The birth of quantum mechanics (partial history)

Transcription

1 Dept of Phys The brth of quantum mechancs (partal hstory) 1902: Lenard s photo-electrc effect (bass of photo-detector) vared the ntensty of carbon arc lght by a factor of 1000 and observed NO effect on the electron energy 1905: Ensten s lght quanta hypothess, E=hn explans photo-electrc effect, but couldn t explan the phenomena of nterference and dffracton 1923: de Brogle s matter wave hypothess whle tryng to explan dffracton usng lght-quanta, realzed that materal partcles mght have wave property. proposed l=h/p (from E=hn and E=cp for photons), easly explans the formula L=nħ n Bohr s model. M.C. Chang

2 1925: Davsson and Germer (Ref: Quantum Mechancs, by Tomonaga) Study the pattern of electron scatterng off N target to determne the electrc feld n the atom before annealng (varaton due to electronc shells? ) after annealng Upon a colleague s (Elasser) advce, they realzed that the angular varaton of the scatterng maybe due to electron wave dffracton.

3 1999: Zelnger s Venna group 2-slt nterference usng C 60 The velocty of 210 m/s corresponds to a de Brogle wavelength for C 60 of l=h/p= 2.5 pm. 3

4 1925: Schrodnger (Ref: Schrodnger: lfe and thought, by Moore) Durng Nov, 1925, Schrodnger gave a semnar on de Brogle s work. One audence (Debye) suggested that there should be a wave equaton. Durng the Chrstmas, Schrodnger started from the usual wave eq. 2 Ψ + k 2 Ψ = 0 ( k = 2π / λ) When l=h/p and p 2 /2m=E-V are used, 2 h 2 the wave eq. becomes Ψ + VΨ = EΨ 2m He then obtaned the correct energy spectrum for the hydrogen atom (chap 13), and studed the spectrum of SHO (chap 7), the Stark effect (chap 17), the absorpton and emsson of radaton by an atom (chap 18), all wthn 6 months of hs dscovery. The radaton problem led hm to wrte down the tme-dependent Schrodnger eq. 2 h 2 Ψ + VΨ = ħ Ψ 2m t

5 Low ntensty photon nterference Dualty (the same apples to electrons) 5

6 Whch-path measurement Complementarty: Once we know the path, the nterference dsappears. Partcle and wave propertes are lke 2 sdes of a con. The formalsm of QM says that we won t and shouldn t be able to determne the path and keep the nterference. If we can, then QM s lke statstcal mechancs, and a deeper theory (hdden varable theory) s requred. 6

7 Can we know the path but keep the nterference? A. Ensten s thought experment, 1927 d q Fgure s from Bertet et al, Nature 2001 To observe nteference, we need d snθ < λ However, from uncertanty relaton, we know d p d > h / p = λ / snθ where we have used p screen screen screen > h h = p snθ = snθ λ 7

8 Can the photons be cheated? Delayed choce experment (proposed by J.A. Wheeler, 1978) We decde whether to determne ther paths only after the partcles passed the slt (but before they ht the target) 8

9 Interference usng partcle pars (Dopfer, 1998) , 1 have antparallel momenta 2, 2 have antparallel momenta If we regster the postons of 1 and 2, then no nterference If we regster n such a way that destroys the nfo about the postons of 1 and 2, then nterference appears If we don t regster? Agan NO nterference! You don t need to touch the partcles to destroy the nterference!

10 QM s really more werd than partcles showng wave property, or the exstence of uncertanty relaton. (esp. after we learned about mult-partcle systems) Ensten understood ths serous conflct wth tradtonal physcs long tme ago and thought there s somethng wrong wth the theory. Tll now, all experments are consstent wth the predctons of QM. Everybody knows how to use t, but nobody can gve a satsfyng pcture of the quantum phenomena. In the followng, we start to learn the basc rules of QM. (slghtly more general then Shankar s.)

11 Three postulate of quantum mechancs Y W w Y Note: I. The state of a system s represented by a vector Y> n a Hlbert space In the x> bass, <x Y>= Y(x) s the famlar wave functon n (1-dm) real space, whch s complex-valued. In the k> bass, <k Y>= Y(k) s the Fourertransformed wave functon n k-space. Y> and c Y> (c s a constant) descrbe the same state for a physcal system, so we always choose the normalzed state.

12 W Note: II. To every physcal observable, there s a correspondng Hermtan operator W For example, the operators x and p=ħ/ (d/dx) are the poston and momentum operators (n 1-dm) An operator can have very dfferent forms on dfferent bases. E.g., < x x Ψ >= x < x Ψ > d < p x Ψ >= ħ < p Ψ > dp It s not always easy to know the operator for an observable, some can be constructed from x and p (e.g. L=x p), some cannot (e.g. spn S) Hermtan operator physcal observable Untary operator transformaton of a state (space rotaton, tme evoluton...) Nether of the above: ant-untary operator for tme reversal, creaton/annhlaton operators a, a + (chap 7) etc

13 w III. Assume { w >} s the complete set of egenstates of the the physcal observable W. The state of the system can be expanded as Ψ >= ω >< ω Ψ > ( ω can be dscrete or contnuous) (a) Born s rule: When we measure W expermentally, we ll get one of the egenvalues w, wth the probablty Note: P(w )= < w Y> 2 (S P(w )=1) (b) The state of the system s changed from Y> to the egenstate w > as a result of the measurement!! The expectaton value of the physcal observable W after many measurements: < Ω >= = < Ψ 2 ω P( ω ) ω ω > =< Ψ Ω Ψ > The uncertanty of the measurement s defned by 2 2 Ω = < Ω > < Ω > (standard devaton)

14 Example. (dm of Hlbert space = ) the state of an electron s descrbed by Y> (postulate I) the poston of an electron s a physcal observable the correspondng hermtan operator s x (postulate II) ts egenvalues are x, wth egenstates x> when we measure the poston, we get one partcular egenvalue x 1 (a dot on the screen), wth probablty P(x 1 )= <x 1 Y> 2 = Y(x 1 ) 2 (postulate III a) and Y> x 1 > (postulate III b) Note that the new wave functon s <x x 1 >=d(x-x 1 ) ( collapse of the wave functon)

15 A note on the wave functon: (Ref: Introducton to QM, by Grffth) Y(x) 2 Y(x 1 ) 2 dx The partcle s found at x 1 x Q: Where was the partcle just before we made the measurement? 1. The realst vew (shared by Ensten, de Brogle, Schrodnger...) The partcle was at x. 2. The orthodox vew (the Copenhagen nterpretaton, shared by Bohr, Hesenberg, Born...) The partcle wasn t really anywhere, t s the measurement that produces the result. 3. The agnostc vew Refuse to answer. It makes no sense to talk about thngs before the measurement. 1964: Bell s nequalty, we can dstngush 1 and 2 by experment!! 1982: The Aspect experment, vew 2 wns, as expected.

16 Measurng two dfferent physcal observables (Ref: Sakura, Modern QM, chap 1) 1) Compatble observables, [W,L] = 0 Can have a complete set of smultaneous egenstates { w, l j >} In general, we can expand Ψ >= C, j ω, λ j > Ψ > measure Ω get ω measure Λ C, j ω, λ j > C j get λ, j ω, λ j > j,j stll get measure Ω ω, λ j > Ψ > measure Λ get λ j C, j, j ω λ > measure get ω Ω C, j ω, λ j >

17 2) Incompatble observables [W,L] π 0 Do not have a complete set of smultaneous egenstates (If they do, then they wll commute. Prove t!) Note: can have a subset of smultaneous egenstates In general, we can expand Ψ >= or = ω > < ω Ψ > λ > < λ Ψ > Ψ > measure Ω measure Λ ω >< ω Ψ > λ j >< λ j ω >< ω Ψ > get ω get λ j Ψ > measure Λ get λ j measure λ j >< λ j Ψ > get ω Ω ω >< ω λ >< λ Ψ > j j The fnal state depends on the order of the measurement

18 Spn system (Ref: Chap 1, Modern QM, by Sakura; Chap 5, 6 n Feynman s lectures, vol. 3) SG experment Sequental SG experment

19 More sequental experments and results Z Z 100% +> Z X 50% Sx,+> 50% Sx,-> Z X Z 50% +> 50% -> The measurement of S x completely destroys the nfo about S z \S z and S x cannot be determned smultaneously (do not commute) Smlarly for S z and S y (ncompatble observables)

20 Analogy wth the polarzaton of lght y y x y x x S z ± atoms x, y polarzed lght S ± atoms x', y' polarzed lght x S S y y + atoms rght? crcularly polarzed atoms? left crcularly polarzed 1 1 E0x 'cos( kz ωt ) = E x 0 cos( kz ωt) + y cos( kz ωt), E0y 'cos( kz ωt ) = E0 x cos( kz ωt) + y cos( kz ωt). 2 2 kz t 1 1 kz t Re E0e ( ω ) = Re E0 ± x + y e ( ω or ) Sx; ± >= ± Sz; + > + Sz; > F ± HG I KJ 1 1 E ( z, t) = E x cos( kz t) y 0 ω ± cos kz ωt + π R L = Re E 0 1 ( kz ωt) x ± y e Sy ; ± >= Sz; + > ± Sz ; > F HG 1 ± I KJ

21 Another of Ensten s attack on the uncertanty prncple: EPR paradox (Ensten, Podolsky, and Rosen, 1935) 1 2 [x 1 - x 2, p 1 +p 2 ] = 0 So x 1 - x 2 and p 1 +p 2 can both be measured precsely Therefore, x 1 and p 1 (va p 2 ) can both be determned precsely \ Dx 1 Dp 1 = 0! Spooky acton at a dstance s requred! 23

22 Bohm s verson of EPR paradox electrons have zero total angular momentum f we measure S 1x, then we know S 2x (= -S 1x ) f we measure S 2y, then we know S 1y (= -S 2y ) \ DS 1x DS 1y = 0! Quantum nonlocalty, or entanglement 24