Physics 137A: Quantum Mechanics I, Spring 2006Final ExamDirections: The allotted time is 3 hours. The 6 problems count equally. Two sides of your ownnotes are allowed. No books or calculators are allowed, and please ask for help only if a question’smeaning is unclear.1. Consider a particle in the one-dimensional infinite square wellV (x) =0 if 0 < x < a∞ otherwise(1)The energy eigenstates areψn=r2asin(nπxa). (2)The initial state of the particle isψ(x, 0) =1√2(ψ1(x) + ψ2(x)). (3)Hint for doing integrals: cos(x + y) = cos x cos y − sin x sin y.(a) Find the wavefunction for all later times t.(b) Find the probability that the particle is in the right half of the well for t = 0.(c) Find the probability that the particle is in the right half of the well for general time t.(d) What is the first time after t = 0 at which the expected value hxi(t) is the same as at t = 0?You do not need to calculate hxi explicitly.2. (a) Give the expectation values of the operators L2and Lzfor an electron in the stateψnlm= ψ210of hydrogen.(b) List all possible values from independent measurements of the op erators Szand S2of thiselectron, where S is the electron spin. (Note that the spin state has not been specified.)(c) List all possible values from independent measurements o f the operators Jzand J2on thiselectron, where J = L + S is the total angular momentum.(d) Write an expression for the z component of the magnetic moment for an electron in thefollowing two l = 1 states: the first state has ml= 1 and ms= −1/2, and the se cond state hasml= 0 and ms= 1/2. Which has a larger magnetic moment?3. (a) In one dimension, find the lowest-energy eigenstate for the potentialV (x) = −V0δ(x), (4)where V0is a positive constant.(b) Suppose that some other potential has energy eigenstates ψ1with energy E1, and ψ2withenergy E2. Give the outcomes and probabilities of an energy measurement on the stateψ =1√3ψ1+r23ψ2. (5)(c) Suppose that instead of measuring energy, an obse rvableˆO is measured, with eigenstatesφ1=1√2ψ1+1√2ψ2, with eigenvalue u1(6)andφ2=1√2ψ1−1√2ψ2, with eigenvalue u2. (7)Give the probability that u1is measured on the initial state in (b) above.(d) Suppose that first observableˆO was measured with the outcome u1. What is the probabilitythat an energy measureme nt will then give outcome E1?4. (a) Derive the probability current operator for a particle in the three-dimensional Schr¨odingerequation, i.e., find a vector operator j(r) such that∂|ψ(r)|2∂t+ ∇ · j(r) = 0. (8)You may wish to check that your answer has the proper units.(b) Find the value of this operator at the origin (x = y = z = 0) in the state ψ(r) =1√Veik·r.5. Consider a helium atom (one point nucleus, two identical electrons) in which there is noelectron-electron repulsion.(a) Write the Hamiltonian for this system, including only electron kinetic energies and theCoulomb field of the nucleus (which has charge +2e) and show that it is of the form H = H1+ H2,where H1and H2contain terms only involving electron 1 and electron 2.(b) Write a wavefunction including spin that describes two electrons that are both in theorbital ground state.(c) Suppose that E1is the orbital energy of 1 electron in the ground state. In units of E1, whatis the energy of the state in (b)?(d) In units of E1, what is the minimum energy of a Slater determinant for two electrons thatboth have spin up?(e) Now suppose that the Hamiltonian has an additional term JS1· S2that couples the spinsof the two electrons. Calculate how the energies in (c) and (d) are modified.6. (a) Suppose that an electron is in the hydrogen ground stateψ100=1qπa30e−r/a0(9)wherea0=(4π0)¯h2me2. (10)Compute the expectation value of kinetic and potential energy of this state. Useful integrals:Z∞−∞dx e−x2=√π,Z∞0dx xe−x2= 1/2,Z∞−∞dx x2e−x2=√π/2. (11)(b) Show that for any time-independent operatorˆO, the commutator with the Hamiltonianvanishes in an energy eigenstate |ψEi:hψE|[H,ˆO]|ψEi = 0. (12)(c) Use the fact that for a central potential in 3D (you do not need to prove this),[r · p, H] = 2i¯hT − i¯h(r · ∇V ) (13)where T is the kinetic energy and V is the central potential, to show that2hT i = hr · ∇V i (14)in an energy eigenstate. Is this consistent with your answer in