Quantum Harmonic Oscillator

Transcription

1 Quu roc Oscllor Quu roc Oscllor 6 Quu Mccs Prof. Y. F. C

2 Quu roc Oscllor Quu roc Oscllor D S..O.:lr rsorg forc F k, k s forc cos & prbolc pol. V k A prcl oscllg roc pol roc pol s u po of sbly sys 6 Quu Mccs Prof. Y. F. C

3 E: posos of os for crysl r sblzd by prsc of pol s locl loco of c o o poso s sblzd by pol, locl rsuls frs drvv of srs pso locl V s oly pprod by qudrc fuco of.o. 6 Quu Mccs Prof. Y. F. C Quu roc Oscllor Quu roc Oscllor o d V d V o L o o o d V d d dv V V o o L o o d V d V V o

4 for.o. pol, -dp Scrödgr wv q.: us & kg subsuo clld r fucos. 6 Quu Mccs Prof. Y. F. C Scrödgr Wv Eq. for D roc Oscllor Quu roc Oscllor V E d d ε E ε d d ε d d d d

5 Quu roc Oscllor r Fucos O por clss of orogol polyols courd QM & lsr pyscs s r polyols, wc c b dfd by forul, frs fw r polyols r: d d,,,l 3,, 4, 8 3 grl: [ ] k. k k k 6 Quu Mccs Prof. Y. F. C

6 r polyols co fro grg fuco:. Tylor srs:. subsug o : rcurrc rlo: 6 Quu Mccs Prof. Y. F. C r Fucos Quu roc Oscllor < g,, < g g,, u u d u d g, g g g,l,,

7 Quu roc Oscllor r Fucos g subsug g, o g : rcurrc rlo: d d,,,l d w & d d-ordr ordry dffrl quo for d d d d gvlus of D quu.o.: ε E 6 Quu Mccs Prof. Y. F. C

8 Quu roc Oscllor Sory Ss of D roc Oscllor gfucos of D.O.: C w lp of [ ] d, fd orlzo C cos, [ ] d CM, oscllor s forbdd o go byod pol, byod urg pos wr s kc rgy urs gv. quu wv fucos d byod pol, d us r s f probbly for oscllor o b foud clssclly forbdd rgo 6 Quu Mccs Prof. Y. F. C

9 Quu roc Oscllor Sory Ss of D roc Oscllor Quu Mccs Prof. Y. F. C

10 Quu roc Oscllor Sory Ss of D roc Oscllor clsscl probbly of fdg prcl sd rgo Δ : P cl Δ Δ T Δ. v vlocy v As c b prssd s fuco of : v A P cl Δ Δ A 6 Quu Mccs Prof. Y. F. C

11 Quu roc Oscllor Sory Ss of D roc Oscllor dffrc bw wo probbls for s rly srkg r s o zro-po rgy CM quu d clsscl probbly dsrbuos cocd w quu ubr bcos lrg s s vdc of Bor s corrspodc prcpl 3 6 Quu Mccs Prof. Y. F. C

12 Quu roc Oscllor Sory Ss of D roc Oscllor clssclly, oo of.o. s suc r poso of prcl cgs fro o o o or. owvr, loug r s probbly dsrbuo for y gs QM, s dsrbuo s dp of sory ss 3 v so, Erfs or rvls cor suprposo of ubr of gss,.., so-clld wv pck s, wll ld o clsscl bvor 6 Quu Mccs Prof. Y. F. C

13 sow : usg gro fuco, w c v orogoly propry, gro lds o s cosquc, w c ob 6 Quu Mccs Prof. Y. F. C Sory Ss of D roc Oscllor Quu roc Oscllor [ ] d s s s [ ] s s s d s d [ ] d s s [ ] d

14 Quu roc Oscllor T Posso Dsrbuo gv r of occurrc r of vs rlv rvl, Posso dsrbuo gvs probbly P X cly vs wll occur for sll rvl Δ probbly of rcvg cll s r Δ. probbly of rcvg o cll durg s y rvl Δ s gv by r Δ. probbly of rcvg cly clls ol rvl Δ s gv by P r Δ P r Δ Δ P 6 Quu Mccs Prof. Y. F. C

15 Quu roc Oscllor T Posso Dsrbuo rrrgg P Δ P r Δ P r Δ, dvdg roug by, Δ d lg Δ, dffrl rcurrc q. c b foud d wr s dp d r P r P for : dp d r P wc c b grd o ld o P P r w fc probbly P of rcvg o clls zro rvl us b qul o uy: P r 6 Quu Mccs Prof. Y. F. C

16 Quu roc Oscllor T Posso Dsrbuo r dp subsug P o r P r P for : d r P r, rpg s procss, P c b foud o b P r r su of probbls s uy: P r r r r r r of Posso dsrbuo: < r r r r r > P r 6 Quu Mccs Prof. Y. F. C

17 Quu roc Oscllor T Posso Dsrbuo or words, Posso dsrbuo w of s gv by: λ P λ λ 6 Quu Mccs Prof. Y. F. C

18 Quu roc Oscllor Scrödgr Cor Ss of D.O. T Scrödgr cor wv pck s c b grlzd s Ψ, c E w c φ α α c b foud or squr of coffc c s cly s s Posso dsrbuo w of α 6 Quu Mccs Prof. Y. F. C

19 Quu roc Oscllor Scrödgr Cor Ss of D.O. subsug E &, c% : Ψ E φ α Ψ, α φ α α 4 o usg g, : { } α φ φ Ψ, p α α 4 α φ φ p 4 { α α } 6 Quu Mccs Prof. Y. F. C

20 Quu roc Oscllor Scrödgr Cor Ss of D.O. s rsul, probbly dsrbuo of cor s s gv by: { } P Ψ Ψ α φ α φ p{ α cos φ α cos φ } p { [ α cos φ] } α,,, p cos[ ] cos c b clrly s cr of wv pck ovs p of clsscl oo α cos φ 6 Quu Mccs Prof. Y. F. C

21 w, & opror cg o gs 6 Quu Mccs Prof. Y. F. C Cro & Alo Oprors Quu roc Oscllor % [ ]

22 slr wy, opror cg o gs 6 Quu Mccs Prof. Y. F. C Cro & Alo Oprors Quu roc Oscllor p [ ] [ ] p

23 & cosquly, s cov o df w oprors: & 6 Quu Mccs Prof. Y. F. C Cro & Alo Oprors Quu roc Oscllor p p p p

24 Quu roc Oscllor Cro & Alo Oprors opror â s crsg cro opror: s s oprg w â o - sory ss ylds s, wc s proporol o gr - s opror â s lowrg lo opror: s s oprg w â o - sory ss ylds s, wc s proporol o gr -- s 6 Quu Mccs Prof. Y. F. C

25 rs of &, oprors & c b prssd s: & w c fd couor of s lddr oprors: wc s so-clld cocl couo rlo 6 Quu Mccs Prof. Y. F. C Cro & Alo Oprors Quu roc Oscllor â â p p [ ] [ ],,, ] [, p p p p

26 Quu roc Oscllor Cro & Alo Oprors â s r cojug â : proof:? p? p? p? p 6 Quu Mccs Prof. Y. F. C

27 w, & opror cg o gs 6 Quu Mccs Prof. Y. F. C Cro & Alo Oprors % [ ] Quu roc Oscllor

28 slr wy, opror cg o gs 6 Quu Mccs Prof. Y. F. C Cro & Alo Oprors p [ ] [ ] p Quu roc Oscllor

29 & cosquly, s cov o df w oprors: & 6 Quu Mccs Prof. Y. F. C Cro & Alo Oprors p p p p Quu roc Oscllor

30 Quu roc Oscllor Cro & Alo Oprors opror â s crsg cro opror: s s oprg w â o - sory ss ylds s, wc s proporol o gr - s opror â s lowrg lo opror: s s oprg w â o - sory ss ylds s, wc s proporol o gr -- s 6 Quu Mccs Prof. Y. F. C

31 rs of &, oprors & c b prssd s: & w c fd couor of s lddr oprors: wc s so-clld cocl couo rlo 6 Quu Mccs Prof. Y. F. C Cro & Alo Oprors â â p p [ ] [ ],,, ] [, p p p p Quu roc Oscllor

32 Quu roc Oscllor Cro & Alo Oprors â s r cojug â : proof:? p? p? p? p 6 Quu Mccs Prof. Y. F. C

33 w & usg couo rlo df so-clld ubr opror:.o. lo ks for: 6 Quu Mccs Prof. Y. F. C Cro & Alo Oprors 4 4 p 4 4 p ], [ N N Quu roc Oscllor

34 Quu roc Oscllor Cro & Alo Oprors gss of â c b foud o b cor ss Ψ,; α : Ψ,; α α α α α cor ss v u ucry Ψ,; α? Ψ,; α Ψ,; α Ψ,; α α α cosφ Ψ,; α? Ψ,; α Ψ,; α Ψ,; α α α αα αα Δ Ψ α? Ψ α Ψ α Ψ α,;,;,;,; 6 Quu Mccs Prof. Y. F. C

35 Quu roc Oscllor Cro & Alo Oprors Ψ α p? Ψ α Ψ α Ψ α,;,;,;,; α α sφ Ψ p? Ψ Ψ Ψ Δ,; α,; α,; α,; α p α α αα αα,;,;,; Ψ α p Ψ α Ψ α p Ψ,; α s cosquc, w ob u ucry s: Δ Δ p 6 Quu Mccs Prof. Y. F. C