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HOWARD UNIVERSITY WASHINGTON, DC, 20059 DEPARTMENT OF PHYSICS AND ASTRONOMY (202)-806-6245 (Main Office) 2355 Sixth Str., NW (202)-806-5830 (FAX) —Quantum Mechanics— 1. Develop a variational principle from the Schrödinger equation − h̄2 ~ 2 ∇ ψλ + V (~r)ψλ = Eλ ψλ , 2m where ψλ is the wave-function of the stationary state with energy Eλ , and V (~r) is a nonnegative real potential. R a. Using that d3 ~rψλ∗ ψλ0 = δλ,λ0 , obtain an integral expression for Eλ , in terms of ψλ , ~ λ , their conjugates and V (~r), but no higher derivatives. Show that Eλ ≥ 0. ∇ψ b. Vary this integral expression for Eλ with respect to a small change ψλ → ψλ + δψ, ~ λ. and determine δEλ as an integral expression quadratic in δψ and ∇ψ © ª P c. Using the completeness of the set ψλ , expand δψ = µ cµ ψµ and integrate the ∗ expression for δEλ to obtain δEλ as a function of cµ , cµ , Eµ and Eλ . d. Which ψλ (i.e., which Eλ ) can be determined by minimizing δEλ ? 2. Consider a particle of mass M constrained to move on a circle of radius a in the x, y-plane. a. Write down the Schrödinger equation in terms of the usual cylindrical-polar angle φ. b. Determine the complete set of states, the corresponding energy spectrum and orthonormalize the stationary states. c. Assume now that the particle has charge q and is placed in a small electric field ~ = Eêx . Determine the first non-zero perturbative correction to the energy levels. E ~ = Bêz . Determine the first d. Instead of the electric field, apply a small magnetic field B non-zero perturbative correction to the energy levels. e. What is the degeneracy of the unperturbed system (the one with E = 0 = B)? And with E = 6 0 = B? And with E = 0 6= B? 1 3. For a simple harmonic oscillator of mass m and spring constant k = mω 2 (where ω is the classical frequency), the wave equation is h̄2 d2 ψ 1 2 − + 2 kx ψ = Eψ . 2m dx2 a. Simplify the equation to −ψ 00 + y 2 ψ = 2²ψ, where primes denote differentiation with respect to y: define x = ay and E = b² and the determine a, b. 1 b. For this simplified equation, show that ψ approaches e− 2 y for large y. 1 2 c. Let ψ = e− 2 y f (y), and determine the differential equation for f (y). 2 d. Show that f must be either an even or an odd function of y. e. Show that f cannot be an infinite power series of y. f. Try f0 = 1, f1 = y and f2 = y 2 − c and determine c. Show that, corresponding to these three solutions, ²0 = 12 , ²1 = 32 and ²2 = 52 , ~ def 4. Consider the rôle of the angular momentum in quantum mechanics, where L = ~r ×~ p def ~ = −ih̄ ~r ×∇. ~ becomes L ~ generates infinitesimal rotations when applied to functions of ~r. a. Show that L b. Use this to show further that the angular momentum of a particle, in a central force field, is conserved. (Hint: rotation by a small angle δφ in the direction n̂ produces a ~r → ~r + δφ n̂ ×~r.) 2