2770:
wave before the observation and B) consider the particle aspect after the detection (this is called the
Heisenberg–von Neumann collapse postulate). Indeed, since one could only observe the photon in one point of space (a photon can not be absorbed twice) this implies that the meaning of the wave function is essentially statistical and cannot be confused with a classical wave (such as those that occur in air or water).
2769:
For the relation to be a precise formulation of Bohr complementarity, one must introduce wave–particle duality in the discussion. This means one must consider both wave and particle behavior of light on an equal footing. Wave–particle duality implies that one must A) use the unitary evolution of the
2761:
The mathematical discussion presented above does not require quantum mechanics at its heart. In particular, the derivation is essentially valid for waves of any sort. With slight modifications to account for the squaring of amplitudes, the derivation could be applied to, for example, sound waves or
2773:
In this context the direct observation of a photon in the aperture plane precludes the following recording of the same photon in the focal plane (F). Reciprocally the observation in (F) means that we did not absorb the photon before. If both holes are open this implies that we don't know where we
392:. The formulation is in terms of the diffraction and interference of waves. The culmination of the development is a presentation of two numbers that characterizes the visibility of the interference fringes in the experiment, linked together as the
1463:
1614:
1750:
2406:
876:
2025:
1300:
1172:
491:
230:, in 1996, derived a related relation dealing with experimentally acquiring knowledge of the two paths using an apparatus, as opposed to predicting the path based on initial preparation. This relation is
3227:
Wootters, William K., and
Wojciech H. Zurek. "Complementarity in the double-slit experiment: Quantum nonseparability and a quantitative statement of Bohr's principle." Physical Review D 19.2 (1979): 473.
316:, says that the wave and particle aspects of quantum objects cannot be observed at the same time. The wave–particle duality relations makes Bohr's statement more quantitative – an experiment can yield
320:
information about the wave and particle aspects of a photon simultaneously, but the more information a particular experiment gives about one, the less it will give about the other. The predictability
2268:
2175:
1918:
1834:
578:
1676:
380:
is a measure of the wave information. The relations shows that they are inversely related, as one goes up, the other goes down. Fringes are visible over a wide range of distinguishability.
1520:
for a single aperture (perfect distinguishability). In the far-field of the two pinholes the two waves interfere and produce fringes. The intensity of the interference pattern at a point
3326:
Demonstrates that quantum interference effects are destroyed by irreversible object-apparatus correlations ("measurement"), not by
Heisenberg's uncertainty principle itself. See also
360:
which is the degree to which one can experimentally acquire information about the path of the particle, are measures of the particle information, while the visibility of the fringes
2698:
275:
192:
138:
975:
771:
2865:
means that both holes are open and play a symmetric role. If we detect the photon at (F), we don't know where the photon would have been detected in the aperture plane and
2459:
2464:
There are two extremal cases with a straightforward intuitive interpretation: In a single hole experiment, the fringe visibility is zero (as there are no fringes). That is,
2751:
2645:
726:
2573:
2082:
2055:
1856:
is the distance between the aperture screen and the far field analysis plane. If a lens is used to observe the fringes in the rear focal plane, the angle is given by
2516:
since we know (by definition) which hole the photon passed through. On the other hand, for a two slit configuration, where the two slits are indistinguishable with
1032:
1005:
690:
663:
612:
1311:
17:
2996:
2970:
2944:
2918:
2889:
2863:
2829:
2599:
2540:
2514:
2488:
1518:
1492:
909:
3184:
143:
Although it is treated as a single relation, it actually involves two separate relations, which mathematically look very similar. The first relation, derived by
3274:
3254:
2792:
1938:
1854:
1770:
636:
378:
358:
338:
295:
224:
84:
60:
1534:
1685:
914:
To distinguish which pinhole a photon passed through, one needs some measure of the distinguishability between pinholes. Such a measure is given by
2279:
779:
3188:
206:
in 1995. This relation involves correctly guessing which of the two paths the particle would have taken, based on the initial preparation. Here
3081:
195:
396:. The next section will discuss the orthodox quantum mechanical interpretation of the duality relation in terms of wave–particle duality.
1953:
1183:
1055:
2084:
denote the maximum and minimum intensity of the fringes respectively. By the rules of constructive and destructive interference we have
2946:
and this means that a statistical accumulation of photons at (F) builds up an interference pattern with maximal visibility. Conversely,
732:; the pinhole shape is irrelevant, and the pinholes are considered to be idealized. The wave is taken to have a fixed incident momentum
413:
340:
which expresses the degree of probability with which path of the particle can be correctly guessed, and the distinguishability
728:
is the single hole wave function for an aperture centered on the origin. The single-hole wave-function is taken to be that of
300:
The significance of the relations is that they express quantitatively the complementarity of wave and particle viewpoints in
2183:
2090:
2650:
The above presentation was limited to a pure quantum state. More generally, for a mixture of quantum states, one will have
1859:
1775:
502:
1622:
3290:
Englert, Berthold-Georg; Scully, Marlan O.; Walther, Herbert (1991). "Quantum
Optical Tests of Complementarity".
3362:
2656:
233:
150:
96:
920:
735:
2835:) is open. If now we detect the photon at (F), we know that that photon would have been detected in
2417:
305:
2710:
2604:
695:
3385:
3142:
2545:
3137:
2060:
2033:
729:
404:
389:
301:
227:
3040:
Bera, Manabendra Nath; Qureshi, Tabish; Siddiqui, Mohd Asad; Pati, Arun Kumar (22 July 2015).
3020:
1458:{\displaystyle P=\left|\;{\frac {|C_{A}|^{2}-|C_{B}|^{2}}{|C_{A}|^{2}+|C_{B}|^{2}}}\,\right|}
3301:
3202:
3150:
3103:
3015:
1010:
983:
668:
641:
590:
8:
3330:
2975:
2949:
2923:
2897:
2868:
2842:
2808:
2578:
2519:
2493:
2467:
1497:
1471:
884:
194:. It was later extended to, providing an equality for the case of pure quantum states by
63:
3305:
3206:
3154:
3107:
1772:
is the separation between the two pinholes. The angle α from the horizontal is given by
3340:
3317:
3259:
3239:
3053:
2777:
1923:
1839:
1755:
621:
363:
343:
323:
280:
209:
144:
69:
45:
3041:
3214:
3166:
3119:
3010:
3001:
The above treatment formalizes wave particle duality for the double-slit experiment.
309:
2411:
And hence we get, for a single photon in a pure quantum state, the duality relation
3321:
3309:
3292:
3210:
3158:
3111:
3063:
1046:
1609:{\displaystyle I(y)\propto 1+V\cos \left({\frac {p_{y}d}{\hbar }}+\varphi \right)}
3193:
3094:
3191:(1988). "Simultaneous wave and particle knowledge in a neutron interferometer".
3162:
1745:{\displaystyle \varphi =\operatorname {Arg} (C_{A})-\operatorname {Arg} (C_{B})}
3067:
87:
2998:
and thus, no fringes appear after a statistical recording of several photons.
3379:
3367:
3085:
2401:{\displaystyle V=2{\frac {|C_{A}\cdot C_{B}^{*}|}{|C_{A}|^{2}+|C_{B}|^{2}}}.}
871:{\displaystyle \Psi _{0}(x)\propto {\frac {e^{ip_{0}\cdot |x|/\hbar }}{|x|}}}
400:
199:
3170:
3115:
1941:
1034:
are the probabilities of finding that the particle passed through aperture
3123:
3345:
3089:
2763:
1525:
203:
692:
are proportionality factors for the corresponding wave amplitudes, and
313:
3140:(1996). "Fringe Visibility and Which-Way Information: An Inequality".
2703:
For the remainder of the development, we assume the light source is a
3313:
3339:
Drezet, Aurelien (2005). "Complementarity and Afshar's experiment".
2020:{\displaystyle V={\frac {I_{\max }-I_{\min }}{I_{\max }+I_{\min }}}}
1295:{\displaystyle P_{B}={\frac {|C_{B}|^{2}}{|C_{A}|^{2}+|C_{B}|^{2}}}}
1167:{\displaystyle P_{A}={\frac {|C_{A}|^{2}}{|C_{A}|^{2}+|C_{B}|^{2}}}}
3058:
486:{\displaystyle \Psi _{\text{Total}}(x)=\Psi _{A}(x)+\Psi _{B}(x).}
2753:
holds, following from the coherence properties of laser light.
638:
is a position in space downstream of the slits. The constants
2704:
3042:"Duality of quantum coherence and path distinguishability"
383:
3360:
388:
This section reviews the mathematical formulation of the
3039:
2774:
would have detected the photon in the aperture plane.
2263:{\displaystyle I_{\min }\propto ||C_{A}|-|C_{B}||^{2}}
2170:{\displaystyle I_{\max }\propto ||C_{A}|+|C_{B}||^{2}}
3363:"Wave–particle duality quantified for the first time"
3262:
3242:
2978:
2952:
2926:
2900:
2871:
2845:
2811:
2780:
2713:
2659:
2607:
2581:
2548:
2522:
2496:
2470:
2420:
2282:
2186:
2093:
2063:
2036:
1956:
1926:
1913:{\displaystyle \sin(\alpha )\simeq \tan(\alpha )=y/f}
1862:
1842:
1829:{\displaystyle \sin(\alpha )\simeq \tan(\alpha )=y/L}
1778:
1758:
1688:
1625:
1537:
1500:
1474:
1314:
1186:
1058:
1013:
986:
923:
887:
782:
738:
698:
671:
644:
624:
593:
505:
416:
366:
346:
326:
283:
236:
212:
153:
99:
72:
48:
3289:
3080:
3074:
583:
is the wave function associated with the pinhole at
573:{\displaystyle \Psi _{A}(x)=C_{A}\Psi _{0}(x-x_{A})}
2601:. Hence in both these extremal cases we also have
1671:{\displaystyle p_{y}=h/\lambda \cdot \sin(\alpha )}
3268:
3248:
2990:
2964:
2938:
2912:
2883:
2857:
2823:
2786:
2745:
2692:
2639:
2593:
2567:
2534:
2508:
2482:
2453:
2400:
2262:
2169:
2076:
2049:
2019:
1932:
1912:
1848:
1828:
1764:
1744:
1670:
1608:
1512:
1486:
1457:
1294:
1166:
1026:
999:
969:
903:
870:
765:
720:
684:
657:
630:
606:
572:
485:
372:
352:
332:
289:
269:
218:
186:
132:
78:
54:
3256:" here is usually referred to as "predictability
3092:(1995). "Two interferometric complementarities".
2794:defines thus the predictability of the two holes
3377:
3183:
2554:
2192:
2099:
2069:
2042:
2009:
1996:
1984:
1971:
226:can be called the predictability. A year later
3230:
3361:Karmela Padavic-Callaghan (1 September 2021).
66:with the definiteness, or distinguishability,
3236:Actually, what is called "distinguishability
1947:The visibility of the fringes is defined by
147:and Allaine Yasin in 1988, is expressed as
3177:
1678:is the momentum of the particle along the
1326:
36:Englert–Greenberger–Yasin duality relation
3344:
3057:
2689:
2450:
1449:
966:
911:is the radial distance from the pinhole.
266:
183:
129:
3130:
3136:
3026:
614:; a similar relation holds for pinhole
384:The mathematics of two-slit diffraction
14:
3378:
3338:
2273:Equivalently, this can be written as
2693:{\displaystyle V^{2}+P^{2}\leq 1.\,}
394:Englert–Greenberger duality relation
18:Englert–Greenberger duality relation
3328:"The Duality in Matter and Light".
270:{\displaystyle D^{2}+V^{2}\leq 1\,}
187:{\displaystyle P^{2}+V^{2}\leq 1\,}
133:{\displaystyle D^{2}+V^{2}\leq 1\,}
24:
3283:
2805:A maximal value of predictability
2756:
2542:, one has perfect visibility with
970:{\displaystyle P=|P_{A}-P_{B}|,\,}
784:
700:
539:
507:
462:
440:
418:
297:is called the distinguishability.
25:
3397:
3354:
1590:
846:
766:{\displaystyle p_{0}=h/\lambda }
405:Young double-aperture experiment
2454:{\displaystyle V^{2}+P^{2}=1\,}
3221:
3033:
2831:means that only one hole (say
2382:
2366:
2352:
2336:
2329:
2296:
2250:
2244:
2229:
2221:
2206:
2201:
2157:
2151:
2136:
2128:
2113:
2108:
1893:
1887:
1875:
1869:
1809:
1803:
1791:
1785:
1739:
1726:
1714:
1701:
1665:
1659:
1547:
1541:
1436:
1420:
1406:
1390:
1377:
1361:
1347:
1331:
1279:
1263:
1249:
1233:
1220:
1204:
1151:
1135:
1121:
1105:
1092:
1076:
959:
931:
897:
889:
861:
853:
837:
829:
799:
793:
715:
709:
567:
548:
522:
516:
477:
471:
455:
449:
433:
427:
32:wave–particle duality relation
13:
1:
2891:characterizes our ignorance.
2746:{\displaystyle V^{2}+P^{2}=1}
2640:{\displaystyle V^{2}+P^{2}=1}
3215:10.1016/0375-9601(88)90114-4
1752:is a fixed phase shift, and
1494:for two symmetric holes and
721:{\displaystyle \Psi _{0}(x)}
40:Englert–Greenberger relation
27:A relation of quantum optics
7:
3163:10.1103/PhysRevLett.77.2154
3004:
2568:{\displaystyle I_{\min }=0}
86:, of the photons' paths in
10:
3402:
3068:10.1103/PhysRevA.92.012118
42:, relates the visibility,
2839:necessarily. Conversely,
2077:{\displaystyle I_{\min }}
2050:{\displaystyle I_{\max }}
306:complementarity principle
2707:, so that we can assume
3138:Englert, Berthold-Georg
302:double-slit experiments
3270:
3250:
3185:Greenberger, Daniel M.
3116:10.1103/PhysRevA.51.54
2992:
2966:
2940:
2914:
2885:
2859:
2825:
2788:
2747:
2694:
2641:
2595:
2569:
2536:
2510:
2484:
2455:
2402:
2264:
2171:
2078:
2051:
2021:
1934:
1914:
1850:
1830:
1766:
1746:
1672:
1610:
1514:
1488:
1468:We have in particular
1459:
1296:
1168:
1028:
1001:
971:
905:
872:
767:
730:Fraunhofer diffraction
722:
686:
659:
632:
608:
574:
487:
390:double-slit experiment
374:
354:
334:
291:
271:
228:Berthold-Georg Englert
220:
188:
134:
80:
56:
3271:
3251:
3021:Quantum indeterminacy
2993:
2967:
2941:
2915:
2886:
2860:
2826:
2789:
2748:
2695:
2642:
2596:
2570:
2537:
2511:
2485:
2456:
2403:
2265:
2172:
2079:
2052:
2022:
1935:
1915:
1851:
1831:
1767:
1747:
1673:
1611:
1515:
1489:
1460:
1297:
1169:
1029:
1027:{\displaystyle P_{B}}
1002:
1000:{\displaystyle P_{A}}
972:
906:
873:
768:
723:
687:
685:{\displaystyle C_{B}}
660:
658:{\displaystyle C_{A}}
633:
609:
607:{\displaystyle x_{A}}
575:
488:
375:
355:
335:
292:
272:
221:
189:
135:
81:
57:
3260:
3240:
3027:References and notes
3016:Quantum entanglement
2976:
2950:
2924:
2898:
2869:
2843:
2809:
2778:
2711:
2657:
2605:
2579:
2546:
2520:
2494:
2468:
2418:
2280:
2184:
2091:
2061:
2034:
1954:
1924:
1860:
1840:
1776:
1756:
1686:
1623:
1535:
1498:
1472:
1312:
1184:
1056:
1049:measure is given by
1011:
984:
921:
885:
780:
736:
696:
669:
642:
622:
591:
503:
414:
364:
344:
324:
281:
234:
210:
151:
97:
90:. As an inequality:
70:
64:interference fringes
46:
3331:Scientific American
3306:1991Natur.351..111S
3207:1988PhLA..128..391G
3155:1996PhRvL..77.2154E
3108:1995PhRvA..51...54J
2991:{\displaystyle V=0}
2965:{\displaystyle P=1}
2939:{\displaystyle V=1}
2913:{\displaystyle P=0}
2884:{\displaystyle P=0}
2858:{\displaystyle P=0}
2824:{\displaystyle P=1}
2594:{\displaystyle V=1}
2535:{\displaystyle P=0}
2509:{\displaystyle P=1}
2483:{\displaystyle V=0}
2327:
1513:{\displaystyle P=1}
1487:{\displaystyle P=0}
904:{\displaystyle |x|}
3266:
3246:
2988:
2962:
2936:
2910:
2881:
2855:
2821:
2784:
2743:
2690:
2637:
2591:
2565:
2532:
2506:
2480:
2451:
2398:
2313:
2260:
2167:
2074:
2047:
2017:
1930:
1910:
1846:
1826:
1762:
1742:
1668:
1606:
1510:
1484:
1455:
1292:
1164:
1024:
997:
967:
901:
868:
763:
718:
682:
655:
628:
604:
570:
483:
407:can be written as
370:
350:
330:
287:
267:
216:
184:
145:Daniel Greenberger
130:
76:
52:
34:, also called the
3300:(6322): 111–116.
3269:{\displaystyle P}
3249:{\displaystyle D}
3149:(11): 2154–2157.
3046:Physical Review A
3011:Afshar experiment
2787:{\displaystyle P}
2762:water waves in a
2393:
2015:
1933:{\displaystyle f}
1849:{\displaystyle L}
1765:{\displaystyle d}
1593:
1447:
1290:
1162:
866:
631:{\displaystyle x}
424:
373:{\displaystyle V}
353:{\displaystyle D}
333:{\displaystyle P}
310:quantum mechanics
290:{\displaystyle D}
219:{\displaystyle P}
79:{\displaystyle D}
55:{\displaystyle V}
16:(Redirected from
3393:
3372:
3350:
3348:
3346:quant-ph/0508091
3335:
3334:. December 1994.
3325:
3314:10.1038/351111a0
3277:
3275:
3273:
3272:
3267:
3255:
3253:
3252:
3247:
3234:
3228:
3225:
3219:
3218:
3181:
3175:
3174:
3143:Phys. Rev. Lett.
3134:
3128:
3127:
3078:
3072:
3071:
3061:
3037:
2997:
2995:
2994:
2989:
2971:
2969:
2968:
2963:
2945:
2943:
2942:
2937:
2919:
2917:
2916:
2911:
2890:
2888:
2887:
2882:
2864:
2862:
2861:
2856:
2830:
2828:
2827:
2822:
2793:
2791:
2790:
2785:
2752:
2750:
2749:
2744:
2736:
2735:
2723:
2722:
2699:
2697:
2696:
2691:
2682:
2681:
2669:
2668:
2646:
2644:
2643:
2638:
2630:
2629:
2617:
2616:
2600:
2598:
2597:
2592:
2574:
2572:
2571:
2566:
2558:
2557:
2541:
2539:
2538:
2533:
2515:
2513:
2512:
2507:
2489:
2487:
2486:
2481:
2460:
2458:
2457:
2452:
2443:
2442:
2430:
2429:
2407:
2405:
2404:
2399:
2394:
2392:
2391:
2390:
2385:
2379:
2378:
2369:
2361:
2360:
2355:
2349:
2348:
2339:
2333:
2332:
2326:
2321:
2309:
2308:
2299:
2293:
2269:
2267:
2266:
2261:
2259:
2258:
2253:
2247:
2242:
2241:
2232:
2224:
2219:
2218:
2209:
2204:
2196:
2195:
2176:
2174:
2173:
2168:
2166:
2165:
2160:
2154:
2149:
2148:
2139:
2131:
2126:
2125:
2116:
2111:
2103:
2102:
2083:
2081:
2080:
2075:
2073:
2072:
2056:
2054:
2053:
2048:
2046:
2045:
2026:
2024:
2023:
2018:
2016:
2014:
2013:
2012:
2000:
1999:
1989:
1988:
1987:
1975:
1974:
1964:
1939:
1937:
1936:
1931:
1919:
1917:
1916:
1911:
1906:
1855:
1853:
1852:
1847:
1835:
1833:
1832:
1827:
1822:
1771:
1769:
1768:
1763:
1751:
1749:
1748:
1743:
1738:
1737:
1713:
1712:
1677:
1675:
1674:
1669:
1646:
1635:
1634:
1615:
1613:
1612:
1607:
1605:
1601:
1594:
1589:
1585:
1584:
1574:
1519:
1517:
1516:
1511:
1493:
1491:
1490:
1485:
1464:
1462:
1461:
1456:
1454:
1450:
1448:
1446:
1445:
1444:
1439:
1433:
1432:
1423:
1415:
1414:
1409:
1403:
1402:
1393:
1387:
1386:
1385:
1380:
1374:
1373:
1364:
1356:
1355:
1350:
1344:
1343:
1334:
1328:
1301:
1299:
1298:
1293:
1291:
1289:
1288:
1287:
1282:
1276:
1275:
1266:
1258:
1257:
1252:
1246:
1245:
1236:
1230:
1229:
1228:
1223:
1217:
1216:
1207:
1201:
1196:
1195:
1173:
1171:
1170:
1165:
1163:
1161:
1160:
1159:
1154:
1148:
1147:
1138:
1130:
1129:
1124:
1118:
1117:
1108:
1102:
1101:
1100:
1095:
1089:
1088:
1079:
1073:
1068:
1067:
1047:Born probability
1033:
1031:
1030:
1025:
1023:
1022:
1006:
1004:
1003:
998:
996:
995:
976:
974:
973:
968:
962:
957:
956:
944:
943:
934:
910:
908:
907:
902:
900:
892:
877:
875:
874:
869:
867:
865:
864:
856:
850:
849:
845:
840:
832:
824:
823:
806:
792:
791:
772:
770:
769:
764:
759:
748:
747:
727:
725:
724:
719:
708:
707:
691:
689:
688:
683:
681:
680:
664:
662:
661:
656:
654:
653:
637:
635:
634:
629:
618:. The variable
613:
611:
610:
605:
603:
602:
579:
577:
576:
571:
566:
565:
547:
546:
537:
536:
515:
514:
492:
490:
489:
484:
470:
469:
448:
447:
426:
425:
422:
379:
377:
376:
371:
359:
357:
356:
351:
339:
337:
336:
331:
312:, formulated by
296:
294:
293:
288:
276:
274:
273:
268:
259:
258:
246:
245:
225:
223:
222:
217:
193:
191:
190:
185:
176:
175:
163:
162:
139:
137:
136:
131:
122:
121:
109:
108:
85:
83:
82:
77:
61:
59:
58:
53:
21:
3401:
3400:
3396:
3395:
3394:
3392:
3391:
3390:
3376:
3375:
3357:
3327:
3286:
3284:Further reading
3281:
3280:
3261:
3258:
3257:
3241:
3238:
3237:
3235:
3231:
3226:
3222:
3182:
3178:
3135:
3131:
3079:
3075:
3038:
3034:
3029:
3007:
2977:
2974:
2973:
2951:
2948:
2947:
2925:
2922:
2921:
2899:
2896:
2895:
2870:
2867:
2866:
2844:
2841:
2840:
2810:
2807:
2806:
2779:
2776:
2775:
2759:
2757:Complementarity
2731:
2727:
2718:
2714:
2712:
2709:
2708:
2677:
2673:
2664:
2660:
2658:
2655:
2654:
2625:
2621:
2612:
2608:
2606:
2603:
2602:
2580:
2577:
2576:
2553:
2549:
2547:
2544:
2543:
2521:
2518:
2517:
2495:
2492:
2491:
2469:
2466:
2465:
2438:
2434:
2425:
2421:
2419:
2416:
2415:
2386:
2381:
2380:
2374:
2370:
2365:
2356:
2351:
2350:
2344:
2340:
2335:
2334:
2328:
2322:
2317:
2304:
2300:
2295:
2294:
2292:
2281:
2278:
2277:
2254:
2249:
2248:
2243:
2237:
2233:
2228:
2220:
2214:
2210:
2205:
2200:
2191:
2187:
2185:
2182:
2181:
2161:
2156:
2155:
2150:
2144:
2140:
2135:
2127:
2121:
2117:
2112:
2107:
2098:
2094:
2092:
2089:
2088:
2068:
2064:
2062:
2059:
2058:
2041:
2037:
2035:
2032:
2031:
2008:
2004:
1995:
1991:
1990:
1983:
1979:
1970:
1966:
1965:
1963:
1955:
1952:
1951:
1925:
1922:
1921:
1902:
1861:
1858:
1857:
1841:
1838:
1837:
1818:
1777:
1774:
1773:
1757:
1754:
1753:
1733:
1729:
1708:
1704:
1687:
1684:
1683:
1642:
1630:
1626:
1624:
1621:
1620:
1580:
1576:
1575:
1573:
1572:
1568:
1536:
1533:
1532:
1499:
1496:
1495:
1473:
1470:
1469:
1440:
1435:
1434:
1428:
1424:
1419:
1410:
1405:
1404:
1398:
1394:
1389:
1388:
1381:
1376:
1375:
1369:
1365:
1360:
1351:
1346:
1345:
1339:
1335:
1330:
1329:
1327:
1325:
1321:
1313:
1310:
1309:
1283:
1278:
1277:
1271:
1267:
1262:
1253:
1248:
1247:
1241:
1237:
1232:
1231:
1224:
1219:
1218:
1212:
1208:
1203:
1202:
1200:
1191:
1187:
1185:
1182:
1181:
1155:
1150:
1149:
1143:
1139:
1134:
1125:
1120:
1119:
1113:
1109:
1104:
1103:
1096:
1091:
1090:
1084:
1080:
1075:
1074:
1072:
1063:
1059:
1057:
1054:
1053:
1018:
1014:
1012:
1009:
1008:
991:
987:
985:
982:
981:
958:
952:
948:
939:
935:
930:
922:
919:
918:
896:
888:
886:
883:
882:
860:
852:
851:
841:
836:
828:
819:
815:
811:
807:
805:
787:
783:
781:
778:
777:
755:
743:
739:
737:
734:
733:
703:
699:
697:
694:
693:
676:
672:
670:
667:
666:
649:
645:
643:
640:
639:
623:
620:
619:
598:
594:
592:
589:
588:
561:
557:
542:
538:
532:
528:
510:
506:
504:
501:
500:
465:
461:
443:
439:
421:
417:
415:
412:
411:
386:
365:
362:
361:
345:
342:
341:
325:
322:
321:
282:
279:
278:
254:
250:
241:
237:
235:
232:
231:
211:
208:
207:
171:
167:
158:
154:
152:
149:
148:
117:
113:
104:
100:
98:
95:
94:
71:
68:
67:
47:
44:
43:
28:
23:
22:
15:
12:
11:
5:
3399:
3389:
3388:
3386:Quantum optics
3374:
3373:
3356:
3355:External links
3353:
3352:
3351:
3336:
3285:
3282:
3279:
3278:
3265:
3245:
3229:
3220:
3201:(8): 391–394.
3189:Yasin, Allaine
3176:
3129:
3086:Shimony, Abner
3073:
3031:
3030:
3028:
3025:
3024:
3023:
3018:
3013:
3006:
3003:
2987:
2984:
2981:
2961:
2958:
2955:
2935:
2932:
2929:
2909:
2906:
2903:
2894:Similarly, if
2880:
2877:
2874:
2854:
2851:
2848:
2820:
2817:
2814:
2783:
2758:
2755:
2742:
2739:
2734:
2730:
2726:
2721:
2717:
2701:
2700:
2688:
2685:
2680:
2676:
2672:
2667:
2663:
2636:
2633:
2628:
2624:
2620:
2615:
2611:
2590:
2587:
2584:
2564:
2561:
2556:
2552:
2531:
2528:
2525:
2505:
2502:
2499:
2479:
2476:
2473:
2462:
2461:
2449:
2446:
2441:
2437:
2433:
2428:
2424:
2409:
2408:
2397:
2389:
2384:
2377:
2373:
2368:
2364:
2359:
2354:
2347:
2343:
2338:
2331:
2325:
2320:
2316:
2312:
2307:
2303:
2298:
2291:
2288:
2285:
2271:
2270:
2257:
2252:
2246:
2240:
2236:
2231:
2227:
2223:
2217:
2213:
2208:
2203:
2199:
2194:
2190:
2178:
2177:
2164:
2159:
2153:
2147:
2143:
2138:
2134:
2130:
2124:
2120:
2115:
2110:
2106:
2101:
2097:
2071:
2067:
2044:
2040:
2028:
2027:
2011:
2007:
2003:
1998:
1994:
1986:
1982:
1978:
1973:
1969:
1962:
1959:
1929:
1909:
1905:
1901:
1898:
1895:
1892:
1889:
1886:
1883:
1880:
1877:
1874:
1871:
1868:
1865:
1845:
1825:
1821:
1817:
1814:
1811:
1808:
1805:
1802:
1799:
1796:
1793:
1790:
1787:
1784:
1781:
1761:
1741:
1736:
1732:
1728:
1725:
1722:
1719:
1716:
1711:
1707:
1703:
1700:
1697:
1694:
1691:
1667:
1664:
1661:
1658:
1655:
1652:
1649:
1645:
1641:
1638:
1633:
1629:
1617:
1616:
1604:
1600:
1597:
1592:
1588:
1583:
1579:
1571:
1567:
1564:
1561:
1558:
1555:
1552:
1549:
1546:
1543:
1540:
1509:
1506:
1503:
1483:
1480:
1477:
1466:
1465:
1453:
1443:
1438:
1431:
1427:
1422:
1418:
1413:
1408:
1401:
1397:
1392:
1384:
1379:
1372:
1368:
1363:
1359:
1354:
1349:
1342:
1338:
1333:
1324:
1320:
1317:
1303:
1302:
1286:
1281:
1274:
1270:
1265:
1261:
1256:
1251:
1244:
1240:
1235:
1227:
1222:
1215:
1211:
1206:
1199:
1194:
1190:
1175:
1174:
1158:
1153:
1146:
1142:
1137:
1133:
1128:
1123:
1116:
1112:
1107:
1099:
1094:
1087:
1083:
1078:
1071:
1066:
1062:
1042:respectively.
1021:
1017:
994:
990:
978:
977:
965:
961:
955:
951:
947:
942:
938:
933:
929:
926:
899:
895:
891:
879:
878:
863:
859:
855:
848:
844:
839:
835:
831:
827:
822:
818:
814:
810:
804:
801:
798:
795:
790:
786:
762:
758:
754:
751:
746:
742:
717:
714:
711:
706:
702:
679:
675:
652:
648:
627:
601:
597:
581:
580:
569:
564:
560:
556:
553:
550:
545:
541:
535:
531:
527:
524:
521:
518:
513:
509:
494:
493:
482:
479:
476:
473:
468:
464:
460:
457:
454:
451:
446:
442:
438:
435:
432:
429:
420:
385:
382:
369:
349:
329:
286:
265:
262:
257:
253:
249:
244:
240:
215:
182:
179:
174:
170:
166:
161:
157:
141:
140:
128:
125:
120:
116:
112:
107:
103:
88:quantum optics
75:
51:
26:
9:
6:
4:
3:
2:
3398:
3387:
3384:
3383:
3381:
3370:
3369:
3368:Physics World
3364:
3359:
3358:
3347:
3342:
3337:
3333:
3332:
3323:
3319:
3315:
3311:
3307:
3303:
3299:
3295:
3294:
3288:
3287:
3263:
3243:
3233:
3224:
3216:
3212:
3208:
3204:
3200:
3196:
3195:
3194:Phys. Lett. A
3190:
3186:
3180:
3172:
3168:
3164:
3160:
3156:
3152:
3148:
3145:
3144:
3139:
3133:
3125:
3121:
3117:
3113:
3109:
3105:
3101:
3097:
3096:
3091:
3087:
3083:
3082:Jaeger, Gregg
3077:
3069:
3065:
3060:
3055:
3052:(1): 012118.
3051:
3047:
3043:
3036:
3032:
3022:
3019:
3017:
3014:
3012:
3009:
3008:
3002:
2999:
2985:
2982:
2979:
2959:
2956:
2953:
2933:
2930:
2927:
2907:
2904:
2901:
2892:
2878:
2875:
2872:
2852:
2849:
2846:
2838:
2834:
2818:
2815:
2812:
2803:
2801:
2797:
2781:
2771:
2767:
2765:
2754:
2740:
2737:
2732:
2728:
2724:
2719:
2715:
2706:
2686:
2683:
2678:
2674:
2670:
2665:
2661:
2653:
2652:
2651:
2648:
2634:
2631:
2626:
2622:
2618:
2613:
2609:
2588:
2585:
2582:
2562:
2559:
2550:
2529:
2526:
2523:
2503:
2500:
2497:
2477:
2474:
2471:
2447:
2444:
2439:
2435:
2431:
2426:
2422:
2414:
2413:
2412:
2395:
2387:
2375:
2371:
2362:
2357:
2345:
2341:
2323:
2318:
2314:
2310:
2305:
2301:
2289:
2286:
2283:
2276:
2275:
2274:
2255:
2238:
2234:
2225:
2215:
2211:
2197:
2188:
2180:
2179:
2162:
2145:
2141:
2132:
2122:
2118:
2104:
2095:
2087:
2086:
2085:
2065:
2038:
2005:
2001:
1992:
1980:
1976:
1967:
1960:
1957:
1950:
1949:
1948:
1945:
1944:of the lens.
1943:
1927:
1907:
1903:
1899:
1896:
1890:
1884:
1881:
1878:
1872:
1866:
1863:
1843:
1823:
1819:
1815:
1812:
1806:
1800:
1797:
1794:
1788:
1782:
1779:
1759:
1734:
1730:
1723:
1720:
1717:
1709:
1705:
1698:
1695:
1692:
1689:
1681:
1662:
1656:
1653:
1650:
1647:
1643:
1639:
1636:
1631:
1627:
1602:
1598:
1595:
1586:
1581:
1577:
1569:
1565:
1562:
1559:
1556:
1553:
1550:
1544:
1538:
1531:
1530:
1529:
1527:
1523:
1507:
1504:
1501:
1481:
1478:
1475:
1451:
1441:
1429:
1425:
1416:
1411:
1399:
1395:
1382:
1370:
1366:
1357:
1352:
1340:
1336:
1322:
1318:
1315:
1308:
1307:
1306:
1305:then we get:
1284:
1272:
1268:
1259:
1254:
1242:
1238:
1225:
1213:
1209:
1197:
1192:
1188:
1180:
1179:
1178:
1156:
1144:
1140:
1131:
1126:
1114:
1110:
1097:
1085:
1081:
1069:
1064:
1060:
1052:
1051:
1050:
1048:
1043:
1041:
1038:and aperture
1037:
1019:
1015:
992:
988:
963:
953:
949:
945:
940:
936:
927:
924:
917:
916:
915:
912:
893:
857:
842:
833:
825:
820:
816:
812:
808:
802:
796:
788:
776:
775:
774:
760:
756:
752:
749:
744:
740:
731:
712:
704:
677:
673:
650:
646:
625:
617:
599:
595:
586:
562:
558:
554:
551:
543:
533:
529:
525:
519:
511:
499:
498:
497:
496:The function
480:
474:
466:
458:
452:
444:
436:
430:
410:
409:
408:
406:
402:
401:wave function
397:
395:
391:
381:
367:
347:
327:
319:
315:
311:
307:
303:
298:
284:
263:
260:
255:
251:
247:
242:
238:
229:
213:
205:
201:
200:Abner Shimony
197:
180:
177:
172:
168:
164:
159:
155:
146:
126:
123:
118:
114:
110:
105:
101:
93:
92:
91:
89:
73:
65:
49:
41:
37:
33:
19:
3366:
3329:
3297:
3291:
3232:
3223:
3198:
3192:
3179:
3146:
3141:
3132:
3102:(1): 54–67.
3099:
3095:Phys. Rev. A
3093:
3090:Vaidman, Lev
3076:
3049:
3045:
3035:
3000:
2893:
2836:
2832:
2804:
2799:
2795:
2772:
2768:
2760:
2702:
2649:
2463:
2410:
2272:
2029:
1946:
1942:focal length
1679:
1618:
1528:is given by
1521:
1467:
1304:
1176:
1044:
1039:
1035:
979:
913:
880:
615:
587:centered on
584:
582:
495:
398:
393:
387:
317:
299:
196:Gregg Jaeger
142:
39:
35:
31:
29:
2764:ripple tank
1682:direction,
1526:focal plane
204:Lev Vaidman
3059:1503.02990
2575:and hence
1045:Since the
314:Niels Bohr
2798:and
2684:≤
2324:∗
2311:⋅
2226:−
2198:∝
2105:∝
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