Quantum Mechanics Qualifying Exam - August 2016 Notes and Instructions

Transcription

1 Quntum Mechnics Qulifying Exm - August 016 Notes nd Instructions There re 6 problems. Attempt them ll s prtil credit will be given. Write on only one side of the pper for your solutions. Write your lis on the top of every pge of your solutions. Number ech pge of your solution with the problem number nd pge number (e.g. Problem 3, p. /4 is the second of four pges for the solution to problem 3.) You must show your work to receive full credit. Possibly useful formuls: Spin Opertor S = h σ, σ x = ( ) ( 0 i, σ y = i 0 ) ( 1 0, σ z = 0 1 ) (1) In sphericl coordintes, ψ = 1 r r rψ + 1 r sin θ Hrmonic oscilltor wve functions u 0 (x) = ( mω π h )1/4 e mωx h u 1 (x) = ( mω π h )1/4 mω h mωx xe h ψ (sin θ θ θ ) + 1 r sin θ ψ. () φ

2 Problem 1: Time dependent solutions to Schrodinger s Eqution (10 pts) Consider prticle of mss m in n infinite squre well. { 0, - V (x) = x, x < or x > + The solutions to the time independent Schrodinger Eqution re: H Ψ n = E n Ψ n for n=1,,3,... where E n = n π h nd m x Ψ n = Ψ n (x) = cos(nπx ) n = 1, 3, 5,... Assume t t o, the prticle is in the stte: sin(nπx ) n =, 4, 6,... Ψ(t o = 0) = 3/10 Ψ 1 i 7/10 Ψ 3 Answer the following questions: ) Using Dirc nottion, write down the expression for the time evolution opertor, U(t, t o = 0) in terms of energy eigenvlues nd eigensttes. (1 pt) b) Find Ψ(t) = U(t, t o = 0) Ψ(t o = 0) (1 pt) c) Does your Ψ(t) in prt b) stisfy the time independent Schrodinger Eqution? Demonstrte explicitly. (1 pt) d) Does your Ψ(t) in prt b) stisfy the time dependent Schrodinger Eqution? Demonstrte explicitly. (1 pt) e) Is the uncertinty in the energy E > 0, < 0 or = 0 for Ψ(t)? Discuss. (1 pt) f) Stte whether the following properties re time dependent or time independent for system in the stte Ψ(t). (4 pts) i) E ii) x iii) p iv) P, where P is the prity opertor g) How do your nswers to prt f) chnge fter the energy is mesured t time t nd the result is E = 9π h? (1 pt) m

3 Problem : Hydrogen Atom (10 pts) In this problem you will clculte the reltivistic correction to the energies of the hydrogen tom. The hydrogen tom Hmiltonin is in terms of its electron in the field of the positively chrged nucleus H 0 = p e m e 4πɛ 0 r where p is the electrons momentum, r its position, m e its mss, nd e the chrge. This Hmiltonin is nonreltivistic (p/(mc) 1). The correct reltivistic expression to use for the kinetic energy is T = p c + m ec 4 m e c recll tht r nl = n 0 {1 + 1 l(l + 1) [1 n ]} r nl = n 4 0{1 + 3 l(l + 1) 1/3 [1 n ]} 1 r nl = 1 0 n 1 r nl = 1 0 n3 1 l + 1/ 1 r 3 nl = n3 1 l(l + 1/)(l + 1). Use this informtion to find the first non-zero order correction to the Hmiltonin due to the reltivistic motion of the electron. ( Points) b. Show tht this correction is digonl in the nlm bsis by proving tht it commutes with the ngulr momentum opertor L. Why is it sufficient to prove tht the perturbtion commutes with L to show tht the perturbtion is digonl in the nlm bsis? (4 Points) c. Using the fct tht p = H 0 + e m e 4πɛ 0 r find the reltivistic energy correction to the energy levels of the Hydrogen tom. (4 Points)

4 Problem 3: Angulr momentum (10 pts) One prticle hs spin j 1 nd nother prticle hs spin j. () [1 point] Wht re the good quntum numbers for the two-prticle system with J = J 1 + J in the direct product bsis? Write down the bsis vectors lbelled ccording to their eigenvlues. (b) [1 points] Write down the bsis vectors in the totl j bsis. quntum numbers in this cse? Wht re the good (c) [ points] Write down the completeness reltion for the direct product bsis sttes. (d) [ points] Use the completeness reltion to relte the totl -j bsis to the direct product bsis. Identify the Clebsch-Gordon coefficient. (e) [ points] Write down the reltion between totl-j nd direct product bses for j 1 = 1/ nd j = 1/. Recll. J ± j, m >= h (j m)(j ± m + 1) j, m ± 1 > (f) [ points] Suppose you hve n interction of the form H I = A J 1 J where J = J 1 + J. Which bsis vectors re best to use nd why?

5 Problem 4: 3D Attrctive Potentil (10 pts) Consider prticle tht moves subjected to three dimensionl ttrctive potentil V (x, y, z) = h m [λ 1δ(x) + λ δ(y) + λ 3 δ(z)], where λ 1, λ, λ 3 > 0. ) Find the energy nd the wvefunction of the prticle in this potentil. (4 points) b) Interpret the mening of this stte. Clculte the probbility of finding the prticle inside rectngulr volume centered t the origin, with size l i = 1/λ i, with i = 1,, 3 for the x, y, z directions respectively. ( points) c) Compute the sptil nd momentum uncertinties ( x) nd ( p) for the stte of item ) nd explicitly check Heisenberg s inequlity. (4 points) Hint: d x dx = x x sign(x) d sign(x) = δ(x) dx

6 Problem 5: Expnding Hrmonic Oscilltor (10 pts) Consider prticle of mss m confined in 1D hrmonic oscilltor potentil with frequency ω 0 H = P m + m ω 0 X (1) The rising nd lowering opertors re useful for hrmonic oscilltor problems: where λ = h mω 0 = 1 ( X λ iλ h P ) = 1 ( X λ + iλ h P ) is the length scle for the hrmonic oscilltor: () [ pts] Use the rising nd lowering opertors to derive the ground stte wvefuction, ψ 0 (x), nd the first excited stte wvefuction, ψ 1 (x), for the Hmiltonin H. Be sure to show your work. (b) [1 pt] Consider sudden chnge in the potentil, modeled by chnge in the originl frequency of the oscilltor by some multiplictive vlue f, to the new Hmiltonin: H b = P m + m ω 1 X, ω 1 = fω 0, 0 < f < 1 (3) Sudden in this cse mens tht one cn ignore the time it tkes to chnge the potentil. If φ 0 (x) nd φ 1 (x) re the ground nd first excited stte wvefunctions of H b, wht re the functionl forms for these wvefunctions? Explin your nswer. (c) [3 pts] The oscilltor is in the ground stte ψ 0 (x) when the potentil suddenly chnges. Wht is the expecttion vlue of the energy of the oscilltor fter the potentil chnges? Show your work. (d) [ pts] If the oscilltor is in the stte ψ 0 (x) when the potentil suddenly chnges, wht is the probbility of the oscilltor being in the ground stte of H b fter the potentil chnges? Show your work. (e) [1 pt] If the oscilltor is in the stte ψ 0 (x) when the potentil suddenly chnges, wht is the probbility of the oscilltor being in the first excited stte of H b fter the potentil chnges? Explin your nswer. (f) [1 pt] Finlly, ssume the oscilltor is in the first excited stte of H, ψ 1 (x), when the potentil suddenly chnges. Wht is the expecttion vlue of the energy of the oscilltor fter the potentil chnges? Is the chnge in the expecttion vlue of the energy, from H to H b, for ψ 1 lrger thn, smller thn, or the sme s ψ 0? Explin. Remember tht the Gussin integrls hve the form: () e x dx = x n e xn dx = π (n 1) n π (4)

7 Problem 6: Delt function in 1-D well(10 pts) A prticle of mss m is plced in n ttrctive 1-D delt function potentil V (x) = h λδ(x)/m with positive λ. The prticle nd the potentil re locted in n infinite box with wlls t x=± / (i.e V (/) = V ( /) = ) ) Determine the condition on the prmeters for which the system will hve exctly one bound stte with negtive energy eigenvlue E nd give its wve function (4 pts). b) For the sme system, determine the energy eigenvlues nd eigenvectors for sttes with positive E. (3 pts) c) If the coefficient λ < 0, explin in detil how your results chnge for prts ) nd b) (3 pts)