Intro to QM due: February 8, 2019 Problem Set 12

Transcription

1 Intro to QM du: Fbruary 8, 9 Prob St Prob : Us [ x i, p j ] i δ ij to vrify that th anguar ontu oprators L i jk ɛ ijk x j p k satisfy th coutation rations [ L i, L j ] i k ɛ ijk Lk, [ L i, x j ] i k ɛ ijk x k, [ L i, p j ] i k ɛ ijk p k. Hr i, j, k {,, } and sus run ovr ths vaus. Soution: [ ] [ L i, x j ] ɛ i x k p, x j ɛ i [ x k p, x j ] Do [ L i, x j ] first: ɛ i [ x k, x j ] p x k [ p, x j ] i ɛ i x k δ j i k ɛ ikj x k i k ɛ ijk x k. In th scond in I usd th Libniz proprty of th coutator, and in th ast in I usd th antisytry of th ɛ sybo. An aost idntica cacuation givs th [ L i, p j ] coutator. Thn [ [ L i, L j ] L i, ] ɛ j x k p ɛ j [ L i, x k ] p x k [ L i, p ] i ɛ j ɛ ik x p x k ɛ i p i ɛ j ɛ ik x p ɛ j ɛ i x k p i ɛ j ɛ ik x p i ɛ j ɛ i x k p k k i δ ji δ δ j δ i x p i δ j δ ki δ ji δ k x k p k [ ] i δ ji x p x j p i x i p j δ ji x p i x i p j x j p i i δ i δ j δ j δ i x p i ɛ ijk ɛ x p i ɛ ijk Lk. k k In th fifth in I usd th idntity ɛ jɛ i δ j δ ki δ ji δ k in cass, and thn I usd it again in th sixth in. that was introducd Prob : A diatoic ocu ad of two atos of asss and with nrgy spctru as in quation 9. of th txt aks a pury rotationa transition fro an stat to an stat, itting a photon of frquncy ω so of nrgy ω. What is th intratoic distanc of th two atos in this ocu

2 in trs of,,, and ω? Soution: Th nrgy spctru givn in quation 9. is E n, n ω I, whr n, {,,,...} and I is th ont of inrtia of th diatoic ocu. if d is th intratoic distanc of th two atos in th ocu, w hav Thus, I µd, whr µ :. A pury rotationa transition is on in which th n quantu nubr dos not chang. So fro th nrgy spctru, th nrgy of th ittd photon is Soving for d givs ω E γ E n, E n, [ ] I µd. d µω. Prob : Considr a partic in a stat with wav function ψ Nx y z αr whr N is th noraization factor. Show, by rwriting th Y ±, functions in trs of x, y, z, and r, that / / x ± iy z Y,±, Y, 8π r 4π r. Using this, find th probabiitis PL z? of asuring th th possib vaus of L z for a partic in th stat ψ givn abov. Soution: Fro fro th txt w hav that / / Y,± sin θ ±iφ, Y, cos θ. 8π 4π In sphrica coordinats x ± iy r sin θ ±iφ, and z r cos θ, giving. Thrfor ψ Nx y z αr { i x iy N r { / i 8π N Y { N r i Y i Y Y i x iy r i 8π }, z } r αr r / Y 4π / Y whr N r N4π/ / r αr. Thrfor ψ N c,, with c ± : i /, c :. } r αr

3 Hr N is a radia Hibrt spac stat with wavfunction N r r N, and, ar th usua L, L z anguar ontu ignstats. Th possib vaus of L z that can b obtaind ar its ignvaus,. By th axios of QM, Prob L z ψ P Lz ψ whr th projction oprator on th ignspac is so P Lz r dr r, r,, ProbL z ] r dr r;, ψ r dr r N c,, r dr N r r N c δ, δ, N N c, whr in th scond in I insrtd th xprssion for ψ w found abov, in th third in I usd th orthonoraity of th, stats, and in th ast in I usd th r rd r r coptnss ration. Th condition that ψ is noraizd ipis ψ ψ r dr r;, ψ N N c,, whr in th scond stp I usd a coptnss ration, and th third stp is xacty th sa cacuation as was don abov. Cobining th two cacuations thus givs that ProbL z c c. So ProbL z i i i ProbL z i i i ProbL z i i 4 4, 4 6, 4 6. For probs 4-7, considr a rigid rotator irsd in a unifor agntic fid in th z dirction, with th haitonian Ĥ I L ω Lz

4 whr I and ω ar givn positiv constants. Suppos th wav function of th rotator at ti t is givn by θ, φ ψ sin θ sin φ. 4π Prob 4: What vaus of L z wi b obtaind if a asurnt is carrid out at ti t, and with what probabiity wi ths vaus occur? Soution: Writing sin φ i iφ i iφ, and coparing to xprssions for Y,± θ, φ, w s that ψ i,,. Thrfor, siiary as in th ast prob, w can ony hav L z ± with qua probabiitis ProbL z ±, ± ψ i. Prob 5: If a asurnt of L x is carrid out at ti t, what rsuts can b obtaind and with what probabiitis? Soution: Sinc th stat is in th ignspac of L, th L x ignvaus can ony b, or ± i, just th sa as L z. Rca fro arir in th cours that in th L z ignbasis ordrd as {,,,,, }, th atrix nts of L z and L x ar L z, Lx. S, g, quation.8 of th txt, or prob st 4. Fro this atrix xprssion for L x it is asy to find th noraizd ignvctors L x, L x, L x. Thus ProbL x L x ψ ProbL x L x ψ ProbL x L x ψ i i i 8,, 8.

5 Prob 6: What is θ, φ ψt? Soution: Rca that ψt E iet/ E E ψ whr E ar an orthonora basis of nrgy ignstats. Sinc Ĥ, L I ω Lz, I ω,, w s that th, for an nrgy ignbasis with nrgy ignvaus E, W thn gt ψt, ie,t/,, ψ ie,t/, i,,,, i ie,t/, ie, t/, i i t/i iω t, iωt,, I ω. whr in th scond in I put in th xprssion for ψ w found abov, in th third in I usd th orthonoraity of th anguar ontu ignstats, and in th fourth in I usd th xprssion for th nrgy ignvaus found abov. Thus θ, φ ψt i i t/i iω t θ, φ, iωt θ, φ, i i t/i iω t Y, θ, φ iωt Y, θ, φ i 4 π i t/i iωt sin θ iφ iωt sin θ iφ 4π i t/i sin θ sinφ ω t, whr I usd th xprssion for Y,± θ, φ in th third in. Prob 7: What is L x for this stat at ti t? Soution: L x ψt L x ψt i i t/i iω t, iωt, Lx iω t, iωt, iω t iωt iωt iω t iωt iωt. iωt iωt In th first in I put in th xprssion for ψt found in th ast prob. In th scond in I usd th vctor/atrix xprssion for th stats and oprator as in prob 5.