Download Quantum Mechanics I Final Exam, Spring 2006 - Prof. J. E. Moore and more Exams Quantum Mechanics in PDF only on Docsity! Physics 137A: Quantum Mechanics I, Spring 2006 Final Exam Directions: The allotted time is 3 hours. The 6 problems count equally. Two sides of your own notes are allowed. No books or calculators are allowed, and please ask for help only if a question’s meaning is unclear. 1. Consider a particle in the one-dimensional infinite square well V (x) = { 0 if 0 < x < a ∞ otherwise (1) The energy eigenstates are ψn = √ 2 a sin( nπx a ). (2) The initial state of the particle is ψ(x, 0) = 1√ 2 (ψ1(x) + ψ2(x)). (3) Hint for doing integrals: cos(x+ y) = cosx cos y − sinx 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 〈x〉(t) is the same as at t = 0? You do not need to calculate 〈x〉 explicitly. 2. (a) Give the expectation values of the operators L2 and Lz for an electron in the state ψnlm = ψ210 of hydrogen. (b) List all possible values from independent measurements of the operators Sz and S2 of this electron, where S is the electron spin. (Note that the spin state has not been specified.) (c) List all possible values from independent measurements of the operators Jz and J2 on this electron, 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 the following two l = 1 states: the first state has ml = 1 and ms = −1/2, and the second state has ml = 0 and ms = 1/2. Which has a larger magnetic moment? 3. (a) In one dimension, find the lowest-energy eigenstate for the potential V (x) = −V0δ(x), (4) where V0 is a positive constant. (b) Suppose that some other potential has energy eigenstates ψ1 with energy E1, and ψ2 with energy E2. Give the outcomes and probabilities of an energy measurement on the state ψ = 1√ 3 ψ1 + √ 2 3 ψ2. (5) (c) Suppose that instead of measuring energy, an observable Ô is measured, with eigenstates φ1 = 1√ 2 ψ1 + 1√ 2 ψ2, with eigenvalue u1 (6) and φ2 = 1√ 2 ψ1 − 1√ 2 ψ2, with eigenvalue u2. (7) Give the probability that u1 is measured on the initial state in (b) above. (d) Suppose that first observable Ô was measured with the outcome u1. What is the probability that an energy measurement will then give outcome E1? 4. (a) Derive the probability current operator for a particle in the three-dimensional Schrödinger equation, 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√ V eik·r. 5. Consider a helium atom (one point nucleus, two identical electrons) in which there is no electron-electron repulsion. (a) Write the Hamiltonian for this system, including only electron kinetic energies and the Coulomb field of the nucleus (which has charge +2e) and show that it is of the form H = H1 +H2, where H1 and H2 contain terms only involving electron 1 and electron 2. (b) Write a wavefunction including spin that describes two electrons that are both in the orbital ground state. (c) Suppose that E1 is the orbital energy of 1 electron in the ground state. In units of E1, what is the energy of the state in (b)?
