Fermi · Concept Library · Courses · Books · Problems · Simulations · Math Wiki · High School Fermi › Problems A Modern Approach to Quantum Mechanics · Problem 6.2
The One-Dimensional Harmonic Oscillator — John S. Townsend
Physics — Quantum Mechanics · position-operator-momentum-space
Problem Show that
⟨ p ∣ x ^ ∣ ψ ⟩ = i ℏ ∂ ∂ p ⟨ p ∣ ψ ⟩ \langle p|\hat{x}|\psi\rangle = i\hbar \frac{\partial}{\partial p}\langle p|\psi\rangle ⟨ p ∣ x ^ ∣ ψ ⟩ = i ℏ ∂ p ∂ ⟨ p ∣ ψ ⟩ ⟨ φ ∣ x ^ ∣ ψ ⟩ = ∫ d p ⟨ p ∣ φ ⟩ ∗ i ℏ ∂ ∂ p ⟨ p ∣ ψ ⟩ \langle \varphi|\hat{x}|\psi\rangle = \int dp\, \langle p|\varphi\rangle^* i\hbar \frac{\partial}{\partial p}\langle p|\psi\rangle ⟨ φ ∣ x ^ ∣ ψ ⟩ = ∫ d p ⟨ p ∣ φ ⟩ ∗ i ℏ ∂ p ∂ ⟨ p ∣ ψ ⟩ Browse the problem library