7081:
7077:
7070:
7069:
7065:
7063:
7060:
7059:
7036:
7035:
7033:
7030:
7029:
7010:
7001:
6997:
6995:
6992:
6991:
6974:
6973:
6971:
6968:
6967:
6948:
6940:
6930:
6929:
6925:
6914:
6904:
6903:
6899:
6897:
6894:
6893:
6874:
6864:
6863:
6859:
6852:
6851:
6847:
6836:
6826:
6825:
6821:
6808:
6793:
6783:
6770:
6766:
6765:
6755:
6749:
6745:
6743:
6742:
6738:
6727:
6717:
6716:
6712:
6705:
6704:
6693:
6692:
6691:
6689:
6686:
6685:
6678:
6647:
6627:
6619:
6603:
6595:
6591:
6587:
6579:
6562:
6549:
6544:
6540:
6529:
6528:
6527:
6525:
6522:
6521:
6492:
6488:
6482:
6478:
6472:
6459:
6455:
6449:
6438:
6437:
6436:
6434:
6431:
6430:
6412:
6404:
6400:
6396:
6384:
6380:
6371:
6367:
6356:
6352:
6351:
6347:
6338:
6334:
6325:
6321:
6310:
6306:
6305:
6301:
6292:
6288:
6279:
6275:
6264:
6260:
6259:
6255:
6253:
6250:
6249:
6229:
6225:
6223:
6220:
6219:
6202:
6198:
6196:
6193:
6192:
6175:
6171:
6169:
6166:
6165:
6148:
6144:
6142:
6139:
6138:
6117:
6113:
6106:
6102:
6095:
6093:
6084:
6080:
6078:
6075:
6074:
6058:
6049:
6045:
6043:
6040:
6039:
6022:
6018:
6000:
5996:
5987:
5983:
5977:
5973:
5964:
5960:
5951:
5947:
5941:
5937:
5931:
5927:
5925:
5922:
5921:
5899:
5895:
5886:
5882:
5876:
5872:
5863:
5859:
5855:
5851:
5840:
5836:
5830:
5826:
5820:
5816:
5812:
5808:
5806:
5803:
5802:
5785:
5781:
5779:
5776:
5775:
5754:
5750:
5745:
5741:
5730:
5729:
5728:
5717:
5713:
5704:
5700:
5695:
5691:
5680:
5679:
5678:
5676:
5673:
5672:
5648:
5628:
5623:
5622:
5616:
5612:
5606:
5594:
5593:
5589:
5584:
5581:
5580:
5561:
5546:
5542:
5532:
5526:
5522:
5520:
5503:
5502:
5500:
5497:
5496:
5473:
5468:
5467:
5461:
5457:
5451:
5439:
5437:
5434:
5433:
5416:
5412:
5410:
5407:
5406:
5390:
5388:
5385:
5384:
5361:
5344:
5336:
5316:
5311:
5310:
5304:
5300:
5294:
5282:
5281:
5277:
5272:
5269:
5268:
5243:
5239:
5233:
5229:
5223:
5219:
5215:
5211:
5196:
5192:
5183:
5179:
5170:
5159:
5158:
5157:
5146:
5142:
5141:
5137:
5135:
5132:
5131:
5106:
5103:
5102:
5075:
5071:
5060:
5057:
5056:
5033:
5029:
5027:
5024:
5023:
5007:
5004:
5003:
4966:
4955:
4954:
4953:
4951:
4948:
4947:
4928:
4917:
4915:
4914:
4913:
4907:
4903:
4894:
4883:
4882:
4881:
4879:
4876:
4875:
4858:
4847:
4846:
4845:
4839:
4828:
4827:
4826:
4820:
4809:
4808:
4807:
4793:
4792:
4790:
4787:
4786:
4769:
4764:
4763:
4757:
4753:
4747:
4736:
4724:
4722:
4719:
4718:
4680:
4675:
4656:
4651:
4643:
4633:
4632:
4628:
4617:
4607:
4606:
4602:
4600:
4597:
4596:
4577:
4567:
4566:
4562:
4555:
4547:
4543:
4539:
4528:
4518:
4517:
4513:
4511:
4508:
4507:
4485:
4472:
4464:
4457:
4453:
4442:
4434:
4426:
4409:
4396:
4385:
4384:
4383:
4382:
4380:
4377:
4376:
4360:
4352:
4349:
4348:
4323:
4320:
4319:
4302:
4297:
4296:
4289:
4288:
4284:
4279:
4271:
4268:
4267:
4251:
4242:
4237:
4236:
4228:
4217:
4211:
4207:
4201:
4196:
4195:
4188:
4187:
4183:
4178:
4170:
4161:
4148:
4135:
4134:
4123:
4121:
4120:
4119:
4118:
4114:
4113:
4107:
4103:
4095:
4086:
4081:
4080:
4072:
4061:
4055:
4051:
4043:
4040:
4039:
4023:
4015:
4012:
4011:
3993:
3985:
3981:
3977:
3967:
3966:
3962:
3960:
3957:
3956:
3935:
3930:
3929:
3920:
3915:
3914:
3913:
3909:
3898:
3893:
3892:
3891:
3887:
3879:
3874:
3873:
3872:
3868:
3866:
3863:
3862:
3843:
3835:
3832:
3831:
3812:
3798:
3793:
3792:
3783:
3778:
3777:
3776:
3765:
3763:
3762:
3761:
3749:
3744:
3743:
3738:
3729:
3724:
3723:
3718:
3710:
3693:
3679:
3674:
3673:
3672:
3661:
3659:
3658:
3657:
3649:
3644:
3643:
3642:
3631:
3629:
3628:
3627:
3625:
3622:
3621:
3603:
3602:
3591:
3589:
3588:
3587:
3585:
3582:
3581:
3562:
3549:
3548:
3544:
3533:
3520:
3519:
3508:
3506:
3505:
3504:
3502:
3499:
3498:
3471:
3470:
3459:
3457:
3456:
3455:
3441:
3440:
3435:
3432:
3431:
3412:
3393:
3387:
3376:
3374:
3373:
3372:
3370:
3356:
3355:
3353:
3350:
3349:
3326:
3307:
3306:
3301:
3300:
3292:
3284:
3281:
3280:
3262:
3261:
3255:
3251:
3248:
3247:
3241:
3237:
3234:
3233:
3227:
3223:
3216:
3215:
3207:
3196:
3195:
3189:
3184:
3183:
3181:
3175:
3170:
3169:
3167:
3161:
3156:
3155:
3148:
3147:
3139:
3137:
3134:
3133:
3119:
3118:
3110:
3105:
3097:
3082:
3081:
3072:
3067:
3066:
3060:
3056:
3047:
3042:
3041:
3035:
3031:
3022:
3017:
3016:
3010:
3006:
2998:
2983:
2982:
2972:
2971:
2966:
2965:
2957:
2944:
2936:
2923:
2922:
2911:
2909:
2908:
2907:
2903:
2901:
2898:
2897:
2884:
2882:Using operators
2879:
2857:
2853:
2844:
2840:
2831:
2827:
2826:
2815:
2814:
2813:
2811:
2808:
2807:
2796:
2792:
2789:
2783:
2776:
2770:
2763:
2757:
2751:
2728:
2724:
2715:
2711:
2702:
2698:
2697:
2686:
2685:
2684:
2682:
2679:
2678:
2672:
2649:
2636:
2628:
2624:
2620:
2609:
2601:
2598:
2597:
2593:
2579:
2578:
2567:
2552:
2551:
2543:
2529:
2525:
2515:
2511:
2503:
2499:
2486:
2482:
2475:
2467:
2460:
2456:
2447:
2446:
2440:
2439:
2431:
2417:
2413:
2403:
2399:
2393:
2392:
2386:
2385:
2377:
2372:
2371:
2363:
2356:
2352:
2345:
2337:
2330:
2326:
2320:
2319:
2310:
2309:
2300:
2295:
2294:
2286:
2269:
2264:
2263:
2255:
2244:
2237:
2233:
2226:
2217:
2212:
2211:
2203:
2193:
2191:
2188:
2187:
2164:
2151:
2143:
2136:
2132:
2121:
2113:
2110:
2109:
2106:
2098:
2095:
2089:
2082:
2076:
2069:
2063:
2053:
2047:
2044:
2040:
2021:
2007:
2003:
1993:
1989:
1977:
1972:
1971:
1963:
1955:
1952:
1951:
1948:
1936:
1933:
1926:
1919:
1913:
1910:
1906:
1900:
1893:
1889:
1882:
1863:
1851:
1847:
1835:
1830:
1829:
1821:
1813:
1810:
1809:
1805:
1798:
1793:
1785:
1781:
1772:
1768:
1765:
1759:
1752:
1746:
1739:
1733:
1727:
1704:
1700:
1691:
1687:
1678:
1674:
1673:
1662:
1661:
1660:
1658:
1655:
1654:
1651:
1639:
1631:
1621:
1617:
1608:
1600:
1593:
1584:
1576:
1573:
1566:
1559:
1553:
1541:
1537:
1517:
1512:
1511:
1505:
1501:
1492:
1487:
1486:
1480:
1476:
1467:
1462:
1461:
1455:
1451:
1449:
1446:
1445:
1442:
1435:
1428:
1422:
1412:
1397:
1392:
1387:
1365:
1355:
1354:
1350:
1343:
1335:
1331:
1327:
1316:
1308:
1298:
1297:
1293:
1291:
1288:
1287:
1271:
1269:
1266:
1265:
1249:
1247:
1244:
1243:
1239:
1237:Bloch's theorem
1230:
1207:
1199:
1191:
1181:
1180:
1176:
1165:
1155:
1154:
1150:
1148:
1145:
1144:
1133:
1111:
1098:
1090:
1086:
1082:
1071:
1063:
1060:
1059:
1043:
1040:
1039:
1024:
1022:Bloch's theorem
1018:
1006:
995:
977:
968:
962:
956:
950:
933:
921:
911:
895:
892:
891:
881:
875:
869:
863:
847:
844:
843:
837:
815:
802:
794:
790:
786:
775:
767:
764:
763:
752:
745:
739:
736:
730:
727:
721:
713:
684:
679:
650:
648:
645:
644:
617:
610:
606:
596:
592:
588:
586:
583:
582:
566:
564:
561:
560:
538:
534:
530:
528:
525:
524:
508:
506:
503:
502:
484:
480:
476:
474:
471:
470:
454:
451:
450:
434:
431:
430:
414:
412:
409:
408:
388:
385:
384:
366:
362:
358:
356:
353:
352:
340:(or less often
338:Bloch electrons
311:Bloch functions
287:
284:
283:
263:
260:
259:
239:
237:
234:
233:
210:
207:
206:
186:
183:
182:
166:
164:
161:
160:
157:
138:
125:
117:
113:
109:
98:
90:
87:
86:
61:Bloch's theorem
46:
24:
17:
12:
11:
5:
13676:
13666:
13665:
13660:
13655:
13638:
13637:
13635:
13634:
13622:
13619:Physics Portal
13610:
13598:
13585:
13582:
13581:
13579:
13578:
13573:
13568:
13566:Liquid crystal
13563:
13558:
13552:
13550:
13544:
13543:
13541:
13540:
13535:
13534:
13533:
13528:
13518:
13513:
13508:
13503:
13498:
13493:
13488:
13483:
13477:
13475:
13473:Quasiparticles
13469:
13468:
13466:
13465:
13460:
13455:
13450:
13445:
13440:
13435:
13433:Superdiamagnet
13430:
13425:
13419:
13417:
13413:
13412:
13409:
13408:
13406:
13405:
13400:
13395:
13390:
13385:
13379:
13377:
13373:
13372:
13370:
13369:
13364:
13359:
13357:Superconductor
13354:
13349:
13344:
13339:
13337:Mott insulator
13334:
13328:
13326:
13322:
13321:
13319:
13318:
13313:
13308:
13303:
13298:
13293:
13288:
13283:
13278:
13273:
13268:
13262:
13260:
13256:
13255:
13253:
13252:
13247:
13242:
13237:
13232:
13227:
13222:
13217:
13211:
13209:
13202:
13198:
13197:
13195:
13194:
13189:
13184:
13179:
13173:
13171:
13167:
13166:
13159:
13157:
13155:
13154:
13149:
13144:
13139:
13134:
13129:
13124:
13119:
13114:
13109:
13104:
13098:
13096:
13090:
13089:
13082:
13081:
13074:
13067:
13059:
13052:
13051:
13041:(3): 619–654.
13026:
13017:
13008:
12994:
12979:
12973:
12955:Ashcroft, Neil
12950:
12948:
12945:
12942:
12941:
12914:(3): 633–670.
12898:
12879:(1): 145–156.
12859:
12846:
12839:
12816:
12797:
12770:Gaston Floquet
12761:
12724:
12678:
12666:
12654:
12642:
12627:
12615:
12604:
12586:
12575:
12532:
12520:
12508:
12501:
12483:
12476:
12453:
12443:
12442:
12440:
12437:
12435:
12434:
12429:
12424:
12419:
12414:
12409:
12404:
12399:
12393:
12391:
12388:
12349:
12346:
12343:
12340:
12337:
12334:
12331:
12328:
12325:
12317:
12313:
12309:
12304:
12299:
12295:
12272:Floquet theory
12260:Gaston Floquet
12250:
12247:
12222:
12217:
12213:
12209:
12205:
12201:
12196:
12192:
12189:
12186:
12180:
12177:
12171:
12158:
12136:
12111:
12106:
12102:
12099:
12095:
12091:
12087:
12083:
12079:
12074:
12070:
12067:
12064:
12060:
12056:
12052:
12048:
12044:
12030:charge carrier
12026:
12022:charge carrier
12009:
12005:
12001:
11997:
11972:
11968:
11964:
11936:
11932:
11928:
11922:
11919:
11914:
11910:
11907:
11903:
11899:
11894:
11890:
11884:
11880:
11876:
11872:
11866:
11862:
11858:
11855:
11851:
11846:
11841:
11838:
11834:
11831:
11827:
11822:
11819:
11814:
11808:
11804:
11800:
11797:
11793:
11788:
11784:
11781:
11778:
11775:
11771:
11767:
11763:
11757:
11753:
11749:
11746:
11742:
11737:
11732:
11729:
11725:
11722:
11718:
11713:
11710:
11705:
11699:
11695:
11691:
11688:
11684:
11679:
11675:
11672:
11664:
11661:
11657:
11654:
11649:
11643:
11638:
11633:
11628:
11624:
11618:
11613:
11608:
11605:
11601:
11595:
11590:
11586:
11580:
11572:
11568:
11564:
11559:
11555:
11551:
11546:
11542:
11538:
11533:
11529:
11523:
11519:
11493:
11489:
11466:
11462:
11435:
11430:
11427:
11422:
11418:
11412:
11408:
11404:
11396:
11391:
11386:
11382:
11377:
11374:
11369:
11365:
11362:
11359:
11356:
11353:
11350:
11346:
11340:
11335:
11331:
11324:
11319:
11315:
11312:
11308:
11299:
11296:
11292:
11289:
11284:
11280:
11275:
11271:
11264:
11261:
11255:
11251:
11245:
11240:
11236:
11230:
11226:
11217:
11213:
11209:
11204:
11200:
11196:
11189:
11185:
11179:
11175:
11166:
11163:
11159:
11153:
11150:
11126:
11122:
11118:
11111:
11106:
11102:
11098:
11094:
11091:
11087:
11083:
11079:
11075:
11069:
11065:
11061:
11034:
11029:
11026:
11021:
11017:
11011:
11007:
11003:
10995:
10990:
10982:
10977:
10974:
10969:
10965:
10961:
10957:
10954:
10951:
10948:
10945:
10942:
10938:
10932:
10927:
10923:
10915:
10909:
10905:
10901:
10895:
10891:
10888:
10884:
10875:
10872:
10868:
10865:
10860:
10856:
10851:
10847:
10840:
10837:
10831:
10827:
10821:
10816:
10812:
10806:
10802:
10793:
10789:
10785:
10780:
10776:
10772:
10765:
10761:
10755:
10751:
10742:
10739:
10735:
10729:
10726:
10710:
10694:
10690:
10686:
10680:
10677:
10672:
10668:
10665:
10661:
10657:
10652:
10648:
10642:
10638:
10634:
10630:
10624:
10620:
10616:
10613:
10609:
10604:
10599:
10596:
10592:
10589:
10585:
10580:
10577:
10572:
10566:
10562:
10558:
10555:
10551:
10546:
10542:
10539:
10536:
10533:
10529:
10525:
10521:
10515:
10511:
10507:
10504:
10500:
10495:
10490:
10487:
10483:
10480:
10476:
10471:
10468:
10463:
10457:
10453:
10449:
10446:
10442:
10437:
10433:
10430:
10422:
10419:
10415:
10412:
10407:
10401:
10396:
10391:
10386:
10382:
10376:
10371:
10366:
10363:
10359:
10353:
10348:
10344:
10338:
10330:
10326:
10322:
10317:
10313:
10309:
10304:
10300:
10296:
10291:
10287:
10281:
10277:
10260:
10257:effective mass
10238:
10234:
10230:
10226:
10223:
10220:
10217:
10214:
10209:
10203:
10199:
10195:
10189:
10185:
10182:
10177:
10172:
10168:
10162:
10155:
10151:
10144:
10140:
10136:
10110:
10106:
10102:
10098:
10094:
10090:
10087:
10084:
10081:
10078:
10073:
10067:
10063:
10059:
10053:
10049:
10046:
10041:
10036:
10032:
10026:
10019:
10015:
10008:
10004:
10000:
9965:
9961:
9957:
9951:
9945:
9941:
9937:
9931:
9927:
9924:
9901:
9897:
9893:
9887:
9883:
9877:
9873:
9868:
9864:
9861:
9858:
9855:
9852:
9847:
9842:
9838:
9830:
9824:
9820:
9816:
9810:
9806:
9803:
9798:
9794:
9790:
9785:
9781:
9772:
9768:
9764:
9757:
9753:
9749:
9741:
9737:
9711:
9708:
9705:
9702:
9694:
9688:
9685:
9680:
9676:
9671:
9666:
9662:
9654:
9649:
9642:
9638:
9631:
9628:
9620:
9615:
9611:
9605:
9601:
9598:
9594:
9585:
9582:
9578:
9575:
9570:
9566:
9561:
9557:
9550:
9547:
9539:
9534:
9530:
9524:
9520:
9517:
9514:
9509:
9504:
9500:
9496:
9491:
9487:
9464:
9460:
9453:
9450:
9444:
9440:
9434:
9431:
9427:
9423:
9420:
9417:
9414:
9411:
9408:
9404:
9398:
9393:
9389:
9383:
9377:
9369:
9366:
9359:
9353:
9349:
9345:
9337:
9334:
9307:
9303:
9299:
9291:
9288:
9264:
9260:
9256:
9252:
9248:
9243:
9239:
9218:
9213:
9209:
9205:
9202:
9199:
9194:
9190:
9184:
9180:
9171:
9167:
9163:
9158:
9154:
9150:
9143:
9139:
9133:
9129:
9120:
9117:
9113:
9107:
9104:
9099:
9094:
9090:
9081:
9077:
9073:
9066:
9062:
9058:
9050:
9046:
9042:
9039:
9035:
9031:
9026:
9022:
9018:
9015:
9011:
9007:
9003:
8999:
8994:
8990:
8947:
8943:
8939:
8934:
8930:
8926:
8921:
8917:
8913:
8908:
8904:
8898:
8894:
8866:
8862:
8855:
8851:
8847:
8830:
8817:
8811:
8807:
8800:
8797:
8794:
8791:
8785:
8781:
8774:
8769:
8766:
8761:
8755:
8751:
8747:
8743:
8740:
8737:
8734:
8731:
8726:
8720:
8716:
8712:
8706:
8702:
8699:
8694:
8689:
8685:
8679:
8672:
8668:
8661:
8657:
8653:
8637:
8612:
8608:
8588:
8585:
8582:
8579:
8559:
8555:
8551:
8548:
8545:
8542:
8539:
8536:
8533:
8521:
8517:
8485:
8481:
8477:
8471:
8466:
8462:
8458:
8454:
8451:
8448:
8445:
8441:
8437:
8431:
8426:
8420:
8415:
8410:
8406:
8403:
8400:
8397:
8393:
8385:
8382:
8376:
8372:
8366:
8363:
8361:
8359:
8355:
8351:
8345:
8340:
8333:
8328:
8324:
8323:
8320:
8316:
8312:
8306:
8301:
8294:
8290:
8286:
8282:
8278:
8274:
8270:
8266:
8263:
8260:
8256:
8252:
8248:
8244:
8238:
8233:
8227:
8223:
8216:
8212:
8208:
8204:
8200:
8196:
8193:
8189:
8185:
8179:
8174:
8170:
8164:
8160:
8156:
8152:
8148:
8144:
8140:
8136:
8133:
8130:
8127:
8123:
8119:
8113:
8108:
8101:
8097:
8093:
8089:
8085:
8079:
8074:
8068:
8061:
8058:
8052:
8048:
8042:
8039:
8037:
8035:
8031:
8027:
8021:
8016:
8009:
8005:
8001:
7997:
7993:
7986:
7981:
7977:
7976:
7973:
7969:
7965:
7959:
7954:
7947:
7943:
7939:
7935:
7931:
7927:
7923:
7919:
7916:
7913:
7909:
7905:
7901:
7897:
7891:
7886:
7880:
7876:
7869:
7865:
7861:
7857:
7853:
7849:
7846:
7842:
7838:
7832:
7827:
7823:
7817:
7813:
7809:
7805:
7801:
7797:
7793:
7789:
7786:
7782:
7778:
7774:
7770:
7764:
7759:
7755:
7749:
7745:
7741:
7737:
7733:
7729:
7726:
7722:
7718:
7712:
7707:
7700:
7696:
7692:
7688:
7684:
7679:
7675:
7671:
7667:
7663:
7659:
7655:
7648:
7645:
7638:
7634:
7630:
7624:
7621:
7619:
7617:
7613:
7609:
7603:
7598:
7591:
7587:
7583:
7579:
7575:
7568:
7563:
7559:
7558:
7555:
7551:
7547:
7541:
7536:
7529:
7525:
7521:
7517:
7513:
7509:
7505:
7501:
7498:
7495:
7491:
7487:
7483:
7479:
7473:
7468:
7464:
7458:
7454:
7450:
7446:
7442:
7438:
7435:
7431:
7427:
7421:
7416:
7409:
7405:
7401:
7397:
7393:
7388:
7384:
7380:
7376:
7373:
7367:
7364:
7357:
7353:
7349:
7343:
7340:
7338:
7336:
7332:
7328:
7322:
7317:
7310:
7306:
7302:
7298:
7294:
7287:
7282:
7278:
7277:
7254:
7250:
7246:
7242:
7236:
7231:
7224:
7220:
7216:
7212:
7208:
7203:
7198:
7194:
7190:
7186:
7183:
7180:
7175:
7171:
7164:
7161:
7154:
7150:
7146:
7139:
7135:
7131:
7127:
7123:
7119:
7113:
7108:
7101:
7097:
7093:
7089:
7085:
7080:
7073:
7068:
7054:
7039:
7017:
7013:
7009:
7004:
7000:
6977:
6955:
6951:
6947:
6943:
6939:
6933:
6928:
6924:
6921:
6917:
6913:
6907:
6902:
6881:
6877:
6873:
6867:
6862:
6855:
6850:
6846:
6843:
6839:
6835:
6829:
6824:
6819:
6815:
6811:
6807:
6804:
6801:
6796:
6791:
6786:
6782:
6779:
6776:
6773:
6769:
6761:
6758:
6752:
6748:
6741:
6737:
6734:
6730:
6726:
6720:
6715:
6708:
6700:
6697:
6677:
6674:
6634:
6630:
6626:
6622:
6618:
6615:
6612:
6606:
6602:
6598:
6594:
6590:
6586:
6582:
6578:
6575:
6572:
6569:
6565:
6561:
6558:
6552:
6547:
6543:
6536:
6533:
6509:
6506:
6503:
6498:
6495:
6491:
6485:
6481:
6475:
6471:
6467:
6462:
6458:
6452:
6445:
6442:
6415:
6411:
6407:
6403:
6399:
6395:
6392:
6387:
6383:
6379:
6374:
6370:
6366:
6359:
6355:
6350:
6346:
6341:
6337:
6333:
6328:
6324:
6320:
6313:
6309:
6304:
6300:
6295:
6291:
6287:
6282:
6278:
6274:
6267:
6263:
6258:
6232:
6228:
6205:
6201:
6178:
6174:
6151:
6147:
6120:
6116:
6109:
6105:
6101:
6098:
6092:
6087:
6083:
6061:
6057:
6052:
6048:
6025:
6021:
6017:
6014:
6011:
6008:
6003:
5999:
5995:
5990:
5986:
5980:
5976:
5972:
5967:
5963:
5959:
5954:
5950:
5944:
5940:
5934:
5930:
5907:
5902:
5898:
5894:
5889:
5885:
5879:
5875:
5871:
5866:
5862:
5858:
5854:
5850:
5843:
5839:
5833:
5829:
5823:
5819:
5815:
5811:
5788:
5784:
5757:
5753:
5748:
5744:
5737:
5734:
5727:
5720:
5716:
5712:
5707:
5703:
5698:
5694:
5687:
5684:
5655:
5651:
5647:
5644:
5641:
5637:
5631:
5626:
5619:
5615:
5609:
5605:
5601:
5597:
5592:
5588:
5568:
5564:
5560:
5557:
5554:
5549:
5545:
5538:
5535:
5529:
5525:
5519:
5516:
5510:
5507:
5476:
5471:
5464:
5460:
5454:
5450:
5446:
5442:
5419:
5415:
5393:
5368:
5364:
5360:
5357:
5354:
5351:
5347:
5343:
5339:
5335:
5332:
5329:
5325:
5319:
5314:
5307:
5303:
5297:
5293:
5289:
5285:
5280:
5276:
5246:
5242:
5236:
5232:
5226:
5222:
5218:
5214:
5210:
5207:
5204:
5199:
5195:
5191:
5186:
5182:
5178:
5173:
5166:
5163:
5156:
5149:
5145:
5140:
5116:
5113:
5110:
5086:
5083:
5078:
5074:
5070:
5067:
5064:
5044:
5041:
5036:
5032:
5011:
5000:roots of unity
4996:representation
4969:
4962:
4959:
4931:
4924:
4920:
4910:
4906:
4902:
4897:
4890:
4887:
4861:
4854:
4851:
4842:
4835:
4832:
4823:
4816:
4813:
4806:
4800:
4797:
4772:
4767:
4760:
4756:
4750:
4745:
4742:
4739:
4735:
4731:
4727:
4700:
4679:
4676:
4663:
4659:
4654:
4650:
4646:
4642:
4636:
4631:
4627:
4624:
4620:
4616:
4610:
4605:
4584:
4580:
4576:
4570:
4565:
4558:
4554:
4550:
4546:
4542:
4538:
4535:
4531:
4527:
4521:
4516:
4495:
4492:
4488:
4484:
4481:
4475:
4471:
4467:
4463:
4460:
4456:
4452:
4449:
4445:
4441:
4437:
4433:
4429:
4425:
4422:
4419:
4416:
4412:
4408:
4405:
4399:
4392:
4389:
4363:
4359:
4356:
4336:
4333:
4330:
4327:
4305:
4300:
4292:
4287:
4282:
4278:
4275:
4266:and therefore
4254:
4250:
4245:
4240:
4235:
4231:
4227:
4224:
4220:
4214:
4210:
4204:
4199:
4191:
4186:
4181:
4177:
4173:
4169:
4164:
4159:
4155:
4151:
4147:
4144:
4138:
4130:
4126:
4117:
4110:
4106:
4102:
4098:
4094:
4089:
4084:
4079:
4075:
4071:
4068:
4064:
4058:
4054:
4050:
4047:
4026:
4022:
4019:
3996:
3992:
3988:
3984:
3980:
3976:
3970:
3965:
3938:
3933:
3928:
3923:
3918:
3912:
3908:
3901:
3896:
3890:
3882:
3877:
3871:
3850:
3846:
3842:
3839:
3819:
3815:
3811:
3808:
3801:
3796:
3791:
3786:
3781:
3772:
3768:
3760:
3757:
3752:
3747:
3741:
3737:
3732:
3727:
3721:
3717:
3713:
3709:
3706:
3703:
3700:
3696:
3692:
3689:
3682:
3677:
3668:
3664:
3652:
3647:
3638:
3634:
3606:
3598:
3594:
3569:
3565:
3561:
3558:
3552:
3547:
3543:
3540:
3536:
3532:
3529:
3523:
3515:
3511:
3486:
3483:
3480:
3474:
3466:
3462:
3454:
3448:
3445:
3439:
3419:
3415:
3411:
3408:
3405:
3399:
3396:
3390:
3383:
3379:
3369:
3363:
3360:
3333:
3329:
3325:
3322:
3319:
3316:
3310:
3304:
3299:
3295:
3291:
3288:
3266:
3258:
3254:
3250:
3249:
3244:
3240:
3236:
3235:
3230:
3226:
3222:
3221:
3219:
3214:
3210:
3205:
3200:
3192:
3187:
3182:
3178:
3173:
3168:
3164:
3159:
3154:
3153:
3151:
3146:
3142:
3117:
3113:
3108:
3104:
3100:
3096:
3093:
3090:
3087:
3085:
3083:
3080:
3075:
3070:
3063:
3059:
3055:
3050:
3045:
3038:
3034:
3030:
3025:
3020:
3013:
3009:
3005:
3001:
2997:
2994:
2991:
2988:
2986:
2984:
2981:
2975:
2969:
2964:
2960:
2956:
2953:
2950:
2947:
2945:
2943:
2939:
2935:
2932:
2926:
2918:
2914:
2906:
2905:
2888:
2883:
2880:
2860:
2856:
2852:
2847:
2843:
2839:
2834:
2830:
2822:
2819:
2794:
2787:
2781:
2774:
2768:
2761:
2755:
2731:
2727:
2723:
2718:
2714:
2710:
2705:
2701:
2693:
2690:
2677:As above, let
2659:
2656:
2652:
2648:
2645:
2639:
2635:
2631:
2627:
2623:
2619:
2616:
2612:
2608:
2605:
2577:
2574:
2570:
2566:
2563:
2560:
2557:
2555:
2553:
2550:
2546:
2542:
2539:
2532:
2528:
2524:
2521:
2518:
2514:
2506:
2502:
2498:
2495:
2492:
2489:
2485:
2478:
2474:
2470:
2466:
2463:
2459:
2455:
2452:
2450:
2448:
2443:
2438:
2434:
2430:
2427:
2420:
2416:
2412:
2409:
2406:
2402:
2396:
2389:
2380:
2375:
2370:
2366:
2362:
2359:
2355:
2348:
2344:
2340:
2336:
2333:
2329:
2323:
2318:
2315:
2313:
2311:
2308:
2303:
2298:
2293:
2289:
2285:
2282:
2277:
2272:
2267:
2262:
2258:
2254:
2251:
2247:
2243:
2240:
2236:
2232:
2229:
2227:
2225:
2220:
2215:
2210:
2206:
2202:
2199:
2196:
2195:
2175:
2171:
2167:
2163:
2160:
2154:
2150:
2146:
2142:
2139:
2135:
2131:
2128:
2124:
2120:
2117:
2102:
2093:
2087:
2080:
2074:
2067:
2061:
2042:
2028:
2024:
2020:
2017:
2010:
2006:
2002:
1999:
1996:
1992:
1988:
1985:
1980:
1975:
1970:
1966:
1962:
1959:
1944:
1931:
1924:
1917:
1908:
1891:
1870:
1866:
1862:
1859:
1854:
1850:
1846:
1843:
1838:
1833:
1828:
1824:
1820:
1817:
1801:Proof of Lemma
1799:
1776:
1770:
1763:
1757:
1750:
1744:
1737:
1731:
1707:
1703:
1699:
1694:
1690:
1686:
1681:
1677:
1669:
1666:
1650:
1647:
1635:
1613:
1604:
1589:
1580:
1571:
1564:
1557:
1539:
1525:
1520:
1515:
1508:
1504:
1500:
1495:
1490:
1483:
1479:
1475:
1470:
1465:
1458:
1454:
1440:
1433:
1426:
1411:
1408:
1401:
1396:
1393:
1391:
1388:
1375:
1372:
1368:
1364:
1358:
1353:
1346:
1342:
1338:
1334:
1330:
1326:
1323:
1319:
1315:
1311:
1307:
1301:
1296:
1274:
1252:
1234:
1229:
1228:
1217:
1214:
1210:
1206:
1202:
1198:
1194:
1190:
1184:
1179:
1175:
1172:
1168:
1164:
1158:
1153:
1121:
1118:
1114:
1110:
1107:
1101:
1097:
1093:
1089:
1085:
1081:
1078:
1074:
1070:
1067:
1047:
1036:
1019:
1017:
1014:
1005:
1002:
992:group velocity
899:
851:
825:
822:
818:
814:
811:
805:
801:
797:
793:
789:
785:
782:
778:
774:
771:
750:
743:
734:
725:
712:
709:
683:
680:
678:
675:
667:Brillouin zone
653:
630:
626:
623:
620:
616:
613:
609:
605:
599:
595:
591:
569:
541:
537:
533:
511:
487:
483:
479:
458:
438:
417:
392:
369:
365:
361:
323:wave functions
304:imaginary unit
291:
280:Euler's number
267:
242:
214:
190:
169:
145:
141:
137:
134:
128:
124:
120:
116:
112:
108:
105:
101:
97:
94:
83:Bloch function
81:
35:square modulus
15:
9:
6:
4:
3:
2:
13675:
13664:
13661:
13659:
13656:
13654:
13651:
13650:
13648:
13633:
13632:
13623:
13621:
13620:
13615:
13611:
13609:
13608:
13599:
13597:
13596:
13587:
13586:
13583:
13577:
13574:
13572:
13569:
13567:
13564:
13562:
13559:
13557:
13554:
13553:
13551:
13549:
13545:
13539:
13536:
13532:
13529:
13527:
13524:
13523:
13522:
13519:
13517:
13514:
13512:
13509:
13507:
13504:
13502:
13499:
13497:
13494:
13492:
13489:
13487:
13484:
13482:
13479:
13478:
13476:
13474:
13470:
13464:
13461:
13459:
13456:
13454:
13451:
13449:
13446:
13444:
13441:
13439:
13436:
13434:
13431:
13429:
13426:
13424:
13421:
13420:
13418:
13414:
13404:
13401:
13399:
13396:
13394:
13391:
13389:
13386:
13384:
13381:
13380:
13378:
13374:
13368:
13365:
13363:
13360:
13358:
13355:
13353:
13350:
13348:
13345:
13343:
13342:Semiconductor
13340:
13338:
13335:
13333:
13330:
13329:
13327:
13323:
13317:
13314:
13312:
13311:Hubbard model
13309:
13307:
13304:
13302:
13299:
13297:
13294:
13292:
13289:
13287:
13284:
13282:
13279:
13277:
13274:
13272:
13269:
13267:
13264:
13263:
13261:
13257:
13251:
13248:
13246:
13243:
13241:
13238:
13236:
13233:
13231:
13228:
13226:
13223:
13221:
13218:
13216:
13213:
13212:
13210:
13206:
13203:
13199:
13193:
13190:
13188:
13185:
13183:
13180:
13178:
13175:
13174:
13172:
13168:
13163:
13153:
13150:
13148:
13145:
13143:
13140:
13138:
13135:
13133:
13130:
13128:
13125:
13123:
13120:
13118:
13115:
13113:
13110:
13108:
13105:
13103:
13100:
13099:
13097:
13095:
13091:
13087:
13080:
13075:
13073:
13068:
13066:
13061:
13060:
13057:
13048:
13044:
13040:
13036:
13032:
13027:
13023:
13018:
13014:
13009:
13005:
13001:
12997:
12991:
12987:
12986:
12980:
12976:
12970:
12966:
12965:
12960:
12956:
12952:
12951:
12937:
12933:
12929:
12925:
12921:
12917:
12913:
12909:
12902:
12894:
12890:
12886:
12882:
12878:
12874:
12873:Amer. J. Math
12870:
12867:Katsuda, A.;
12863:
12856:
12850:
12842:
12840:0-486-49565-5
12836:
12832:
12831:
12826:
12820:
12811:
12807:
12801:
12792:
12787:
12783:
12779:
12775:
12771:
12765:
12755:
12750:
12746:
12742:
12738:
12734:
12728:
12720:
12716:
12712:
12708:
12704:
12700:
12696:
12693:(in German).
12692:
12688:
12682:
12676:, p. 227
12675:
12670:
12664:, p. 229
12663:
12658:
12652:, p. 228
12651:
12646:
12639:
12634:
12632:
12625:, p. 140
12624:
12619:
12613:
12608:
12597:
12590:
12584:
12579:
12560:
12556:
12549:
12545:
12539:
12537:
12530:, p. 137
12529:
12524:
12518:, p. 134
12517:
12512:
12504:
12498:
12494:
12487:
12479:
12477:0-471-14286-7
12473:
12469:
12468:
12463:
12457:
12448:
12444:
12433:
12430:
12428:
12425:
12423:
12420:
12418:
12415:
12413:
12410:
12408:
12405:
12403:
12400:
12398:
12395:
12394:
12387:
12385:
12380:
12378:
12374:
12368:
12364:
12347:
12344:
12341:
12338:
12332:
12326:
12323:
12315:
12311:
12307:
12302:
12297:
12293:
12281:
12277:
12273:
12269:
12265:
12261:
12257:
12246:
12244:
12243:Lorentz force
12240:
12234:
12220:
12211:
12203:
12194:
12190:
12187:
12184:
12178:
12175:
12169:
12161:
12157:
12155:
12151:
12123:
12109:
12100:
12081:
12072:
12068:
12065:
12062:
12033:
12031:
12025:
12023:
11970:
11966:
11962:
11950:
11920:
11917:
11912:
11908:
11892:
11888:
11874:
11864:
11856:
11853:
11839:
11836:
11820:
11817:
11806:
11798:
11795:
11782:
11776:
11765:
11755:
11747:
11744:
11730:
11727:
11711:
11708:
11697:
11689:
11686:
11673:
11662:
11659:
11655:
11652:
11647:
11641:
11636:
11631:
11626:
11622:
11616:
11611:
11606:
11603:
11599:
11593:
11588:
11584:
11578:
11570:
11566:
11557:
11553:
11531:
11527:
11521:
11491:
11487:
11464:
11460:
11428:
11425:
11420:
11416:
11406:
11402:
11394:
11375:
11372:
11357:
11354:
11348:
11338:
11333:
11329:
11313:
11297:
11294:
11290:
11287:
11282:
11278:
11273:
11269:
11262:
11259:
11253:
11249:
11243:
11238:
11234:
11228:
11224:
11215:
11211:
11202:
11198:
11187:
11183:
11177:
11164:
11161:
11157:
11151:
11148:
11120:
11116:
11100:
11096:
11092:
11081:
11073:
11063:
11059:
11027:
11024:
11019:
11015:
11005:
11001:
10993:
10975:
10972:
10967:
10955:
10949:
10946:
10940:
10930:
10925:
10921:
10913:
10903:
10899:
10889:
10886:
10873:
10870:
10866:
10863:
10858:
10854:
10849:
10845:
10838:
10835:
10829:
10825:
10819:
10814:
10810:
10804:
10800:
10791:
10787:
10778:
10774:
10763:
10759:
10753:
10740:
10737:
10733:
10727:
10724:
10708:
10678:
10675:
10670:
10666:
10650:
10646:
10632:
10622:
10614:
10611:
10597:
10594:
10578:
10575:
10564:
10556:
10553:
10540:
10534:
10523:
10513:
10505:
10502:
10488:
10485:
10469:
10466:
10455:
10447:
10444:
10431:
10420:
10417:
10413:
10410:
10405:
10399:
10394:
10389:
10384:
10380:
10374:
10369:
10364:
10361:
10357:
10351:
10346:
10342:
10336:
10328:
10324:
10315:
10311:
10289:
10285:
10279:
10263:
10259:
10258:
10252:
10232:
10228:
10218:
10215:
10207:
10197:
10193:
10183:
10180:
10175:
10170:
10166:
10160:
10142:
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9814:
9804:
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9770:
9766:
9755:
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9739:
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9700:
9692:
9686:
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9669:
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9613:
9609:
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9507:
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9412:
9406:
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9332:
9301:
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9211:
9207:
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9097:
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8979:
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8464:
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8424:
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8380:
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6945:
6926:
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6848:
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6789:
6780:
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6695:
6683:
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6666:
6664:
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6624:
6613:
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6489:
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5988:
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5666:
5642:
5639:
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5590:
5586:
5555:
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5514:
5505:
5494:
5489:
5474:
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5355:
5352:
5341:
5330:
5327:
5323:
5317:
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4699:
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4325:
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4092:
4087:
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4052:
4048:
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4017:
3990:
3982:
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3899:
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3869:
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3758:
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3443:
3406:
3403:
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3358:
3347:
3320:
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3297:
3286:
3264:
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3242:
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3198:
3190:
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3162:
3149:
3144:
3102:
3091:
3088:
3086:
3073:
3061:
3057:
3053:
3048:
3036:
3032:
3028:
3023:
3011:
3007:
3003:
2992:
2989:
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2962:
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2948:
2946:
2930:
2894:
2887:
2878:
2858:
2854:
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2832:
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2786:
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2699:
2688:
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2643:
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2617:
2603:
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2537:
2530:
2526:
2522:
2519:
2516:
2512:
2504:
2500:
2496:
2493:
2490:
2487:
2483:
2472:
2464:
2461:
2457:
2453:
2451:
2425:
2418:
2414:
2410:
2407:
2404:
2400:
2378:
2368:
2360:
2357:
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2342:
2334:
2331:
2327:
2316:
2314:
2301:
2291:
2280:
2270:
2260:
2249:
2241:
2238:
2234:
2230:
2228:
2218:
2208:
2197:
2173:
2158:
2148:
2140:
2137:
2133:
2129:
2115:
2105:
2101:
2092:
2086:
2079:
2073:
2066:
2060:
2056:
2050:
2015:
2008:
2004:
2000:
1997:
1994:
1990:
1986:
1978:
1968:
1957:
1947:
1943:
1939:
1930:
1923:
1916:
1903:
1898:
1885:
1857:
1852:
1848:
1844:
1836:
1826:
1815:
1797:
1791:
1775:
1762:
1756:
1749:
1743:
1736:
1730:
1725:
1705:
1701:
1697:
1692:
1688:
1684:
1679:
1675:
1664:
1646:
1644:
1638:
1634:
1628:
1624:
1616:
1612:
1607:
1603:
1597:
1592:
1588:
1583:
1579:
1570:
1563:
1556:
1551:
1550:
1544:
1523:
1518:
1506:
1502:
1498:
1493:
1481:
1477:
1473:
1468:
1456:
1452:
1439:
1432:
1425:
1421:
1416:
1407:
1400:
1386:
1373:
1351:
1340:
1332:
1328:
1324:
1313:
1294:
1233:
1215:
1204:
1196:
1177:
1173:
1151:
1140:
1136:
1119:
1105:
1095:
1087:
1083:
1079:
1065:
1045:
1037:
1034:
1033:
1032:
1030:
1013:
1011:
1001:
998:
993:
989:
985:
980:
974:
971:
965:
959:
953:
948:
943:
941:
936:
929:
925:
919:
914:
897:
889:
884:
878:
872:
866:
849:
840:
823:
809:
799:
791:
787:
783:
769:
757:
749:
742:
733:
724:
717:
708:
706:
702:
698:
694:
691:structure in
690:
682:Applicability
674:
672:
668:
611:
607:
603:
593:
589:
558:
535:
531:
481:
477:
456:
436:
406:
390:
363:
359:
349:
347:
343:
339:
334:
332:
328:
324:
320:
316:
312:
307:
305:
289:
281:
265:
257:
232:
228:
212:
204:
203:wave function
188:
181:is position,
156:
132:
122:
114:
110:
106:
92:
84:
80:
78:
74:
71:modulated by
70:
66:
62:
58:
49:
43:
36:
32:
28:
22:
13629:
13617:
13605:
13593:
13511:Pines' demon
13280:
13250:Kondo effect
13152:Time crystal
13038:
13034:
13021:
12984:
12963:
12911:
12907:
12901:
12876:
12872:
12862:
12854:
12849:
12829:
12819:
12809:
12800:
12781:
12777:
12764:
12744:
12740:
12727:
12694:
12690:
12681:
12669:
12657:
12645:
12618:
12607:
12589:
12578:
12568:12 September
12566:. Retrieved
12554:
12523:
12511:
12492:
12486:
12465:
12456:
12447:
12381:
12366:
12362:
12275:
12262:(1883), and
12252:
12236:
12162:
12159:
12150:acceleration
12125:
12034:
12027:
11952:
11452:Eliminating
10714:
10264:
10261:
10254:
9984:
9981:
9724:
8977:
8971:
8965:
8834:
8641:
8638:
8634:
8502:
7267:
7058:
6679:
6667:
6648:
6247:
6136:
5773:
5669:
5667:
5490:
5380:
5262:
5129:
5099:cyclic group
4985:
4945:
4707:translations
4704:
4684:space groups
4681:
3954:
2895:
2892:
2885:
2784:
2778:
2771:
2765:
2758:
2752:
2747:
2676:
2673:
2103:
2099:
2090:
2084:
2077:
2071:
2064:
2058:
2054:
2048:
1945:
1941:
1937:
1928:
1921:
1914:
1901:
1883:
1803:
1777:
1760:
1754:
1747:
1741:
1734:
1728:
1652:
1636:
1632:
1626:
1622:
1614:
1610:
1605:
1601:
1595:
1590:
1586:
1581:
1577:
1568:
1561:
1554:
1547:
1545:
1437:
1430:
1423:
1419:
1417:
1413:
1405:
1398:
1286:such that:
1235:
1231:
1138:
1134:
1020:
1007:
996:
978:
975:
969:
963:
957:
951:
944:
934:
927:
923:
917:
912:
887:
882:
876:
870:
864:
838:
761:
747:
740:
731:
722:
685:
350:
341:
337:
335:
315:Bloch states
314:
310:
308:
158:
85:
82:
60:
54:
47:
13548:Soft matter
13448:Ferromagnet
13266:Drude model
13235:Berry phase
13215:Hall effect
13035:Wave Motion
12687:Felix Bloch
11051:Again with
6663:point group
4692:point group
4688:translation
4375:. Finally,
2039:Again, the
1897:eigenvalues
1767:(as above,
711:Wave vector
687:a periodic
342:Bloch Waves
231:wave vector
77:Felix Bloch
69:plane waves
13647:Categories
13463:Spin glass
13458:Metamagnet
13438:Paramagnet
13325:Conduction
13301:BCS theory
13142:Superfluid
13137:Supersolid
12502:0521297338
12439:References
12024:in a band
9988:to obtain
6670:characters
6659:characters
1790:eigenstate
729:(left) or
689:dielectric
405:band index
31:Isosurface
13521:Polariton
13428:Diamagnet
13376:Couplings
13352:Conductor
13347:Semimetal
13332:Insulator
13208:Phenomena
13132:Fermi gas
13011:H. Föll.
13004:692760083
12936:121065949
12869:Sunada, T
12825:Magnus, W
12784:: 47–88.
12741:Acta Math
12719:120668259
12212:×
12188:−
12179:˙
12170:ℏ
12101:×
12066:∓
12032:in a band
11967:ℏ
11913:ε
11909:−
11889:ε
11883:⟩
11861:∇
11854:−
11833:⟨
11830:⟩
11803:∇
11796:−
11780:⟨
11774:⟩
11752:∇
11745:−
11724:⟨
11721:⟩
11694:∇
11687:−
11671:⟨
11660:≠
11648:∑
11623:ℏ
11600:δ
11585:ℏ
11563:∂
11550:∂
11528:ε
11518:∂
11421:ε
11417:−
11403:ε
11385:⟩
11361:∇
11355:−
11349:⋅
11330:ℏ
11311:⟨
11295:≠
11283:∑
11250:ℏ
11208:∂
11195:∂
11184:ε
11174:∂
11158:∑
11090:⟩
11060:ψ
11020:ε
11016:−
11002:ε
10953:∇
10947:−
10941:⋅
10922:ℏ
10914:∗
10887:∫
10871:≠
10859:∑
10826:ℏ
10784:∂
10771:∂
10760:ε
10750:∂
10734:∑
10671:ε
10667:−
10647:ε
10641:⟩
10619:∇
10612:−
10591:⟨
10588:⟩
10561:∇
10554:−
10538:⟨
10532:⟩
10510:∇
10503:−
10482:⟨
10479:⟩
10452:∇
10445:−
10429:⟨
10418:≠
10406:∑
10381:ℏ
10358:δ
10343:ℏ
10321:∂
10308:∂
10286:ε
10276:∂
10229:ψ
10222:∇
10216:−
10208:∗
10194:ψ
10181:∫
10167:ℏ
10150:∂
10139:ε
10135:∂
10086:∇
10080:−
10072:∗
10045:∫
10031:ℏ
10014:∂
10003:ε
9999:∂
9950:∗
9923:∫
9860:∇
9854:−
9837:ℏ
9829:∗
9802:∫
9793:∑
9763:∂
9752:ε
9748:∂
9736:∑
9675:−
9637:ψ
9630:^
9619:∗
9610:ψ
9597:∫
9581:≠
9569:∑
9556:ψ
9549:^
9538:∗
9529:ψ
9516:∫
9439:ℏ
9419:∇
9413:−
9407:⋅
9388:ℏ
9368:^
9336:^
9290:^
9238:ε
9162:∂
9149:∂
9138:ε
9128:∂
9112:∑
9072:∂
9061:ε
9057:∂
9045:∑
9021:ε
8989:ε
8938:∂
8925:∂
8903:ε
8893:∂
8861:∂
8850:ε
8846:∂
8816:⟩
8810:^
8799:⟨
8796:ℏ
8790:⟩
8784:^
8773:⟨
8765:ℏ
8746:ψ
8739:∇
8733:−
8725:∗
8711:ψ
8698:∫
8684:ℏ
8667:∂
8656:ε
8652:∂
8607:ℏ
8587:∇
8584:ℏ
8578:−
8550:ℏ
8544:∇
8541:ℏ
8535:−
8520:^
8402:∇
8396:−
8371:ℏ
8289:⋅
8222:∇
8211:⋅
8195:−
8169:∇
8159:⋅
8143:⋅
8129:−
8096:⋅
8047:ℏ
8004:⋅
7942:⋅
7875:∇
7864:⋅
7822:∇
7812:⋅
7796:⋅
7754:∇
7744:⋅
7695:⋅
7666:⋅
7633:ℏ
7629:−
7586:⋅
7524:⋅
7463:∇
7453:⋅
7404:⋅
7375:⋅
7372:∇
7352:ℏ
7348:−
7305:⋅
7219:⋅
7170:∇
7149:ℏ
7145:−
7096:⋅
6999:ε
6849:ε
6778:∇
6772:−
6747:ℏ
6699:^
6629:τ
6614:ψ
6605:τ
6601:⋅
6574:ψ
6557:ψ
6551:τ
6542:ε
6535:^
6494:α
6490:χ
6484:α
6480:ψ
6474:α
6470:∑
6457:ψ
6444:^
6414:τ
6410:⋅
6349:χ
6303:χ
6257:χ
6100:π
6056:∈
6016:π
6010:−
5752:τ
5743:ε
5736:^
5702:τ
5693:ε
5686:^
5643:ψ
5604:∑
5587:ψ
5544:∇
5524:ℏ
5518:−
5509:^
5449:∑
5292:∑
5165:^
5162:τ
5139:χ
5115:∞
5112:→
5069:γ
5063:χ
5031:γ
5010:γ
4988:character
4961:^
4958:τ
4923:^
4889:^
4886:τ
4853:^
4850:τ
4834:^
4831:τ
4815:^
4812:τ
4799:^
4796:τ
4734:∑
4726:τ
4553:⋅
4515:ψ
4480:ψ
4470:⋅
4440:⋅
4421:ψ
4404:ψ
4391:^
4358:∈
4286:λ
4223:ψ
4209:∫
4185:λ
4143:ψ
4129:^
4105:∫
4067:ψ
4053:∫
4021:∈
3991:⋅
3964:λ
3911:λ
3889:λ
3870:λ
3838:ψ
3807:ψ
3771:^
3705:ψ
3688:ψ
3667:^
3637:^
3597:^
3557:ψ
3546:λ
3528:ψ
3514:^
3465:^
3447:^
3382:^
3362:^
3092:ψ
2993:ψ
2952:ψ
2931:ψ
2917:^
2821:^
2746:denote a
2692:^
2634:⋅
2604:ψ
2538:ψ
2527:θ
2520:π
2501:θ
2494:π
2488:−
2473:⋅
2462:−
2426:ψ
2415:θ
2408:π
2369:⋅
2358:−
2343:⋅
2332:−
2281:ψ
2250:⋅
2239:−
2159:ψ
2149:⋅
2138:−
2052:. Define
2016:ψ
2005:θ
1998:π
1958:ψ
1886:= 1, 2, 3
1858:ψ
1816:ψ
1722:denote a
1668:^
1352:ψ
1341:⋅
1295:ψ
1205:⋅
1096:⋅
1066:ψ
1046:ψ
1016:Statement
916:, it can
898:ψ
850:ψ
800:⋅
770:ψ
695:leads to
608:ψ
590:ψ
532:ψ
478:ψ
360:ψ
189:ψ
123:⋅
93:ψ
13595:Category
13576:Colloids
12961:(1976).
12808:(1992).
12772:(1883).
12747:: 1–36.
12735:(1886).
12559:Archived
12546:(2002).
12464:(1996).
12390:See also
12258:(1877),
11921:′
11840:′
11821:′
11731:′
11712:′
11656:′
11429:′
11376:′
11291:′
11028:′
10976:′
10867:′
10679:′
10598:′
10579:′
10489:′
10470:′
10414:′
10255:For the
9687:′
9577:′
3861:we have
3344:and the
2893:Source:
2791:, where
2097:, where
1888:, where
1406:Source:
932:, where
383:, where
321:for the
13607:Commons
13571:Polymer
13538:Polaron
13516:Plasmon
13496:Exciton
12916:Bibcode
12893:2374542
12699:Bibcode
4994:of the
4715:abelian
4711:unitary
938:is any
754:) is a
559:vector
302:is the
254:is the
201:is the
33:of the
13506:Phonon
13501:Magnon
13259:Theory
13117:Plasma
13107:Liquid
13002:
12992:
12971:
12934:
12891:
12837:
12717:
12499:
12474:
12360:where
12148:is an
12126:where
9229:Given
8969:where
6191:, and
6038:where
5432:where
5379:where
4874:where
4690:and a
4347:where
4010:where
3580:Given
1806:ψ
1794:ψ
1788:is an
1786:ψ
1641:, see
1599:, but
1536:where
1132:where
862:, not
836:where
581:, or,
327:states
282:, and
159:where
13481:Anyon
13102:Solid
12932:S2CID
12889:JSTOR
12715:S2CID
12599:(PDF)
12562:(PDF)
12551:(PDF)
10712:Proof
8832:Proof
7056:Proof
4992:trace
4696:basis
4595:with
3132:with
2186:Then
1935:with
1779:Lemma
1620:when
1390:Proof
1029:basis
976:When
319:basis
225:is a
13491:Hole
13000:OCLC
12990:ISBN
12969:ISBN
12835:ISBN
12570:2020
12497:ISBN
12472:ISBN
12375:and
11479:and
8881:and
4713:and
4709:are
4705:All
4318:and
1881:for
1653:Let
945:The
918:also
886:are
880:and
13112:Gas
13043:doi
12924:doi
12912:209
12881:doi
12877:110
12786:doi
12749:doi
12707:doi
12555:MIT
8527:eff
1645:.)
1618:= 0
1594:= 2
888:not
868:or
469:),
325:or
313:or
278:is
55:In
13649::
13039:50
13037:.
13033:.
12998:.
12957:;
12930:.
12922:.
12910:.
12887:.
12875:.
12782:12
12780:.
12776:.
12743:.
12739:.
12713:.
12705:.
12695:52
12630:^
12557:.
12553:.
12535:^
12386:.
12379:.
12282::
12245:.
6665:.
5801::
2777:+
2764:+
2083:+
2070:+
2057:=
1950::
1940:=
1927:,
1920:,
1753:+
1740:+
1625:≠
1609:·
1585:·
1567:,
1560:,
1436:,
1429:,
1012:.
926:+
746:−
707:.
673:.
348:.
333:.
306:.
258:,
205:,
59:,
13078:e
13071:t
13064:v
13049:.
13045::
13006:.
12977:.
12938:.
12926::
12918::
12895:.
12883::
12843:.
12794:.
12788::
12757:.
12751::
12745:8
12721:.
12709::
12701::
12572:.
12505:.
12480:.
12369:)
12367:t
12365:(
12363:f
12348:,
12345:0
12342:=
12339:y
12336:)
12333:t
12330:(
12327:f
12324:+
12316:2
12312:t
12308:d
12303:y
12298:2
12294:d
12221:)
12216:B
12208:v
12204:+
12200:E
12195:(
12191:e
12185:=
12176:k
12135:a
12110:)
12105:B
12098:)
12094:k
12090:(
12086:v
12082:+
12078:E
12073:(
12069:e
12063:=
12059:a
12055:)
12051:k
12047:(
12043:M
12008:)
12004:k
12000:(
11996:M
11971:2
11963:1
11935:)
11931:k
11927:(
11918:n
11906:)
11902:k
11898:(
11893:n
11879:k
11875:n
11871:|
11865:i
11857:i
11850:|
11845:k
11837:n
11826:k
11818:n
11813:|
11807:j
11799:i
11792:|
11787:k
11783:n
11777:+
11770:k
11766:n
11762:|
11756:j
11748:i
11741:|
11736:k
11728:n
11717:k
11709:n
11704:|
11698:i
11690:i
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11678:k
11674:n
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11653:n
11642:2
11637:)
11632:m
11627:2
11617:(
11612:+
11607:j
11604:i
11594:m
11589:2
11579:=
11571:j
11567:k
11558:i
11554:k
11545:)
11541:k
11537:(
11532:n
11522:2
11492:j
11488:q
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11390:|
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11364:)
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11352:(
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11318:k
11314:n
11307:|
11298:n
11288:n
11279:+
11274:2
11270:q
11263:m
11260:2
11254:2
11244:=
11239:j
11235:q
11229:i
11225:q
11216:j
11212:k
11203:i
11199:k
11188:n
11178:2
11165:j
11162:i
11152:2
11149:1
11125:k
11121:n
11117:u
11110:x
11105:k
11101:i
11097:e
11093:=
11086:k
11082:n
11078:|
11074:=
11068:k
11064:n
11033:k
11025:n
11010:k
11006:n
10994:2
10989:|
10981:k
10973:n
10968:u
10964:)
10960:k
10956:+
10950:i
10944:(
10937:q
10931:m
10926:2
10908:k
10904:n
10900:u
10894:r
10890:d
10883:|
10874:n
10864:n
10855:+
10850:2
10846:q
10839:m
10836:2
10830:2
10820:=
10815:j
10811:q
10805:i
10801:q
10792:j
10788:k
10779:i
10775:k
10764:n
10754:2
10741:j
10738:i
10728:2
10725:1
10693:)
10689:k
10685:(
10676:n
10664:)
10660:k
10656:(
10651:n
10637:k
10633:n
10629:|
10623:i
10615:i
10608:|
10603:k
10595:n
10584:k
10576:n
10571:|
10565:j
10557:i
10550:|
10545:k
10541:n
10535:+
10528:k
10524:n
10520:|
10514:j
10506:i
10499:|
10494:k
10486:n
10475:k
10467:n
10462:|
10456:i
10448:i
10441:|
10436:k
10432:n
10421:n
10411:n
10400:2
10395:)
10390:m
10385:2
10375:(
10370:+
10365:j
10362:i
10352:m
10347:2
10337:=
10329:j
10325:k
10316:i
10312:k
10303:)
10299:k
10295:(
10290:n
10280:2
10237:k
10233:n
10225:)
10219:i
10213:(
10202:k
10198:n
10188:r
10184:d
10176:m
10171:2
10161:=
10154:k
10143:n
10109:k
10105:n
10101:u
10097:)
10093:k
10089:+
10083:i
10077:(
10066:k
10062:n
10058:u
10052:r
10048:d
10040:m
10035:2
10025:=
10018:k
10007:n
9985:q
9964:k
9960:n
9956:u
9944:k
9940:n
9936:u
9930:r
9926:d
9900:k
9896:n
9892:u
9886:i
9882:q
9876:i
9872:)
9867:k
9863:+
9857:i
9851:(
9846:m
9841:2
9823:k
9819:n
9815:u
9809:r
9805:d
9797:i
9789:=
9784:i
9780:q
9771:i
9767:k
9756:n
9740:i
9725:q
9710:.
9707:.
9704:.
9701:+
9693:0
9684:n
9679:E
9670:0
9665:n
9661:E
9653:2
9648:|
9641:n
9627:V
9614:n
9604:r
9600:d
9593:|
9584:n
9574:n
9565:+
9560:n
9546:V
9533:n
9523:r
9519:d
9513:+
9508:0
9503:n
9499:E
9495:=
9490:n
9486:E
9463:2
9459:q
9452:m
9449:2
9443:2
9433:+
9430:)
9426:k
9422:+
9416:i
9410:(
9403:q
9397:m
9392:2
9382:+
9376:k
9365:H
9358:=
9352:q
9348:+
9344:k
9333:H
9306:q
9302:+
9298:k
9287:H
9263:)
9259:q
9255:+
9251:k
9247:(
9242:n
9217:)
9212:3
9208:q
9204:(
9201:O
9198:+
9193:j
9189:q
9183:i
9179:q
9170:j
9166:k
9157:i
9153:k
9142:n
9132:2
9119:j
9116:i
9106:2
9103:1
9098:+
9093:i
9089:q
9080:i
9076:k
9065:n
9049:i
9041:+
9038:)
9034:k
9030:(
9025:n
9017:=
9014:)
9010:q
9006:+
9002:k
8998:(
8993:n
8978:k
8972:q
8966:q
8946:j
8942:k
8933:i
8929:k
8920:)
8916:k
8912:(
8907:n
8897:2
8865:k
8854:n
8806:v
8793:=
8780:p
8768:m
8760:=
8754:k
8750:n
8742:)
8736:i
8730:(
8719:k
8715:n
8705:r
8701:d
8693:m
8688:2
8678:=
8671:k
8660:n
8611:k
8581:i
8558:,
8554:k
8547:+
8538:i
8532:=
8516:p
8484:)
8480:x
8476:(
8470:k
8465:u
8461:)
8457:x
8453:(
8450:U
8447:+
8444:)
8440:x
8436:(
8430:k
8425:u
8419:2
8414:)
8409:k
8405:+
8399:i
8392:(
8384:m
8381:2
8375:2
8365:=
8358:)
8354:x
8350:(
8344:k
8339:u
8332:k
8327:E
8319:)
8315:x
8311:(
8305:k
8300:u
8293:x
8285:k
8281:i
8277:e
8273:)
8269:x
8265:(
8262:U
8259:+
8255:)
8251:)
8247:x
8243:(
8237:k
8232:u
8226:2
8215:x
8207:k
8203:i
8199:e
8192:)
8188:x
8184:(
8178:k
8173:u
8163:x
8155:k
8151:i
8147:e
8139:k
8135:i
8132:2
8126:)
8122:x
8118:(
8112:k
8107:u
8100:x
8092:k
8088:i
8084:e
8078:2
8073:k
8067:(
8060:m
8057:2
8051:2
8041:=
8034:)
8030:x
8026:(
8020:k
8015:u
8008:x
8000:k
7996:i
7992:e
7985:k
7980:E
7972:)
7968:x
7964:(
7958:k
7953:u
7946:x
7938:k
7934:i
7930:e
7926:)
7922:x
7918:(
7915:U
7912:+
7908:)
7904:)
7900:x
7896:(
7890:k
7885:u
7879:2
7868:x
7860:k
7856:i
7852:e
7848:+
7845:)
7841:x
7837:(
7831:k
7826:u
7816:x
7808:k
7804:i
7800:e
7792:k
7788:i
7785:+
7781:)
7777:)
7773:x
7769:(
7763:k
7758:u
7748:x
7740:k
7736:i
7732:e
7728:+
7725:)
7721:x
7717:(
7711:k
7706:u
7699:x
7691:k
7687:i
7683:e
7678:k
7674:i
7670:(
7662:k
7658:i
7654:(
7647:m
7644:2
7637:2
7623:=
7616:)
7612:x
7608:(
7602:k
7597:u
7590:x
7582:k
7578:i
7574:e
7567:k
7562:E
7554:)
7550:x
7546:(
7540:k
7535:u
7528:x
7520:k
7516:i
7512:e
7508:)
7504:x
7500:(
7497:U
7494:+
7490:)
7486:)
7482:x
7478:(
7472:k
7467:u
7457:x
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7445:i
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7437:+
7434:)
7430:x
7426:(
7420:k
7415:u
7408:x
7400:k
7396:i
7392:e
7387:k
7383:i
7379:(
7366:m
7363:2
7356:2
7342:=
7335:)
7331:x
7327:(
7321:k
7316:u
7309:x
7301:k
7297:i
7293:e
7286:k
7281:E
7253:)
7249:)
7245:x
7241:(
7235:k
7230:u
7223:x
7215:k
7211:i
7207:e
7202:(
7197:]
7193:)
7189:x
7185:(
7182:U
7179:+
7174:2
7163:m
7160:2
7153:2
7138:[
7134:=
7130:)
7126:)
7122:x
7118:(
7112:k
7107:u
7100:x
7092:k
7088:i
7084:e
7079:(
7072:k
7067:E
7038:k
7016:)
7012:k
7008:(
7003:n
6976:k
6954:)
6950:R
6946:+
6942:r
6938:(
6932:k
6927:u
6923:=
6920:)
6916:r
6912:(
6906:k
6901:u
6880:)
6876:r
6872:(
6866:k
6861:u
6854:k
6845:=
6842:)
6838:r
6834:(
6828:k
6823:u
6818:]
6814:)
6810:r
6806:(
6803:U
6800:+
6795:2
6790:)
6785:k
6781:+
6775:i
6768:(
6760:m
6757:2
6751:2
6740:[
6736:=
6733:)
6729:r
6725:(
6719:k
6714:u
6707:k
6696:H
6633:)
6625:+
6621:r
6617:(
6611:=
6597:k
6593:i
6589:e
6585:)
6581:r
6577:(
6571:=
6568:)
6564:r
6560:(
6546:|
6532:P
6508:)
6505:R
6502:(
6497:j
6466:=
6461:j
6451:R
6441:P
6406:k
6402:i
6398:e
6394:=
6391:)
6386:3
6382:a
6378:,
6373:3
6369:n
6365:(
6358:3
6354:k
6345:)
6340:2
6336:a
6332:,
6327:2
6323:n
6319:(
6312:2
6308:k
6299:)
6294:1
6290:a
6286:,
6281:1
6277:n
6273:(
6266:1
6262:k
6231:1
6227:a
6204:1
6200:L
6177:1
6173:m
6150:1
6146:k
6119:1
6115:L
6108:1
6104:m
6097:2
6091:=
6086:1
6082:k
6060:Z
6051:1
6047:m
6024:1
6020:m
6013:2
6007:)
6002:1
5998:L
5994:+
5989:1
5985:a
5979:1
5975:n
5971:(
5966:1
5962:k
5958:=
5953:1
5949:a
5943:1
5939:n
5933:1
5929:k
5906:)
5901:1
5897:L
5893:+
5888:1
5884:a
5878:1
5874:n
5870:(
5865:1
5861:k
5857:i
5853:e
5849:=
5842:1
5838:a
5832:1
5828:n
5822:1
5818:k
5814:i
5810:e
5787:i
5783:L
5756:i
5747:|
5733:P
5726:=
5719:i
5715:L
5711:+
5706:i
5697:|
5683:P
5670:L
5654:)
5650:r
5646:(
5640:=
5636:)
5630:i
5625:a
5618:i
5614:N
5608:i
5600:+
5596:r
5591:(
5567:)
5563:r
5559:(
5556:V
5553:+
5548:2
5537:m
5534:2
5528:2
5515:=
5506:H
5475:i
5470:a
5463:i
5459:N
5453:i
5445:=
5441:L
5418:i
5414:a
5392:a
5381:L
5367:)
5363:r
5359:(
5356:V
5353:=
5350:)
5346:L
5342:+
5338:r
5334:(
5331:V
5328:=
5324:)
5318:i
5313:a
5306:i
5302:N
5296:i
5288:+
5284:r
5279:(
5275:V
5245:1
5241:a
5235:1
5231:n
5225:1
5221:k
5217:i
5213:e
5209:=
5206:)
5203:)
5198:1
5194:a
5190:,
5185:1
5181:n
5177:(
5172:1
5155:(
5148:1
5144:k
5109:n
5085:1
5082:=
5077:n
5073:)
5066:(
5043:1
5040:=
5035:n
4968:i
4930:i
4919:a
4909:i
4905:n
4901:=
4896:i
4860:3
4841:2
4822:1
4805:=
4771:i
4766:a
4759:i
4755:n
4749:3
4744:1
4741:=
4738:i
4730:=
4662:)
4658:n
4653:A
4649:+
4645:x
4641:(
4635:k
4630:u
4626:=
4623:)
4619:x
4615:(
4609:k
4604:u
4583:)
4579:x
4575:(
4569:k
4564:u
4557:x
4549:k
4545:i
4541:e
4537:=
4534:)
4530:x
4526:(
4520:k
4494:,
4491:)
4487:x
4483:(
4474:a
4466:n
4462:k
4459:i
4455:e
4451:=
4448:)
4444:a
4436:n
4432:+
4428:x
4424:(
4418:=
4415:)
4411:x
4407:(
4398:n
4388:T
4362:R
4355:k
4335:k
4332:i
4329:=
4326:s
4304:2
4299:|
4291:n
4281:|
4277:=
4274:1
4253:x
4249:d
4244:2
4239:|
4234:)
4230:x
4226:(
4219:|
4213:V
4203:2
4198:|
4190:n
4180:|
4176:=
4172:x
4168:d
4163:2
4158:|
4154:)
4150:x
4146:(
4137:n
4125:T
4116:|
4109:V
4101:=
4097:x
4093:d
4088:2
4083:|
4078:)
4074:x
4070:(
4063:|
4057:V
4049:=
4046:1
4025:C
4018:s
3995:a
3987:n
3983:s
3979:e
3975:=
3969:n
3937:2
3932:n
3927:+
3922:1
3917:n
3907:=
3900:2
3895:n
3881:1
3876:n
3849:)
3845:x
3841:(
3818:)
3814:x
3810:(
3800:2
3795:n
3790:+
3785:1
3780:n
3767:T
3759:=
3756:)
3751:2
3746:n
3740:A
3736:+
3731:1
3726:n
3720:A
3716:+
3712:x
3708:(
3702:=
3699:)
3695:x
3691:(
3681:2
3676:n
3663:T
3651:1
3646:n
3633:T
3605:n
3593:T
3568:)
3564:x
3560:(
3551:n
3542:=
3539:)
3535:x
3531:(
3522:n
3510:T
3485:0
3482:=
3479:]
3473:n
3461:T
3453:,
3444:H
3438:[
3418:)
3414:x
3410:(
3407:U
3404:+
3398:m
3395:2
3389:2
3378:p
3368:=
3359:H
3332:)
3328:x
3324:(
3321:U
3318:=
3315:)
3309:n
3303:T
3298:+
3294:x
3290:(
3287:U
3265:)
3257:3
3253:n
3243:2
3239:n
3229:1
3225:n
3218:(
3213:=
3209:n
3204:,
3199:]
3191:3
3186:a
3177:2
3172:a
3163:1
3158:a
3150:[
3145:=
3141:A
3116:)
3112:n
3107:A
3103:+
3099:r
3095:(
3089:=
3079:)
3074:3
3069:a
3062:3
3058:n
3054:+
3049:2
3044:a
3037:2
3033:n
3029:+
3024:1
3019:a
3012:1
3008:n
3004:+
3000:r
2996:(
2990:=
2980:)
2974:n
2968:T
2963:+
2959:r
2955:(
2949:=
2942:)
2938:r
2934:(
2925:n
2913:T
2859:3
2855:n
2851:,
2846:2
2842:n
2838:,
2833:1
2829:n
2818:T
2795:i
2793:n
2788:3
2785:a
2782:3
2779:n
2775:2
2772:a
2769:2
2766:n
2762:1
2759:a
2756:1
2753:n
2730:3
2726:n
2722:,
2717:2
2713:n
2709:,
2704:1
2700:n
2689:T
2658:,
2655:)
2651:r
2647:(
2644:u
2638:r
2630:k
2626:i
2622:e
2618:=
2615:)
2611:r
2607:(
2594:u
2576:.
2573:)
2569:r
2565:(
2562:u
2559:=
2549:)
2545:r
2541:(
2531:j
2523:i
2517:2
2513:e
2505:j
2497:i
2491:2
2484:e
2477:r
2469:k
2465:i
2458:e
2454:=
2442:)
2437:)
2433:r
2429:(
2419:j
2411:i
2405:2
2401:e
2395:(
2388:)
2379:j
2374:a
2365:k
2361:i
2354:e
2347:r
2339:k
2335:i
2328:e
2322:(
2317:=
2307:)
2302:j
2297:a
2292:+
2288:r
2284:(
2276:)
2271:j
2266:a
2261:+
2257:r
2253:(
2246:k
2242:i
2235:e
2231:=
2224:)
2219:j
2214:a
2209:+
2205:r
2201:(
2198:u
2174:.
2170:)
2166:r
2162:(
2153:r
2145:k
2141:i
2134:e
2130:=
2127:)
2123:r
2119:(
2116:u
2104:j
2100:b
2094:3
2091:b
2088:3
2085:θ
2081:2
2078:b
2075:2
2072:θ
2068:1
2065:b
2062:1
2059:θ
2055:k
2049:r
2043:j
2041:θ
2027:)
2023:r
2019:(
2009:j
2001:i
1995:2
1991:e
1987:=
1984:)
1979:j
1974:a
1969:+
1965:r
1961:(
1946:j
1942:C
1938:e
1932:3
1929:θ
1925:2
1922:θ
1918:1
1915:θ
1909:j
1907:C
1902:r
1892:j
1890:C
1884:j
1869:)
1865:r
1861:(
1853:j
1849:C
1845:=
1842:)
1837:j
1832:a
1827:+
1823:r
1819:(
1771:j
1769:n
1764:3
1761:a
1758:3
1755:n
1751:2
1748:a
1745:2
1742:n
1738:1
1735:a
1732:1
1729:n
1706:3
1702:n
1698:,
1693:2
1689:n
1685:,
1680:1
1676:n
1665:T
1637:i
1633:b
1627:j
1623:i
1615:j
1611:b
1606:i
1602:a
1596:π
1591:i
1587:b
1582:i
1578:a
1572:3
1569:b
1565:2
1562:b
1558:1
1555:b
1540:i
1538:n
1524:,
1519:3
1514:a
1507:3
1503:n
1499:+
1494:2
1489:a
1482:2
1478:n
1474:+
1469:1
1464:a
1457:1
1453:n
1441:3
1438:a
1434:2
1431:a
1427:1
1424:a
1374:.
1371:)
1367:x
1363:(
1357:k
1345:a
1337:k
1333:i
1329:e
1325:=
1322:)
1318:a
1314:+
1310:x
1306:(
1300:k
1273:k
1251:a
1216:.
1213:)
1209:a
1201:n
1197:+
1193:x
1189:(
1183:k
1178:u
1174:=
1171:)
1167:x
1163:(
1157:k
1152:u
1141:)
1139:r
1137:(
1135:u
1120:,
1117:)
1113:r
1109:(
1106:u
1100:r
1092:k
1088:i
1084:e
1080:=
1077:)
1073:r
1069:(
997:k
979:k
970:k
964:k
958:k
952:k
935:K
930:)
928:K
924:k
922:(
913:k
883:u
877:k
871:u
865:k
839:u
824:,
821:)
817:r
813:(
810:u
804:r
796:k
792:i
788:e
784:=
781:)
777:r
773:(
751:2
748:k
744:1
741:k
735:2
732:k
726:1
723:k
652:k
629:)
625:K
622:+
619:k
615:(
612:n
604:=
598:k
594:n
568:K
540:k
536:n
510:k
486:k
482:n
457:n
437:u
416:k
391:n
368:k
364:n
290:i
266:e
241:k
213:u
168:r
144:)
140:r
136:(
133:u
127:r
119:k
115:i
111:e
107:=
104:)
100:r
96:(
48:e
23:.
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