Hilbert Space Methods Used in a First Course in Quantum Mechanics

Transcription

1 Hilbert Space Methods Used i a First Course i Quatum Mechaics Victor Poliger Physics/Mathematics Bellevue College 03/07/3-04//3

2 Outlie The Ifiite Square Well: A Follow-Up Timelie of basic evets Statistical iterpretatio of the wave fuctio. Movie. Three approaches i Quatum Theory. Basic postulates of Schrődiger s Wave Mechaics. Time-depedet Schrӧdiger Equatio i D case. Separatio of variables. Statioary Schrӧdiger Equatio ad its matrix equivalet. The Hilbert space as the tool of Quatum Theory. A Bite of Solid-State Physics. Phoos.

3 The Problem of Ifiite Square Well: A Follow-Up De Broglie s waves: h p p Groud state: I the groud state: a x, x, x x xdx 0 a,, 3 0 x x x x x dx a x x x dx 0 a p p p a, 3 d p, p, p i x xdx 0, a d p, p, p x x dx a a a 0 x p.36 3 ie t x / x, t e x, x si a a h p a, p, E m Excited states: / x t e ie t x x,, x si, a a E E,,3,4,... The Ucertaity Priciple: x p

4 Itroductio: Timelie of Basic Evets - I , B. Stewart (Eglad) ad G. Kirchhoff (Prussia). Black-body radiatio. Light icidet ito a hole i a cavity is lost forever. For a observer, the hole looks like a black body Max Plak (Germay),December 4, 900, : Electromagetic eergy does ot follow the classical descriptio, but ca oly be emitted i discrete packets of eergy, hν, proportioal to the frequecy, ν. 905, Lord Rayleigh ad J. Jeas (Eglad) The ultraviolet catastrophe, cost I 4 Plak s empirical formula: I c 5 hc/ kt e h c is speed of light, h J s is the Plak costat. I Lati, quatus meas how much

5 Itroductio: Timelie of Basic Evets - 90, P. Leard (Germay). Photoelectric Effect. Maximum eergy of the ejected electros did ot deped o light itesity but did deped o the color -- the higher-frequecy light caused electros to be ejected with more eergy (Nobel Prize of 905). Leard joied Nazi Party at a early stage. Together with J. Stark he leaded the core campaig to label Eistei s theory as Jewish Physics 905, A. Eistei (Switzerlad) solved this apparet paradox by describig light as composed of discrete quata, ow called photos, rather tha cotiuous waves. Eergy carried by oe photo is E = hν. h W KE KE h W 94, Louis de Broglie (Frace). Wave-Particle Duality. Nobel Prize of 99 for the idea of wave-particle duality coceivig the wave mechaics If electromagetic waves ca maifest features of particles, the a movig particle may be associated with a wave (Nobel Prize of 99). I his ow words, My essetial idea was to exted to all particles the coexistece of waves ad particles discovered by Eistei i 905 i the case of light ad photos. Nobel Prize of 93 for explaatio of the photoeffect h p

6 Statistical Iterpretatio of the Wave Fuctio 96, Max Bor (Germay). Statistical iterpretatio (Nobel Prize of 954) x, t dx Nobel prize of 954 for fudametal research i quatum mechaics, especially for statistical iterpretatio of the wave fuctio probability of fidig the particle i the iterval (x, x + dx) at time t 97, The Davisso-Germer s Experimet (Bell Labs,USA). Usig the surface of the crystal lattice of ickel as diffractio gratig, they observed diffractio patter from a beam of icidet electros 989, Akira Toomura Double-slit experimet o electro diffractio i Hitachi Lab, Japa Nobel prize of 937 (awarded to Davisso ad Thomso) for experimetal discovery of electro diffractio

7 Dr. Quatum Explais it All (watch the movie)

8 Three Approaches i Quatum Theory. Matrix Mechaics (95) M.Bor, W. Heiseberg, ad P. Jorda i Gőttige, Germay (The Drei-Mäer Arbeit ). I 98 all three authors were omiated for Nobel Prize by A. Eistei. Oly Heiseberg wo it i 93 Their equivalecy was prove by Schrődiger i 96 Wave Mechaics (96) E. Schrődiger, Zűrich, Switzerlad. W. Heiseberg, Nobel prize of 93 For creatio of Quatum Mechaics The Path Itegral Formulatio (948) R. Feyma, Priceto, NJ, USA. Their equivalecy was prove by R. Feyma i 948 E. Schrődiger Nobel Prize of 933 (shared with P. A. M. Dirac). R. Feyma Nobel Prize of 965 (shared with J. Schwiger ad S.-I. Tomoaga)

9 Basic postulates of Schrődiger s Wave Mechaics Mathematically rigorous formulatio of quatum mechaics was developed by Paul Dirac, David Hilbert, Joh vo Neuma, ad Herma Weyl.. Every mechaical system is completely described by a wave fuctio, Ψ(x, t), represetig the state of the system. The wave fuctio is double-itegrable over all real umbers; i the case of a fiite motio, the itegral *,,, x t x t x x dx dx with a smooth core, φ(x, x ), exists (coverges). [Ψ(x, t) 0 as x ± ]. The descriptio of ature is essetially probabilistic. Ψ(x, t) is the amplitude of probability: Ψ(x, t) dx represets the probability of fidig the particle i the iterval (x, x + dx) at the istat t. 3. For a fiite motio, the wave fuctio, Ψ(x, t), is ormalized to : x, t dx [ the ecessary coditio: Ψ(x, t) 0 as x ± ] 4. I the case of a fiite ad smooth potetial eergy, V(x), the wave fuctio, Ψ(x, t), ad its first derivatives, Ψ(x, t)/ x ad Ψ(x, t)/ t, is cotiuous, sigle-valued, ad fiite.

10 Basic postulates of Schrődiger s Wave Mechaics -II 5. The Priciple of Superpositio: If Ψ (x, t) is a state with a observable value L ad Ψ (x, t) is aother state of the same system with the observable value L, the ay liear combiatio, Ψ(x, t) = c Ψ (x, t) + c Ψ (x, t), is a state for which the same measuremet results i either L or L with the respective probabilities c ad c. Ψ(x, t) is a solutio of a liear equatio. 6. Ay physical observable is associated with a self-adjoit liear operator. The operators must yield real eigevalues. Operators must be Hermitia. Fˆ x K x, x x dx 7. The Correspodece Priciple (N. Bohr, 90): quatum mechaical descriptio of systems must reproduce predictios of classical mechaics i the limit of large quatum umbers or, equivaletly, whe ħ 0. For large systems, the quatum mechaical descriptio must closely approximate the classical descriptio

11 Time-depedet Schrӧdiger Equatio i D case. Separatio of variables. 8. For ay coservative system, its wave fuctio is a solutio of Schrӧdiger s Equatio: i V x t m x Time-depedet Schrődiger equatio ˆx x Itroducig the operators: ˆp i x (liear mometum) ad, Schrӧdiger s Equatio ca be preseted as i Hˆ with ˆ pˆ The Hamiltoia is the H V x t m operator of total eergy Separatio of variables: x, t xt Time-depedet i x t t Hˆ x Schrődiger equatio t Left-multiplyig each side by gives: xt i Fuctio of t t Hˆ x t t x Fuctio of x

12 i t t Statioary Schrӧdiger Equatio t iet/ t Ae E i t E t t iet/ x, t Ae x Statioary Schrӧdiger Equatio Hˆ x E x Hˆ x E x ˆ * * x H x dx E x x dx If x * x dx * x Hˆ x dx E Example: Simple Harmoic Motio (the harmoic oscillator problem) Hˆ Hˆ E d kx m dx d m dx kx E E, mx / x Ae H x k m m the E is the average value of the Hamiltoia, H

13 A Brief Review Time-depedet Schrődiger Equatio: i t Hˆ Meaig of operators i quatum theory: Hˆ pˆ V x m Meaig of the wave fuctio: If the Hamiltoia, Ĥ, is ot a fuctio of time, eergy coserves, ad the time-depedet Schrődiger Equatio is separable: iet / x, t Ae x, where Hˆ x E x Operator of liear mometum of the movig particle is the (statistical ) average of the respective observable value x, t dx probability of fidig the particle i the iterval (x, x + dx) at time t is the operator of total eergy, the Hamiltoia pˆ i d dx *, ˆ, A t x t A x t dx

14 A Importat Example: Simple Harmoic Motio (the problem of a harmoic oscillator) Solvig the statioary Schrődiger equatio, with Hˆ pˆ V x m d p i, p dx ˆ d H kx The the Schrődiger equatio becomes kx d m dx m x At x, we have: kx 0 So / x e with m dx Therefore, substitute m x / x e x ito m dx d m dx d dx kx d E. This gives: E Ĥ x E x k m m me m 0 with x cost as x Its solutio is x AH x, E, 0,,,3,... m x H x are Hermite polyomials. Thus / x Ae H x

15 Harmoic Oscillators i a Solid Matter Average eergy is As E E 0 Ee e E E Boltzma s probability of fidig a oscillatig atom i th excited state is proportioal to / kt E / kt e e e E e 0 E e with kt 0 0 E / /, we have: e e e e e e e e / / /kt / kt e e e Plak s formula for black-body radiatio: I 5 hc/ kt c e h c h

16 Hilbert Space as a Tool Solvig the statioary Schrődiger equatio, Ĥ x E x We come to a complete set of eigefuctios,, that ca be used as a basis set for ay other fuctio: x x c x Pluggig it ito the origial equatio, it ca be coverted ito the algebraic eigevalue problem Ĥc Ec Here Ĥ is a matrix with matrix elemets For a solid state, its oscillator states are: x x x x H * ˆ m x H x dx