A Modern Approach to Quantum Mechanics · Problem 7.4

Path Integrals — John S. Townsend

Physics — Quantum Mechanics · harmonic-oscillator-momentum-space-ground-state

Problem

Use a^∣0⟩=0\hat{a}|0\rangle = 0 and therefore ⟨p∣a^∣0⟩=0\langle p|\hat{a}|0\rangle = 0 to solve directly for ⟨p∣0⟩\langle p|0\rangle, the ground-state wave function of the harmonic oscillator in momentum space. Normalize the wave function. Hint: Recall the result of Problem 6.2. ⟨p∣x^∣ψ⟩=iℏ∂∂p⟨p∣ψ⟩\langle p|\hat{x}|\psi\rangle = i\hbar\frac{\partial}{\partial p}\langle p|\psi\rangle

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