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PHY4605–Introduction to Quantum Mechanics II Spring 2004 Test 2 March 17, 2004 No other materials or calculators allowed. If you can’t do one part of a problem, solve subsequent parts in terms of unknown answer–define clearly. All parts 10 pts., max=70. Problem 1 required, attempt 1 of remaining 3 problems; circle which ones you want graded. Possibly helpful formulae and constants Z π/2 −π/2 dy y cos y sin 2y = 8/9 δ|ni = Z a/2 nπx a cos2 = a 2 −a/2 Z a/2 nπx a sin2 = a 2 −a/2 iθ 2 |e − 1| = 4 sin2 θ/2 me4 1 En = 2h̄2 n2 H = p̂2 /2m + mω 2 x2 /2 µ ¶ 1 En = h̄ω n + 2 Hψ = Eψ ∂ψ Hψ = ih̄ ∂t h̄2 2 H=− ∇ + V (r) 2m δE (1) = hn|V̂ |ni δE (2) = m6=n En − Em 1 Y00 = √ 4π s 3 Y10 = cos θ 4π s 3 sin θe±iφ 8π Ã ! ∂ ∂ L̂± = h̄e±iφ ± + i cot θ ∂θ ∂φ hφ|H|φi ≥ E0 hφ|φi ∂ L̂z = −ih̄ ∂φ L̂± = L̂x ± iL̂y Y1±1 = ∓ q L̂+ ψ`m = h̄ `(` + 1) − m(m + 1)ψ`m+1 q L̂− ψ`m = h̄ `(` + 1) − m(m − 1)ψ`m−1 X |hn|V̂ |mi|2 m6=n X hm|V̂ |ni|mi L̂2 = L̂21 + L̂22 + 2L̂1z L̂2z + En − Em +L̂1+ L̂2− + L̂1− L̂2+ 1 1. Short Answer. Must attempt (only) 3 of 6. Circle answers to be graded. (a) Given the exact normalized eigenfunctions ψn of a quantum system, Hψn = En ψn , prove the variational principle, i.e. show that for any normalized φ, hφ|H|φi ≥ E0 , where E0 is the exact ground state energy. (b) A chemistry book lists the electronic configuration of an N atom in its ground state as 1s2 2s2 2p3 and states that the wave function for this state is simply the product of hydrogenic orbitals corresponding to the 7 electrons in the levels designated. Criticize this statement. (c) An atomic state is referred to by its spectroscopic term, 3 P2 . What are its quantum numbers S, L, J? Using hydrogenic wavefunctions ψn`m and 2-particle spin states |SMS i, write down an appropriate wave function for 2 electrons which might correspond to this term. 2 (d) Explain the physical origin of fine structure and hyperfine structure in Hydrogen. (e) Angular momentum. Given two angular momenta L1 and L2 , with eigenvalues of the operators L21 and L22 given by 6h̄2 and 15h̄2 /4, what are the allowed possible eigenvalues of the total angular momentum operator squared L2 = (L1 + L2 )2 ? (f) H2 molecule. Sketch the effective potential for the two protons as a function of the distance between them for the ground state and first excited state. In the same picture, sketch the vibrational levels and explain (do not calculate) how you would estimate them. 3 2. Ring molecule. In calculating electronic structure of ring-shaped molecules, one is led to consider simplified models. For example, take two identical fermions of mass m and spin 1/2, moving on a ring of radius a, with spins constrained to point up. The particles can interact through a potential of form V (φ1 , φ2 ) = A cos(φ1 − φ2 ) (1) where φ1 and φ2 are angular positions of the two particles. First assume A = 0. (a) Find all allowed energy levels of this system. (Hint: the only dynamics in this system are the constrained motion (kinetic energy) of a particle going in a circle. The Hamiltonian may therefore be written as a rigid rotator H = L2z /2mr2 ). (b) Find the ground and 1st excited state wave functions. (c) State the degeneracies of the ground and 1st excited states. Now assume A is nonzero but weak, A ¿ h̄2 /(2ma2 ). (d) Find the energies and degeneracies of the states in part c) to first order in A. 4 3. H − Ion. Consider an H − ion, which is a hydrogen atom with two electrons. (a) Write down the Hamiltonian for this system. You may assume that the proton is infinitely heavy. (b) Show that, if you neglect the Coulomb interaction between the electrons, the Hamiltonian separates into equations for each electron. (c) Continue to neglect the Coulomb interaction between the electrons, and assume that both electrons are in the ground (n = 1) state. Write a valid wavefunction Ψ1s1s for the electrons, including specification of the possible spin states. Do the same for one electron in a 1s state and the other in a 2s state, Ψ1s2s . (d) Including the Coulomb interaction between electrons, would you expect spin triplet states to be lower or higher in energy than spin singlet states with the same orbital quantum numbers? Why? 5 4. Angular momentum. (a) Sketch the angular and radial forms of the hydrogenic wavefunctions φn`m for the cases (n = 2, ` = 1, m) and (n = 3, ` = 1, m). Make sure you make a separate sketch for each m value. (b) Given eigenstates of angular two independent angular momenta L1 and L2 designated by |`1 , m1 , `2 , m2 i, consider the state 1 1 1 |Ψi = √ |2, 2, 1, −1i + √ |2, 1, 1, 0i − √ |2, 0, 1, 1i. 3 6 2 (2) Show that this state is an eigenstate of total Lz ≡ L1z + L2z with m = 1. (c) Show that |ψi is an eigenstate of total L ≡ L1 + L2 with L = 2. (d) Use the lowering operator to find the eigenstate of L2 and Lz with L = 2 and m = 0 (call it |` = 2, m = 0i) in terms of the |`1 , m1 , `2 , m2 i basis. 6