p137bmt1sol - Physics 137B, Fall 2007 Quantum Mechanics II...

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Physics 137B, Fall 2007 Quantum Mechanics II Midterm I 10/9/2007, 9:40-11:00 a.m. No books, notes, or calculators are allowed. Please start each of the 4 problems on a fresh side. You should make an effort to answer every problem. I. Consider a one-dimensional harmonic oscillator: H = p x 2 2 m + kx 2 2 . (1) Some useful facts: the energy levels of the one-dimensional harmonic oscillator are E = ( n +1 / 2)¯ , n = 0 , 1 , . . . , and the lowest two wavefunctions are of the form ψ 0 = α 1 exp( - βx 2 ) , ψ 1 = α 2 x exp( - βx 2 ) (2) where α 1 , α 2 , β are positive constants. All the following questions are for a single spinless particle. (a) Using the exact values for the energy levels and ω = p k/m , find the change in the ground state energy induced by a small increase in the mass m m + dm , to first order in dm . The ground state energy is ¯ hω/ 2 = ¯ h p k/ ( m + dm ) / 2 ¯ h p ( k/m )(1 - dm/m ) / 2 h/ 2) p k/m (1 - dm/ 2 m ), where means that we have kept only linear order in dm . So the change is ¯ h p k/m ( - dm/ 4 m ). (b) Explain how you would use first-order perturbation theory for the change in (a) by writing the change induced by dm as a perturbation. Write, but do not calculate, the integral that will give the first-order perturbation theory result for the energy change. The perturbation Hamiltonian is - ( p 2 / 2 m )( dm/m ). The first-order energy correction is then h ψ 0 | - ( p 2 / 2 m )( dm/m ) | ψ 0 i (3) . You can check that this is indeed equal to the above result, since the expected kinetic energy in the harmonic oscillator is half the total energy. (c) Is the second order correction to the ground state energy positive, negative, or zero? The second-order correction from the ground state is always either negative or zero. Since all even states are connected by the perturbation Hamiltonian in this case, the second-order shift is negative. II. (a) Consider the 2p levels of the hydrogen atom and ignore fine structure corrections. Including spin, how many levels are there (for a single electron)? There are six levels with ` = 1 and s = 1 / 2, which we can label by two quantum numbers m ` = - 1 , 0 , 1 and m s = - 1 / 2 , 1 / 2. Without fine structure, these are all degenerate.
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p137bmt1sol - Physics 137B, Fall 2007 Quantum Mechanics II...

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