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p137bp5 - 2(a What is the density of states ρ E of a 1D...

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Physics 137B, Fall 2007 Problem set 5 : approximations for time-dependent problems Assigned Friday, 28 September. Due in box 5 pm Friday, 5 October. 1. The goal of this problem is to compare time-dependent perturbation theory with the sudden approximation. Suppose that a stationary electron is in the ground state | ↓ of the Hamiltonian (the same as in problem 4 last week) H 0 = B B ¯ h S z (1) at time t 0, and that for times t > 0, there is an additional perturbation H = B B ¯ h S x (2) with B B . (a) Use first-order time-dependent perturbation theory to find the time evolution of the following three quantities for t 0: the transition probability P ↑↓ ( t ) for the system to be found in the state | ↑ ; the expectation value of the original energy function H 0 ; and the expectation value of the final energy function H 0 + H . (b) Use the sudden approximation to find the state of the system for t 0. Convert this back to the t < 0 basis to get the transition probability P ↑↓ ( t ) in terms of the “unperturbed” eigenstates.
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Unformatted text preview: 2. (a) What is the density of states ρ ( E ) of a 1D harmonic oscillator of frequency ω ? Hint: first calculate N ( E ), the number of states of energy less than E , and replace the resulting step-like function by a continuous (in this case, linear) function that agrees on average when N is large. Then use ρ ( E ) = dN/dE . (b) What is the density of states of a 2D harmonic oscillator? You should start by convincing yourself that the energy levels are, for nonnegative integers n x and n y , E n x ,n y = ¯ hω (1 + n x + n y ) . (3) 3. Bransden 9.1. (The goal is to calculate the total transition probability out of the ground state from this perturbation.) 4. Check your answer to problem 3 in the limit τ → ∞ by using the sudden approximation. (Hint: can you make the final Hamiltonian into another harmonic oscillator?) 1...
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