PS6_v2 - Physics 115A Problem Set 6 Due Tuesday November 3...

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Physics 115A, Problem Set 6 Due Tuesday, November 3 @ 5pm in the box outside Broida 1019 (PSR) Suggested reading: Griffiths Section 2.5, 2.6 1 SHO in Momentum Space The harmonic oscillator Hamiltonian in any representation is ˆ H = ˆ p 2 2 m + 1 2 2 ˆ x 2 . So let’s consider the simple harmonic oscillator in momentum space, where we define the position and momentum operators to be ˆ x = i ~ ∂p and ˆ p = p . 1. Show that the harmonic oscillator Hamiltonian in momentum space can be re-written in terms of raising and lowering operators a + , a - as ˆ H = ~ ω ( a + a - + 1 2 ). What are a + , a - in terms of p and d dp ? 2. Argue that there is a ground state φ 0 ( p ) annihilated by a - , and find φ 0 explicitly as a function of p . 3. Fourier transform your ground state wavefunction φ 0 ( p ) to position space (you can use p = ~ k to employ our familiar Fourier transform relating x and k ). Show that this is simply our familiar position-space ground state wavefunction ψ 0 ( x ).
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