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42 THE CITY UNIVERSITY OF NEW YORK First Examination for Ph.D. Candidates in Physics Quantum Mechanics January 21, 2010 Instructions: a) PUT YOUR IDENTIFICATION NUMBER ON EACH PAGE. b) Solve two of the following three problems. Each problem is worth 25 points. c) Start each problem on a new page. d) Indicate clearly on the first sheet which two problems you choose to solve. If you do not indicate which problems you wish to be graded, the first two problems will be graded. 1. (a) [3] For a harmonic oscillator potential suppose the wave function at t = 0 is given by H = p2 2m kx2 ψ(x, 0) = A a n u n (x) where A is a normalizing constant, a n are given constants, and u n (x) are the harmonic oscillator eigenfunctions. Find A and write ψ(x, t). n=0 (b) [6] Show that the expected value of position is given by x t = 1 [ n + 1 α A2 a n a 2 n+1e iωt + n=0 ] n 2 a n 1e iωt (1) and obtain a similar expression for p t. Also find H t where H is the Hamiltonian. (c) [4] Heisenberg s equation of motion for the expectation value of an operator is d dt O = 1 i [O, H]. Show that Eq. (1) and your solution for p t satisfies Heisenberg s equation for x t and p t. (d) [8] Consider the following case : a N = a N+1 =... = a N+K 1 = 1 for some N and K and all other a n are zero. Also assume N >> K >> 1. Show that x t = a cos ωt and find the constant a. Also find p t. Comment on your answers. (e) [4] Consider the wave function given by ψ(x, 0) = x 2 e α2 x 2 /2 where as usual α = mω. Put ψ(x, 0) in the form given by Eq. (1) Find the coefficients a n and the normalizing constant.a. 1

43 2. (a) [8] For a beam of particles traveling in the z direction and that are scattered by a short range symmetrical potential centered at the origin the main idea is to write the total wave function as ψ(r) = e ikz + f(θ) r eikr where z = r cos θ and where f(θ) is the scattering amplitude. Write explicitly in the appropriate coordinate system the expression for the total current corresponding to the total wave function, ψ(r). Do not try to simplify it. From the expression you wrote pick out the current for the the incoming wave, e ikz, and the current in the radial direction for the outgoing wave f(θ) r eikr. Also, obtain the relationship between the outgoing and incoming current. (b) [9] The Born approximation is that f(θ) = m 2π 2 V (r)e i(k k ) r d 3 r where k and k are the momentum of the incoming and outgoing particles for a spherically symmetric potential. Show that f(θ) = 2m 2 rv (r) sin(kr)dr K where K = k k and where K = K = 2k sin(θ/2) (c) [8] Suppose the potential is where η is a constant. Calculate f(θ). 3. Parts a, b, c are independent of each other. 0 V (r) = ηe αr (a) [8] A harmonic oscillator is in the ground state. This spring constant is suddenly increased by a factor of c 4. That is k new = c 4 k old. Find the probability that it will be in the ground state of the new Hamiltonian and also the probability that it will be in an excited state. Also, find what happens in the limits as c 0 and c. (b) [10] For the potential use the trial function V (x) = { kx 2 /2 x > 0 x 0 ψ(x) x e αx/2 x > 0 (where α is the variational parameter) to estimate the ground state energy. (c) [7] Suppose the Hydrogen atom potential is replaced by the following potential { e2 V (r) = r r < r 0 e2 r e η(r r 0) r > r 0 where η is a small positive number. We want to use stationary state perturbation theory to find the correction to the ground state energy. To do so note that the perturbation can be taken to be H (r) = e2 r ( 1 e η(r r0)) r > r 0 Use stationary state perturbation theory to get the first order correction to the ground state. 2

44 = ˆr r + ˆθ 1 r Equations θ + ˆφ 1 r sin θ φ quantum current = 2mi (Ψ Ψ Ψ Ψ ) = m Im(Ψ Ψ) = Re(Ψ im Ψ) π e ax2 +bx = 2 /(4a) xe ax2 +bx = b 2a x 2 e ax2 +bx = 0 R R a eb π 2a + b2 4a 2 x n e sx dx = n! s n+1 xe sx dx= sr + 1 s 2 e sx dx= 1 s e sr 2 a eb /(4a) π 2 a eb /(4a) e sr ********************************************************************** Harmonic oscillator: u n (x) = N n e α2 x 2 /2 H n (αx) ( 1/2 α u n (x) = e π2n n!) α2 x 2 /2 H n (αx) E n = (n ) ω α = mω = ( ) 1/4 mk k 2 ω = m ( α N n = π2n n! ) 1/2 H 0 (x) = 1 H 1 (x) = 2x H 2 (x) = 4x 2 2 ( ) α 2 1/4 u 0 (x) = e α2 x 2 /2 u 1 (x) = ( ) α 2 1/4 2α e α2 x 2 /2 π π [ u n (x)xu m (x) = 1 ] n + 1 n δ m,n+1 + α 2 2 δ m,n 1 [ ] n + 1 n u n (x)pu m (x) = iα δ m,n δ m,n 1 ************************************************************* Hydrogen atom: a 0 = 2 me 2 E n = Z2 e 2 2a 0 n 2 = me 4 Z2 2 2 n 2 Ground state wave function: 1 π ( 1 a) 3/2 e r/a Ground state energy: e2 2a = me

45 Instructions: THE CITY UNIVERSITY OF NEW YORK First Examination for Ph.D. Candidates in Physics Quantum Mechanics June 17, 2010 a) PUT YOUR IDENTIFICATION NUMBER ON EACH PAGE. b) There are two parts to this exam, parts A and B. Do two questions but you must chose one problem from each part. Each problem is worth 25 points. c) Start each problem on a new page. d) Indicate clearly which two problems you choose to solve. PART A 1. Parts a, b, and c are independent of each other but you may use the results of part a to do parts b and c (a) [5] If the wave function at t = 0 is given by ψ(x, 0) show that ψ(x, t) = e iht/ ψ(x, 0) satisfies use Schrodinger time dependent equation where H is a time independent Hamiltonian. Also by expanding ψ(x, 0) in terms of the the eigenfunctions of the Hamiltonian ψ(x, 0) = n=0 c nu n (x) show that ψ(x, t) = c n e ient/ u n (x) where u n and E n are the eigenfunctions and eigenvalues of the Hamiltonian. n=0 (b) [13] Suppose we are dealing with a harmonic oscillator where and we have the following wave function at time zero ( mω ) 1/4 ψ(x, 0) = e mω 2 (x β)2 π Find ψ(x, t). Hint: calculate c n and substitute in Eq. () and then do the infinite summation using the generating function which is given in the equation sheet. In calculating c n a difficult integral that arises but the answer is given in the formula sheet. (c) [7] As in part a) ψ(x, t) = e iht/ ψ(x, 0) Assume that t is small and expand e iht/ to second order in time. Then, calculate P (t) = ψ (x, 0)ψ(x, t)dx 2 = P (t) which is called the survival probability. Find P (t) to second order in time. That is, find A, B, C where P (t) A + Bt + Ct 2. [The coefficients are constants and what enters in A,B,C are either numerical values or expectation values of the Hamiltonian at time 1

46 2. Parts d and e are independent of a,b,c (a) [5] Consider the delta function potential V (x) = λδ(x) Find the bound states and corresponding energies. Hint try: u(x) = Ae α x /2. Also you will need the fact that the discontinuity of the eigenfunctions satisfy du dx du 0+ dx = λ 2m 0 u(0) 2 You should prove this if you use it. (b) [5] Calculate x, p, x 2, p 2 and the uncertainty product x p for the solution you obtain. You may use the fact that (c) [5] Now consider the two dimensional potential d 2 δ(x) = 1 2 dx x 2 V (x, y) = λ 1 δ(x) λ 2 δ(y) Use separation of variables to find the energy eigenvalues and eigenfunctions. (d) [5] If we have a Hamiltonian that depends on a parameter, say θ; then when one solves the eigenvalue problem H(x, p; θ)u(x; θ) = E(θ)u(x; θ) the energy eigenvalues ad eigenfunctions will contain the parameter. Show that ( ) E H θ = u (x; θ) u(x; θ)dx θ where u(x; θ) is first normalized to one. This is called the Hellmann -Feynman theorem. (e) [5] Consider the harmonic oscillator Hamiltonian d 2 H = 2 2m dx kx2 and take the parameter to be the mass m. Calculate u (x; θ) ( H m) u(x; θ)dx for the ground state of the harmonic oscillator. Since we know the exact answer for the ground state check your answer by calculating E 0 m where E 0 is the ground state energy. 2

47 PART B 3. A relativisitic electron moves in one dimension along the z-axis and interacts with the potential V (x) = mω2 z 2 (I + β) 4 where I is a 4x4 unit matrix and β is a standard Dirac matrix. (Note that the electron is 3- dimensional, but its wavefunction only depends on the z-direction). (a) [3] Write down the Dirac equation governing the motion. (b) [3] What is the physical meaning of the quantity ψ + ψ (c) [6] Write the four-component wave function in the form ( ) u ψ = v where u and v are two-component wave functions. Solve for v in terms of u. (d) [7] Find ψ for the ground state assuming that the electron is in a spin-up state along the z-axis. (e) [3] Find an algebraic formula to determine the energy eigenvalue E. You need not solve for E. (f) [3] Solve for the energy in the non-relativistic limit and comment on its value. 4. A particle of mass m travels with wave vector of magnitude k along the z-direction and scatters off the potential given by the spherical well V ( r) = { V 0 if r < a 0 if r > a (a) [8] Find the scattering amplitude f(q) in the first Born approximation, where q is the magnitude of the wave-vector transfer. (b) [4] Find the corresponding differential scattering cross section dσ dω (c) [7] Next consider a different scattering potential given by a cubic well { V 0 if x < a and y < a and z < a V (x, y, z) = 0 otherwise Find the scattering amplitude f(q, θ, ϕ), where the wave-vector transfer q is expressed in its spherical coordinate form. (d) [6] Find the corresponding differential scattering cross section dσ. Compare the scattering cross dω sections for parts b and d in the limit where qa << 1. 3

48 ( 2 Formulas d 2 2m dx + 1 ) 2 2 mω2 x 2 u n (x) = E n u n (x) (1) ( ) 1/2 u 0 (x) = α π e α 2 x 2 /2 u n (x) = N n e α2 x 2 /2 H n (αx) (2) ( ) 1/2 α u 0 (x) = e α2 x 2 /2 (3) π ( 1/2 α u n (x) = e π2n n!) α2 x 2 /2 H n (αx) (4) E n = (n ) ω α = mω = ( ) 1/4 mk ω = 2 k m N n = ( α π2n n!) 1/2 (5) ( mω ) 1/4 e mω 2 (x a)2 u n (x)dx = π e 2xt t2 = 0 n=0 ( mω ) n/2 a n e mω 4 a2 (6) 2 n! H n (x) t n (7) n! x n e sx dx = n! s n+1 (8) *********************************** ( α = ( σ x = ) ( ) 0 σ I 0 ; β = ; σ 0 0 I ) ( ) ( i ; σ y = ; σ z = 1 0 i ). f( q) = m V ( r)e i q r d 3 r (9) 2π 2 f(θ) = 1 (2l + 1)(e 2iδ l 1)P l (cos θ) (10) 2ik l=0 dσ dω = f 2 σ = 4π k Im(f(0)) (11) 4

49 THE CITY UNIVERSITY OF NEW YORK First Examination for Ph.D. Candidates in Physics Quantum Mechanics January 20, 2011 Instructions: a) PUT YOUR IDENTIFICATION NUMBER ON EACH PAGE. b) There are two parts to this exam, parts A and B. Do two questions but you must chose one problem from each part. Each problem is worth 25 points. c) Start each problem on a new page. d) Indicate clearly which two problems you choose to solve. PART A 1. Consider the following 3-dimensional wave function in the rectangular coordinate system ψ(x, y, z) = A (ix + 3z) e αr2 /2 where α is a constant greater than zero, and where as usual r 2 = x 2 + y 2 + z 2 (a) [4] Normalize this wave function by finding A and calculate x (b) [4] Calculate L z ψ where L z is the angular momentum operator in the z direction. Use the rectangular coordinate system. (c) [7] Write x and z in terms of the spherical harmonics and r. After you do that express the wave functions in terms of r and spherical harmonics. (d) [5] What are the possible values one can measure for the angular momentum in the z direction. and for the square of the of angular momentum. Give the probabilities for obtaining those values. (e) [5] Calculate L z ψ using the wave function obtained in part c 2. Suppose that at time t = 0 the momentum wave function is given by ϕ(p, 0) and suppose that for some potential the solution to Schrödinger s equation in momentum space is given by where ϕ(p, t) = e i S(p,t) ϕ(p ct, 0) S(p, t) = p2 2m t pc 2m t2 + c2 6m t3 (a) [9]Staying in the momentum representation calculate p t and x t expressing your answer in terms of p 0 and x 0. (b) [8] Suppose the initial wave function in position space is Calculate ϕ(p, 0) and then calculate ψ(x, t). ψ(x, 0) = δ(x x 0 ) (c) [8] The Schrödinger s time dependent equation in the momentum representation is i p2 ϕ(p, t) = ϕ(p, t) + V (x)ϕ(p, t) t 2m where x is the position operator in the momentum representation. Substitute ϕ(p, t) into it and find V (x). Hint: calculate i p2 tϕ(p, t) 2mϕ(p, t) and see what V (x) must be. In your work you may use the following identity which can be easily verified but you don t have to do so. ( pc m t c 2m t2) ϕ(p, t) i e i S(p,t) p ϕ(p ct, 0) = i ϕ(p, t) p 1

50 Problem 3. PART B A two-dimensional harmonic oscillator is governed by the hamiltonian H = h ω a + a+ b + b+ ( ) 0 1 where a + and a are raising and lowering operators for states excited along the x- + direction and b and b are raising and lowering operators for states excited along the y- direction. The oscillator is initially in the ground state. It is subjected to a timedependent perturbation + + H1 () t = A( a+ a + b+ b )cos( ωt) exp( αt) where A, ω and α are positive real constants. The full Hamiltonian is Ht () = H + H() t. 0 1 a) (8 points) Find, to first order in perturbation theory, the wave function for the t >> 1/ α. system at long times ( ) b) (7 points) Find the ground state energy of the full system at time t = 0. Calculate the first and second-order perturbation theory corrections. c) (5 points) Find the ground state wave function of the full system at time t = 0. Calculate only the first-order perturbation correction. d) (5 points) Find the exact ground state energy of the full system at time t = 0. Problem 4. A particle of mass m moves along the z-direction with energy E. It encounters a potential field produced by two point scatterers r V( ) V r r ) = 0 δ ( ) + δ ( ak) a) (8 points) Use the first Born approximation to obtain the scattering amplitude for scattering through an angle θ relative to the initial direction of propagation. b) (5 points) Find the differential scattering cross section. c) (3 points) Find the total scattering cross section. d) (4 points) Under what condition will there exist an angle for which there will be no scattering? e) (5 points) Find the S-wave phase shift in the limit of low-energy scattering.

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54 THE CITY UNIVERSITY OF NEW YORK First Examination for Ph.D. Candidates in Physics Quantum Mechanics January 24, 2012 Instructions: a) PUT YOUR IDENTIFICATION NUMBER ON EACH PAGE. b) There are two parts to this exam, parts A and B. Do two questions but you must chose one problem from each part. Each problem is worth 25 points. c) Start each problem on a new page. d) Indicate clearly which two problems you choose to solve. PART A Question 1. Consider the Schrodinger s equation for a central potential Write the solution as 2 2m 2 ψ + V (r)ψ = Eψ ψ = R(r)Y m l (θ, φ) where Yl m (θ, φ) are the standard spherical harmonics. 1a) [4] Find the Equation that R(r) satisfies. 1b) [4]Let Find the Equation that u(r) satisfies. u(r) = rr(r) 1c) [17] Now consider s states, that is, the case where l = 0. The equation that you derived in part b should reduce to 2 d 2 2m dr u + V u = Eu 2 (We are giving you this equation so that you can do this part even if you got parts a) and b) wrong.) Consider the following potential: 0 < r < a V (r) = 0 a < r < b b < r V 0 Obtain the single equation that can be used to calculate the energy eigenvalues. The equation should contain only m,, a, b, V 0, E. Take 0 < E < V 0. Hint: in the region a < r < b write the solutions as functions of r a, and in the region b < r write the solution as functions of r b. 1

55 Question 2. Consider the following Hamiltonian in one dimension H = p2 αx + βp 2m where α and β are constants. The aim is to obtain the time dependent momentum wave function φ(p, t), in terms of φ(p, 0). Write the time dependent Schrodinger equation in momentum space and assume we can write a solution in the form i φ(p, t) = Hφ(p, t) (1) t φ(p, t) = F (p αt)e is(p) (2) where F and S are functions to be determined. Remember that the position operator in the momentum representation is x = i p 1a) [7] Substitute Eq. (2) into Eq. (1). You will find that you can determine S(p). Do so. 1b) [5]. Having determined S(p) let t = 0 in Eq. (2) and determine F (p) in terms of φ(p, 0). Write φ(p, t) explicitly in terms of φ(p, 0). 1c) [13] Using your results find the expressions for < x > t and < p > t in terms so of < x > 0 and < p > 0. Do not guess you must show your work in detail. Relevent Equations 2 = 1 r 2 r r2 r + 1 r 2 sin θ θ sin θ θ + 1 L 2 = 2 [ 1 sin θ θ sin θ θ + 1 sin 2 θ r 2 sin 2 θ ] 2 φ 2 2 φ 2. 2

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58 THE CITY UNIVERSITY OF NEW YORK First Examination for Ph.D. Candidates in Physics Quantum Mechanics June 19, 2012 Instructions: a) PUT YOUR IDENTIFICATION NUMBER ON EACH PAGE. b) There are two parts to this exam, parts A and B. Do two questions but you must chose one problem from each part. Each problem is worth 25 points. c) Start each problem on a new page. d) Indicate clearly which two problems you choose to solve. PART A 1. Consider the following time dependent Hamiltonian where F is a constant and t is time. H = p2 2m F tx (a) [10] Using Heisenberg s equation of motion write and solve the equations for the operators p and x: That is, nd p(t) and x(t) in terms of p(0) and x(0). (b) [10]Suppose the state function at time zero is 1=4 (x; 0) = e x 2 =2+i(x ) 2 =2 where ; ; ; k; are real constants. Normalize the wave function and nd hxi t ; hpi t ; hp 2 i t as functions of time. (c) [5] Find hhi t : After you obtain your answer check you it by taking F = 0. Of course, you should get that hhi t = hp2 0i 2m If you don t it means you made a mistake somewhere. 1

59 2. Parts a) and b) are independent of each other. (a) [15]Consider the following three dimensional wave function. where is a constant and u n00 = 1 4 p 2 u n10 = 1 4 p 2 (r; ; ') = A(u n00 + u n10 ) 1 3=2 2 a 0 3=2 1 a 0 r a 0 e r=2a 0 r a 0 e r=2a 0 cos Normalize the wave function and nd the dipole moment. The dipole moment is de ned as hzi : In your calculations you should take advantage that u n00 and u n10 and are orthogonal. Also take advantage of symmetry. In particular explain why Z z ju n00 j 2 dr = 0 Z z ju n10 j 2 dr = 0 (b) [10] Consider the following wave function (; ') = A(3Y 1 2 (; ') + 4iY 0 2 (; ') + (1 + i)y 1 2 (; ')) Normalize the wave function. What are the possible values one can measure of L 2 and L z and what are the respective probabilities. Also calculate hl 2 i and hl z i : 2

60 PART B Problem 1 A particle of mass m is moving along the z axis with wave vector k. It encounters a spherically symmetric scattering potential given by Θ where 1 ( a r) 0 if r a if r a and ( r a) is the Dirac delta function. Problem 2 a) [15] Find the S wave phase shift. b) [5] Assuming that only the S wave contributes significantly to the scattering find an expression for the scattering amplitude. c) [5] Find the total cross section for the situation described in part b. Useful formula: 1 2i l f( ) (2l 1) e 1 Pl (cos ) 2ik l 0 An electron moves in three dimensions in a harmonic oscillator potential. The hamiltonian is given by a) [9] Derive a formula for the energy eigenvalues. b) [8] Suppose that the following perturbation is introduced =mgz Where z is the vertical component of the displacement vector. Find the corrected energy eigenvalue for the ground state through second order in perturbation theory. c) [8] Find the exact ground state energy for the hamiltonian. Useful formulas:,, 1, 1 1

61 ~ 2 2m r2 + V (r) = E (1) = R(r)Y m ` (; ') (2) ~ 2m r ~2 `(` + 1) R = ER (3) 2mr2 u = rr (4) ~ 2 d V 2 2m dr u + + ~2 `(` + 1) u = Eu 2 2mr2 ********************************************** da dt = [A; H] + (5) ********************************************************************* Z 1 1 Z 1 0 x 2 e ax2 +bx dx = x n e sx dx = n! s n+1 2a + b2 4a 2 r a eb2 =(4a) (6) 3

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65 THE CITY UNIVERSITY OF NEW YORK First Examination for Ph.D. Candidates in Physics Quantum Mechanics June 18, 2013 Instructions: a) PUT YOUR IDENTIFICATION NUMBER ON EACH PAGE. b) There are two parts to this exam, parts A and B. Do two questions but you must chose one from each part. Each question is worth 25 points. c) Start each problem on a new page. d) Indicate clearly which two problems you choose to solve. 1. Parts a,b,c are independent of each other. PART A (a) [10] Consider the following time dependent momentum wave function φ(p, t) = Ne α(p βt)2 /2 ia(p,t)/ A(p, t) = p2 t 2m pβt2 2m + β2 t 3 6m where α, β, are real numbers. Normalize φ(p, t) and calculate x t, p t, x 2 t, p 2 t. Also, calculate the uncertainty product ( x)( p) and show that it is greater then /2 for all time. (b) [10] Consider the following Hamiltonian H = p2 2m + αp where α is a real constant. Obtain the Greens function that is the G(x, x, t) for which ψ(x, t) = G(x, x, t)ψ(x, 0)dx Hint: follow the same derivation that is standardly done for the free particle. (c) [5] Suppose the function ψ(x, t) satisfies Schrödinger time dependent equation Now consider the wave function i ψ(x, t) = Hψ(x, t) (1) t u(x, t) = R(x, t)ψ(x, t) where R is an operator that depends on space and time. Derive the operator equation that R must satisfy so that u(x, t) is also a solution of Eq. (1). [ This was first done by Lewis and Reisenfeld in 1969] 1

66 2. Parts a,b,c are independent of each other. (a) [9] Consider the potential V (r) = e2 r ω2 r 2 which is the potential for the hydrogen atom plus the isotropic harmonic oscillator. Considering 1 2 ωr2 as a perturbation on the hydrogen atom use first order perturbation theory to find the ω that gives zero total energy for the ground state energy. (b) [8] Consider the following potential ( V (r) = 2 5α 2m r ) 1 where α is a fixed constant and the following wave function r 2 ψ(θ, φ, r) = Yl m (θ, φ)r β e αr with m = 0 and l = 1. Is there a value for β which makes u(θ, φ, r) an eigenfunction of the Hamiltonian? If there is find it and the corresponding energy eigenvalue. (c) [8] Suppose ψ n are the eigenfunctions of the Hamiltonian H with corresponding energy eigenvalues Hψ n = E n ψ n and suppose the wave function ψ is defined by ψ = A(ψ n + λu) where λ is a real small parameter and u is a function. Show that the expected value of the Hamiltonian taken with ψ is second order in λ. For convenience you may assume that ψ n and u are real and normalized and define α = ψ n udx. Hint: Be sure to normalize ψ. 2

67 0 Equations π e ax2 +bx = xe ax2 +bx = b π 2a x 2 e ax2 +bx = a eb2 /(4a) 2a + b2 4a 2 a eb2 /(4a) π a eb2 /(4a) x 4 e ax2 = 3 π 4a 5/2 (5) x n e ax dx = n! a n+1 = 1/a n = 0 1/a 2 n = 1 2/a 3 n = 2 6/a 4 n = 3 24/a 5 n = 4 (2) (3) (4) (6) Hydrogen atom ground state: ψ 0 = e r/a 0 πa 3 0, 2 = 1 ( r 2 r 2 ) + r r 2 E 0 = me4 2 2 a 0 = 2 me 2 (7) ( sin θ ) θ θ r 2 sin 2 θ 2 ϕ, (8) 1 r 2 sin θ ) 1 R 2m r 2 r2 (V r r l(l + 1) R = ER (9) 2mr2 For central potentials: If then 2 2m 2 ψ + V (r)ψ = Eψ (10) ψ(r, θ, φ) = R(r)Yl m (θ, φ) (11) [ 1 L 2 = 2 sin θ θ sin θ θ ] sin 2 θ φ 2 (12) ) 1 d dr 2m r 2 r2 (V dr dr + (r) + 2 l(l + 1) R = ER (13) 2mr2 2 2 d 2 2m dr 2 u + u = rr (14) ) (V + 2 l(l + 1) u = Eu (15) 2mr2 3

68 Part B Problem 1 The potential seen by electrons at the surface of a metal placed in a strong electric field has the form shown in the Figure (The potential in the region x 0 is given by Fx,0 x b V( x) Fb, x b Consider an electron of energy 0 (see the Figure) incident on the surface from inside of the metal ( x 0 ). b a) Write down expressions for the quasi-classical wave function of the electron in regions x 0, x 0 x0, x x0, where x 0 is the classical turning point (find it in terms of energy and F ) (5 points) b) Find connections between coefficients of the wave function in each of these three regions using WKB connection formulas. (7 points) c) Calculate the transmission coefficient of this electron neglecting exponentially small reflection at point x b across the barrier in the quasi-classical approximation by applying connection formulas at each of the classical turning points (8 points) d) Determine the limits of applicability of the WKB approximation in this case. (5 points)

69 Problem 2 A hydrogen atom in a state with the principal quantum number n 2 is in a magnetic and an electric field directed perpendicularly to one another so that the interaction Hamiltonian is given by eb H q x L S int E z 2 2mc e Assume that the fields are strong (energy of the electron in the external electric and magnetic fields is much larger than the energy of the spin-orbit interaction). a). Make a list of all degenerate states belonging to this energy level. Are they orthogonal to each other? (3 points) b). Indicate which of the degenerate states are coupled by the perturbation and write down the perturbation matrix in the basis of these states. (8 points) c). Find the energies of all energy levels with n 2 (coupled and uncoupled) in the first order of the perturbation theory. (8 points) d) If the electric and magnetic fields are weak compared to the spin-orbit interaction, list the eigenstates of the zero order Hamiltonian corresponding to n 2, which you would use as an initial basis for the perturbation theory. Indicate which states are degenerate. (6 points) z

70 Useful formulas Normalized wave function of the hydrogen atom with n 2 are: Useful integral: 1 r a 1 r e 4 2 a 200 3/2 a /2 a /2 8a0 0 r e a 0 0 e r/2a0 r/2a0 cos sin e r/2a0 i n x dxx e 0 n! Connection formulas for WKB approximation:

71 THE CITY UNIVERSITY OF NEW YORK First Examination for Ph.D. Candidates in Physics Quantum Mechanics January 21, 2014 Instructions: a) PUT YOUR IDENTIFICATION NUMBER ON EACH PAGE. b) There are two parts to this exam, parts A and B. Do two questions but you must chose one from each part. Each question is worth 25 points. c) Start each problem on a new page. d) Indicate clearly which two problems you choose to solve. 1

72 1. Parts a,b,c are independent of each other. PART A (a) [7] Starting with Schrödinger time dependent equation derive the equation of motion for the expectation value of an operator, A: d A dt = 1 [A, H] + i A t Then, show that for Hamiltonians of the form H = p 2 /2m + V (x), that the expectation values for position and momentum are d x = 1 dt m p d p V = dt x Explain in detail your simplification of the commutators. (b) [5] Suppose we are solving a bound state problem and at a certain point a the potential behaves as V (x) = λδ(x a) Show that the discontinuity of the eigenfunctions at x = a satisfies du dx du a+ dx = λ 2m a 2 u(a) (c) [13] Consider the standard problem of a particle in box ( 0 to L) but with an additional attractive delta function potential V (x) = λδ(x L/2) with λ positive. Derive the algebraic equation that can be used to find the energy eigenvalues. Your equation should contain only m,, L, E, λ. Hint: Write Schrödinger s equation and for the region 0 x L/2 take u(x) = A sin kx + B cos kx and for the region L/2 x L take u(x) = C sin k(x L) + D cos k(x L) 2. Parts a and b are independent of each other. (a) Consider the following wave function ψ(r, 0) = 3u 100 (r) + 4iu 211 (r) where u 100 (r) and u 211 are the eigenfunctions of the hydrogen atom for the n, l, m states. They are given in the equation sheet. i. [8] Normalize the wave and calculate H and L 2 You do not have to do any integrals for this part, just use the fundamental properties of H and L 2. ii. [6] Calculate r. (b) [11]Consider the three dimensional anisotropic harmonic oscillator where the potential V (x, y, z) = 1 2 m ( ω 2 1x 2 + ω 2 2y 2 + ω 2 3z 2) Write the ground state wave function, the ground state energy and calculate r 2 for the ground state. 2

73 Equations for Part A **************************** Harmonic Oscillator: u 100 = 1 π ( 1 a 0 ) 3/2 e r/a 0 u 211 = 1 8 ( ) 1 1 3/2 r e r/2a 0 sin θe iϕ π a 0 a 0 Hydrogen atom energy: E n = e2 2a 0 n 2 = me4 2 2 n 2 ( 2 d 2 2m dx ) 2 mω2 x 2 u n (x) = E n u n (x) u n (x) = N n e α2 x 2 /2 H n (αx) ( α 1/2 u n (x) = π2 n!) n e α2 x 2 /2 H n (αx) E n = (n ) ω ********************** Integrals: α = mω = u 0 (x) = ( ) mk 1/4 k 2 ω = m ( mω ) 1/4 e mω 2 x2 π N n = ( ) α 1/2 π2 n n! π 0 π 0 π 0 0 sin xdx = 2 sin 2 xdx = π/2 sin 3 xdx = 4/3 x n e ax dx = n! a n+1 π e ax2 +bx = xe ax2 +bx = b 2a x 2 e ax2 +bx = a eb2 /(4a) π 2a + b2 4a 2 a eb2 /(4a) π a eb2 /(4a) 3

74 Part B 1. A particle of spin ½ and magnetic moment µ moves in the non-homogeneous magnetic field ( are Cartesian components of the position vector) a) Write down Hamiltonian of the system as a matrix in the basis of the eigenstates of the z-component of the spin including only spin part of interaction with the magnetic field (3 pts) b) Show that upon transformation to the Heisenberg picture, the Hamiltonian does not change (3 pts) c) Write down Heisenberg equations for Cartesian components of the position operator, momentum operator, and spin operators. (7 pts) d) Solve Heisenberg equations for the spin operators neglecting the inhomogeneous part of the magnetic field (6 pts) e) Substituting the found results for the spin matrices into equations for the momentum operators, find their solutions. This will give you solution of the problem in the linear with respect to parameter approximation (6 pts) 2. Consider states of the hydrogen atom with principal quantum number. (Neglect spin) a) Using degenerate perturbation theory find corrections to the energies of all these states due to static uniform electric field directed in z-direction. (7 points) b) Find the correct forms of the zero order wave functions for each of these states (7 points) c) Now imagine that you want to observe the energy splitting induced by the electric field (Stark effect) by measuring optical absorption using monochromatic linearly polarized light. Assuming that the atom is in its ground state and the light is polarized in z- direction, determine frequencies of spectral lines in the absorption spectrum, which will be observed, and their relative strength (7 points) d) Can Stark effect be observed with x-polarized light? Give arguments based on Wigner- Eckart theorem (4 points) Useful formulas for Part B Normalized wave function of the hydrogen atom with n 2 are:

75 1 r a 1 r e 4 2 a 200 3/2 a /2 a /2 8a0 0 r e a 0 0 e r/2a0 r/2a0 cos sin e r/2a0 i Useful integral: n x dxx e 0 n!

76 THE CITY UNIVERSITY OF NEW YORK First Examination for Ph.D. Candidates in Physics Quantum Mechanics June 16, 2014 Instructions: a) PUT YOUR IDENTIFICATION NUMBER ON EACH PAGE. b) There are two parts to this exam, parts A and B. Do two questions but you must chose one from each part. Each question is worth 25 points. c) Start each problem on a new page. d) Indicate clearly which two problems you choose to solve. PART A 1. Parts a and b are independent of each other. (a) [13] The continuity theorem in quantum mechanics states ρ t + j = 0 where ρ is the probability density and j is the quantum mechanical current. There is another continuity equation that applies to energy density ϵ t + S = 0 where the energy density is defined as ϵ(r, t) = 2 2m ψ(r, t) ψ (r, t) + V (r)ψ(r, t)ψ (r, t) and S is the energy density current. Find S the proof goes along the same lines as for the continuity equation. After you obtain your result be sure to show that it is real. You may assume that V is multiplicative. To make it easier do it in one dimension Also, the following relation may be useful ( ψ ) ψ x t x = x ϵ(x, t) = 2 ψ ψ 2m x x + V (x)ψ(x, t)ψ (x, t) ϵ t + S x = 0 ( ψ t ) ψ ψ 2 ψ x t x 2 (b) [12] Consider the following wave function ( ) α 2 1/4 ψ(x) = e α2 x 2 /2+iβx 2 +iγx π under a harmonic oscillator force, where β and γ are real and as usual α = mω. What is the probability that a measurement will find the oscillator in the ground state and what is the probability to find it in an excited state. Make sure you express your final answers in forms that are manifestly real. 1

77 2. Parts a and b are independent of each other. (a) Consider the following operator with α and β real. A = α(l 2 x + L 2 y) + βl 2 z i. [6] Are the spherical Harmonics Yl m (θ, φ) eigenfunctions of the operator? If yes find the eigenvalues. Hint: first express the operator in terms of L 2 and L z. ii. [6] Consider the wave function ψ(θ, φ) = sin θ cos φ Express this wave function in terms of spherical harmonics and determine if it is an eigenfunction of A. If it is find the eigenvalue. Hint write and use the fact that ψ(θ, φ) = sin θ cos φ = sin θ eiφ + e iφ 3 Y 1 ±1 (θ, φ) = 8π e±iφ sin θ (b) [13] Using the variational principle estimate the ground state wave function and energy of the hydrogen atom using the following finite extent trial wave function { 1 r/r if r R, ψ(r) = 0 if r > R. where R is the variational parameter. 2 2

78 Equations for Part A Hydrogen atom energy: **************************** 2 = 1 r 2 r r2 r + 1 r 2 sin θ θ sin θ θ + 1 r 2 sin 2 θ E n = e2 2a 0 n 2 = me4 2 2 n 2 2 φ 2. Harmonic Oscillator: ( 2 d 2 2m dx ) 2 mω2 x 2 u n (x) = E n u n (x) u n (x) = N n e α2 x 2 /2 H n (αx) ( α 1/2 u n (x) = π2 n!) n e α2 x 2 /2 H n (αx) Integrals: E n = (n ) ω α = mω = ( ) mk 1/4 k 2 ω = m ( ) α 2 1/4 u 0 (x) = e α2 x 2 /2 π π 0 π 0 π 0 0 sin xdx = 2 sin 2 xdx = π/2 sin 3 xdx = 4/3 x n e ax dx = n! a n+1 π e ax2 +bx = xe ax2 +bx = b 2a x 2 e ax2 +bx = a eb2 /(4a) π 2a + b2 4a 2 a eb2 /(4a) π a eb2 /(4a) N n = ( ) α 1/2 π2 n n! 3

79 ******************* 1 Y0 0 (θ, φ) = 4π 3 Y1 0 (θ, φ) = 4π cos θ 3 Y 1 ±1 (θ, φ) = 8π e±iφ sin θ 5 Y2 0 (θ, φ) = 16π (3 cos2 θ 1) 15 Y 2 ±1 (θ, φ) = 8π e±iφ sin θ cos θ 15 Y 2 ±2 (θ, φ) = 32π e±2iφ sin 2 θ 4

80 QM, Part B 1. A particle of spin ½ and magnetic moment moves in the uniform magnetic field B0 k,0 t T Bt () 0, t 0, t T ( k is a unit vector in the direction of the field) a) Obtain Heisenberg equations for the components of the spin operator of the particle (5 points) b) By solving the Heisenberg equations, find operators S ˆ () x, y, z t in the Heisenberg picture (10 points) c) If the system is prepared in state described by the spinor written in the basis of the eigenspinors of the operator S z defined at t 0, find the probabilities of obtaining various outcomes, if x-component of the spin is measured at t T (10 points) 2. Consider a quantum system described by the Hamiltonian of the following form 2 pˆ H M x x (1) 2M 2 3! where ˆp and ˆx are canonic operators of momentum and coordinate respectively, M is mass of the particle, and is the frequency of harmonic oscillations. a) Present the Hamiltonian in terms of ladder operators aa ˆ, ˆ of the harmonic oscillator b) Rewrite the Hamiltonian found in a) in the normally ordered form c) Find the expectation value of energy of the particle described by this Hamiltonian if the system is in a coherent state. Useful formulas for Part B m p a x i ; 2 m e daˆ 1 H B; S; Aˆ, Hˆ mc dt i

81 THE CITY UNIVERSITY OF NEW YORK First Examination for Ph.D. Candidates in Physics Quantum Mechanics January 20, 2015 Instructions: a) PUT YOUR IDENTIFICATION NUMBER ON EACH PAGE. b) There are two parts to this exam, parts A and B. Do two questions but you must choose one from each part. Each question is worth 25 points. c) Start each problem on a new page. d) Indicate clearly which two problems you choose to solve. PART A 1. Parts a,b, and c are independent of each other. (a) Consider the following 3 dimensional wave function where ψ(r, θ, φ) = u 1 (r, θ, φ) + ηu 2 (r, θ, φ) u 1 (r, θ, φ) = 1 ( 1 4 2π a 0 ( 1 u 2 (r, θ, φ) = 1 4 2π a 0 ) 3/2 (2 ra0 ) e r/(2a 0) ) 3/2 r a 0 e r/(2a 0) cos θ and where η is a real number. u 1 (r, θ, φ) and u 2 (r, θ, φ) are normalized hydrogen atom eigenfunctions. Normalize the wave function, ψ(r, θ, φ), and calculate the dipole moment µ = z Some of the integrals may be obviously zero if that is the case be sure you explain why they are zero. (b) Consider the one dimensional problem where the potential is given by { λ V (x) = x x > 0 x 0 and where λ is a given positive constant. For the following unnormalized wave function u(x) = x n e αx are there values of α and n which makes u(x) an energy eigenfunction? If yes find α and n and find the energy eigenvalue. (c) Consider the following Hamiltonian H = p2 2m F x + Gp where F and G are real numbers. Solve for the eigenfunctions and eigenvalues of H; Do it in the momentum representation. That is solve Hu E (p) = Eu E (p) for u E (p). Remember that in the momentum representation x = d i dp. Be sure you appropriately normalize the eigenfunctions. Are the eigenvalues continuous or discreet? 1

82 2. Parts a,b and c are independent of each other. (a) Consider the three dimensional potential given by V (r) = kr Use the variational principle to estimate the ground state energy. Use the trail wave function ψ(r, θ, φ) = e αr where α is the variational parameter. (b) Suppose we have a Hamiltonian of the form H = p2 2m + V (x) with corresponding eigenvalues and eigenfunctions E n and u n (x). Define p nk = u n(x)pu k (x)dx ; x nk = u n(x)xu k (x)dx Find an expression that relates p nk to E n, E n and x nk. HINT: first prove that p = im (c) Suppose the spin wave function with respect to the z axis is given by ( ) a + ib ψ = c [H, x] where a, b, c are real numbers. Normalize the wave function and determine the probability of measuring spin up and down. Calculate the average value of spin in the z direction and the standard deviation. 2

83 σ x = ( Relevent Equations da dt = im [H, A] ) ( ) ( 0 i 1 0 ; σ y = ; σ i 0 z = 0 1 1/a n = 0 x n e ax dx = n! 1/a 2 n = 1 0 a n+1 = 2/a 3 n = 2 6/a 4 n = 3 24/a 5 n = 4 π e ax2 +bx = xe ax2 +bx = b 2a x 2 e ax2 +bx = π 0 π 0 π 0 2 = 1 r 2 r r2 r + 1 r 2 sin θ 0 a eb2 /(4a) π 2a + b2 4a 2 sin xdx = 2 sin 2 xdx = π/2 sin 3 xdx = 4/3 x n e ax dx = n! a n+1 a eb2 /(4a) π a eb2 /(4a) θ sin θ θ + 1 r 2 sin 2 θ Hydrogen atom energy: E n = e2 2a 0 n 2 = me4 2 2 n 2 2 φ 2. ). 3

84 QM, Part B Problem 1 a) Use the WKB approximation to determine the bound-state energies of the potential well V( x) x V, x a 0 a 2 h 0, 2 V x a ma b) An electron is inside of a quantum dot modeled as an infinite spherical potential well: 0, r R V () r, r R Using degenerate perturbation theory, find the first order corrections to the energy of states characterized by orbital number l 2 due to a uniform electric field F and corresponding zeroorder wave functions. Problem 2 Consider an exciton in a semiconductor as a particle with spin s 1 and assume that its orbital state is characterized by orbital momentum l 1. a) What are the possible values of the total angular momentum of the exciton J L S 2 (number j characterizing the eigenvalues of the operator J )? b) A measurement of the total angular momentum produced values j 2; m j 2, where m is the quantum number characterizing eigenvalues of the operator J. What are the j probabilities of obtaining various values of m and m, characterizing eigenvalues of operators L z and S z if these quantities are measured immediately after the first measurement? c) Answer the same question as above but assuming that the results of the first measurement produced j 2; m j 0 L S z Some useful formulas

85 Y2, 3cos 1 ; Y2, sin cos e Y 1 15, sin e i 2 Recursion relation for Clebsch-Gordan coefficients j m j m 1 l, s, m, m l, s, j, m 1 l m l m 1 l, s, m 1, m l, s, j, m j j l s j l l l s j s m s m 1 l, s, m, m 1 l, s, j, m s s l s j i l, s, l, s l, s, l + s, l + s 1

86 THE CITY UNIVERSITY OF NEW YORK First Examination for Ph.D. Candidates in Physics Quantum Mechanics June 17, 2015 Instructions: a) PUT YOUR IDENTIFICATION NUMBER ON EACH PAGE. b) There are two parts to this exam, parts A and B. Do two questions but you must chose one from each part. Each question is worth 25 points. c) Start each problem on a new page. d) Indicate clearly which two problems you choose to solve. PART A 1. Parts a, b, and c are independent of each other. (a) [10] A particle is constrained to move on the surface of an infinite cylinder of radius R. The central axis of the cylinder is along the z axis. Find the stationary state energy eigenfunctions and eigenvalues. Hint: the Laplacian in cylindrical coordinates is 2 = 2 r r r r 2 φ z 2. but since it is constrained to r = R we can take it to be 2 = 1 R 2 2 Hence you have to solve Schrödinger s equation for the eigenfunctions and eigenvalues φ z m 2 u(φ, z) = Eu(φ, z) (b) [10] Consider the following angular momentum wave function [ ] ψ(θ, φ) = N αyl m (θ, φ) + iβyl l (θ, φ) with α and β real. Normalize the wave function and calculate the ( L z ) 2 = L 2 z L z 2. After you obtain your answer show that it is manifestly positive, that is ( L z ) 2 = (something real) 2 (c) [5] Suppose the solution to the time dependent Schrödinger equation is given by ψ(x, t) = K x (x, x, t)ψ(x, 0)dx 1

87

88 Relevant Equations 0 0 x n e ax dx = n! a n+1 re αr dr = 1 π 2α α ψ(x, t) = 1 2π **************************** φ(p, t) e ipx/ dp ; φ(p, t) = 1 2π 2 = 1 r 2 r r2 r + 1 r 2 sin θ θ sin θ θ r 2 sin 2 θ φ 2. e i(y y )x dx = 2πδ(y y ) π e ax2 +bx = xe ax2 +bx = b 2a x 2 e ax2 +bx = 2a + b 4a 2 a eb2 /(4a) π a eb2 /(4a) π a eb2 /(4a) 1 Y0 0 (θ, φ) = 4π 3 Y1 0 (θ, φ) = 4π cos θ 3 Y 1 ±1 (θ, φ) = 8π e±iφ sin θ 5 Y2 0 (θ, φ) = 16π (3 cos2 θ 1) 15 Y 2 ±1 (θ, φ) = 8π e±iφ sin θ cos θ 15 Y 2 ±2 (θ, φ) = 32π e±2iφ sin 2 θ ψ(x, t) e ipx/ dp L x = i L y = i L z = i ( y z z ) = ( sin ϕ ) + cot θ cos ϕ y i θ φ ( z x x ) = ( cos ϕ ) cot θ sin ϕ z i θ φ ( x y y ) = x i φ 3

89 L 2 = 2 [ 1 sin θ L 2 = L 2 x + L 2 y + L 2 z = 2 [ 1 sin θ L z = i ϕ θ sin θ θ + 1 sin 2 θ 2 ] ϕ 2 θ sin θ θ + 1 sin 2 θ L z Y m l = m Y m l L 2 Y m l 2 ] φ 2 = l(l + 1) 2 Y m l Central potential Radial Equation: H = 2 2m [ 1 r 2 r r2 r + 1 r 2 sin θ = 2 1 2m r 2 r r2 r + L2 2mr 2 + V (r 2 θ sin θ θ + 1 r 2 sin 2 θ ) 1 R 2m r 2 r2 (V r r l(l + 1) R = ER 2mr2 u = rr d 2 ) (V 2m dr 2 u l(l + 1) u = Eu 2mr2 2 2 ] ϕ 2 + V (r) 4

90 Part B 3. Consider a particle of mass m and charge q in the three-dimensional harmonic potential V = 1 2 mω2 (ˆx ˆx ˆx 2 3) a) [4 points] After defining the ladder operators a i and â i as â i = 1 (ˆp i imωˆx i ) â i = 1 (ˆp i + imωˆx i ) 2mω h 2mω h Show that [â i, â j ] = δ ij and that the hamiltonian of the harmonic oscillator can be written as Ĥ 0 = hω 3 i=1 ( â i âi + 1 ). 2 b) [4 points] What are the three lowest energy levels? What are their degeneracies? List the states associated with each level. c) [3 points] Show that the angular momentum operator ˆL can be written in terms of the raising and lowering operators, as ˆL i = i hɛ ijk â jâk, i.e. ˆL3 = i h(â 1â2 â 2â1). d) [3 points] Show that ˆL built in this manner commutes with Ĥ0. e) [3 points] For the first excited state, find a common set of eigenstates of Ĥ0 and ˆL 3. What are the eigenvalues of ˆL 3? f) [8 points] The particle is perturbed by a uniform time-dependent electric field E(t) = A exp{ (t/τ) 2 } directed along the positive z axis (i.e. in the direction of x 3 ). Compute the probability, at first order approximation, of finding the particle in an excited state at time t = +, if at the time t = it was in the ground state. Which levels of the harmonic oscillator will be accessible, at first order approximation, under this perturbation? Can we observe a transition to the second excited state? 4. Two particles, whose magnetic moments are µ 1 = as 1 and µ 2 = bs 2 respectively, interact with an external magnetic field B, as well as with each other. We take the z-axis in the direction of the field B, so that the Hamiltonian of the system can be written as Ĥ = a S 1z B + b S 2z B + J S 1 S 2. Consider the case s 1 = s 2 = 1/2. Define J = S 1 + S 2 the total angular momentum of the system. a) [5 points] The system can be described either using the basis m 1 m 2 s 1 s 2 m 1 m 2 (eigenstates of S 2 1, S 2 2, S 1z, and S 2z ) or the basis jm j s 1 s 2 jm j (eigenstates of S 2 1, S 2 2, J 2, J z ). Find the expression of the states { j m j } in terms of the appropriate elements of the basis { m l m s }. This part requires the direct construction of all CG coefficients, namely to prove that 1, 1 = +, +, 1, 0 = 1 2 ( +, +, + ), 0, 0 = 1 2 ( +,, + (where we adopted the usual simplified notation +, + + 1/2, +1/2, etc...). ), 1, 1 = b) [20 points] Find the exact energy eigenvalues of the system and sketch the spectrum as a function of the magnetic field. Comment on the degeneracy. 5

91 One-Dimensional Harmonic oscillator: Relevant Formulas for Part B Ĥ 0 = ˆp2 2m mω2ˆx 2 1 â = (ˆp imωˆx) 2mω h â = 1 2mω h (ˆp + imωˆx) [â, â ] = 1 [Ĥ0, â ] = hωâ n 1 â n = n n â n 1 = n Time-dependent Perturbation Theory (first order transition probability): P k s = 1 t h 2 dt k V (t ) s e t 0 where ω ks = E k E s h iω kst 2 Angular Momentum Operators ( h = 1): [J x, J y ] = ij z ([J i, J j ] = iɛ ijk J k ) [J 2, J i ] = 0 J z j m = m j m, J 2 j m = j(j + 1) j m J ± = J x ± ij y J ± j m = j(j + 1) m(m ± 1) j m ± 1 6

92 THE CITY UNIVERSITY OF NEW YORK First Examination for Ph.D. Candidates in Physics Quantum Mechanics January 20, 2016 Instructions: a) PUT YOUR IDENTIFICATION NUMBER ON EACH PAGE. b) There are two parts to this exam, parts A and B. Do two questions but you must chose one from each part. Each question is worth 25 points. c) Start each problem on a new page. d) Indicate clearly which two problems you choose to solve. PART A 1. Parts a, b, c, and d are independent of each other. (a) [6] The equation of continuity in quantum mechanics for real potentials is given by t ψ(r, t) 2 + J = 0 where J is the quantum mechanical current. Suppose the potential is complex, V = V r + iv i. Derive the equation of continuity for this case. (b) [6] Consider the following one dimensional wave function ψ(x) = xe αx/2 0 x where α is a positive real number. Calculate the uncertainty product. The uncertainty product is the product of the standard deviation of position times the standard deviation of momentum. Of course your answer should come out to be greater than /2. (c) [6] Suppose we have an Hermitian operator A that commutes with the angular momentum operator in the z direction L z. Evaluate Yl m [A, L z ]Yl m where Yl m (θ, ϕ) are the usual spherical harmonics. Express your answer in terms of Yl m AYl m. (d) [7] Suppose we have a wave function and we know that the potential is a harmonic oscillator potential. When we measure the total energy we find that the only values are ω/2 or 3 ω/2 with relative frequency of 1/5 and 4/5. Also a measurement of the position gives an expectation value of zero. What is the wave function? Hint: take the wave function to be ψ(x) = A(u 0 (x) + αu 1 (x)) and impose the stated conditions. You can take A to be real but α may be complex. The matrix elements of x are given in the formulas sheet. 1

93 Ph.D. Program in Physics The Graduate Center CUNY 2. Parts a, b, and c are independent of each other (a) [6] A particle in a box of length L has the following wave function ψ(x) = 1 0 x L/2 and zero for L/2 x L. Normalize the wave function and calculate the probability that a measurement of the energy will give a value of E = π2 2 2mL 2 (b) [6] In classical mechanics we have for the force, F = dp dt. Suppose that in quantum mechanics we define the force operator by F = dp dt where p is the quantum mechanical momentum operator. Calculate the expected value of force for a wave function that is an eigenfunction of the Hamiltonian, that is, for a stationary state. (c) [13] Consider the following 3 dimensional wave function u nlm = r 2 e αr Y m l (θ, ϕ) where α is a real positive number. Determine whether there are values of α, l, m that makes this wave function an eigenfunction of the Hydrogen atom Hamiltonian. If there are values list them all, determine n, the principle quantum number, and the corresponding energy. You must show your work. 2

94 Ph.D. Program in Physics The Graduate Center CUNY Relevant Equations ψ(x, t) = 1 2π ψ 2 ψ ψ 2 ψ = (ψ ψ ψ ψ ) **************************** L 2 = 2 [ 1 sin θ 0 x n e ax dx = n! a n+1 ϕ(p, t) e ipx/ dp ; ϕ(p, t) = 1 2π e i(y y )x dx = 2πδ(y y ) π e ax2 +bx = xe ax2 +bx = b 2a x 2 e ax2 +bx = 2a + b 4a 2 L 2 = L 2 x + L 2 y + L 2 z = 2 [ 1 sin θ L z = i φ θ sin θ θ + 1 sin 2 θ 2 ] φ 2 a eb2 /(4a) π a eb2 /(4a) π a eb2 /(4a) θ sin θ θ + 1 sin 2 θ L z Y m l = m Y m l L 2 Y m l ψ(x, t) e ipx/ dp 2 ] ϕ 2 = l(l + 1) 2 Y m l *********************************************************************** Central potential Radial Equation: H = 2 2m [ 1 r 2 r r2 r + 1 r 2 sin θ = 2 1 2m r 2 r r2 r + L2 2mr 2 + V (r 2 θ sin θ θ + 1 r 2 sin 2 θ ) 1 R 2m r 2 r2 (V r r l(l + 1) R = ER 2mr2 u = rr d 2 ) (V 2m dr 2 u l(l + 1) u = Eu 2mr2 2 ***************************************** 3 2 ] φ 2 + V (r)

95 Ph.D. Program in Physics The Graduate Center CUNY Hydrogen atom: 1 2m r 2 r 2 V (r) = e2 r R r2 ( r + e2 r + a 0 = 2 l(l + 1) 2mr2 (1) ) R = ER (2) 2 me 2 (3) E n = e2 2a 0 n 2 = me4 2 2 n 2 (4) l = 0, 1,...n 1 (5) *****************************************************8 One dimensional harmonic oscillator ( p 2 x 2m + 1 ) 2 mω2 x 2 u n (x) = E n u n (x) where and u n (x) = N n e α2 x 2 /2 H n (αx) ( ) mω mk 1/4 k α = = 2 ω = m E n = (n ) ω α = mω u n (x) = N n H n (αx)e 1 2 α2 x 2 N n = ( ) α 1/2 π2 n n! ( ) α 2 1/4 u 0 (x) = e α2 x 2 /2 π u 1 (x) = ( ) α 2 3/4 2π xe αx2 /2 π [ u n (x)xu m (x) = 1 ] n + 1 n δ m,n+1 + α 2 2 δ m,n 1 [ ] n + 1 n u n (x)pu m (x) = iα δ m,n δ m,n 1 (6) (7) **************************************************** For a particle in a box of lenght L the eigenvalues and normalized eigenfunctions are E n = π2 2 2 ( nπ ) 2mL 2 n2 ; u n (x) = L sin x L **************************************************** 4

96 Ph.D. Program in Physics The Graduate Center CUNY Part B 3. We want to combine two angular momenta L and S, and describe the state of the system in the basis of eigenstates of the operators J 2 and J z (where J = L + S). a) [5 points] Assuming that L and S commute, explain why { L 2, S 2, J 2, J z } and { L 2, S 2, L z, S z } are two alternative good sets of compatible observables, while this is not the case for { L z, S z, J 2, J z }. Define the Clebsch-Gordan (CG) coefficients. b) For the case l = 1 and s = 1/2: b.1 [2 points] How many states belong to each of the two orthonormal and complete sets { m l m s } and { j m j }? List all the elements of each set. A quantum system is prepared in the state ψ = 1 3 j = 3 2 m j = j = 1 2 m j = 1 2 b.2 [10 points] Find the expression of the states j m j which appear in ψ in terms of the appropriate elements of the basis { m l m s }. This part requires the direct construction of all CG coefficients relevant to the solution of the problem. b.3 [3 points] What is the probability of measuring m s = 1 2 in the state ψ? What about m s = 1 2? b.4 [3 points] What about a measurement of m l? List the probabilities of finding m l = +1, 0, 1. b.5 [2 points] What is the probability of a combined measurement m l = 1 and m s = 1 2 in ψ? 4. This problem is formed by two independent parts (4a and 4b). 4a. Consider a three-state quantum system described by the Hamiltonian Ĥ = Ĥ0 + Ĥp where Ĥ 0 = a 0 2 0, Ĥ p = b 0 0 2, and b << a. We want to apply non-degenerate perturbation theory to study the full Hamiltonian Ĥ. 1) [10 points] After writing down the eigenstates and eigenvalues of Ĥ0, compute the first- and secondorder corrections to the energy levels due to Ĥp. 2) [5 points] Compute the exact eigenvalues of Ĥ. Check that, by expanding this result in ɛ = b a, you can reproduce your results from part b). 4b. [10 points] Use the WKB approximation to find the allowed energy levels for the one-dimensional harmonic oscillator. 5