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Path Integrals — John S. Townsend
Physics — Quantum Mechanics · coherent-states-ehrenfest
Problem Verify that the expectation values
⟨ α ( t ) ∣ x ^ ∣ α ( t ) ⟩ = ℏ 2 m ω 2 ∣ α ∣ cos ( ω t + δ ) \langle\alpha(t)|\hat{x}|\alpha(t)\rangle = \sqrt{\frac{\hbar}{2m\omega}}\,2|\alpha|\cos(\omega t + \delta) ⟨ α ( t ) ∣ x ^ ∣ α ( t )⟩ = 2 mω ℏ 2∣ α ∣ cos ( ω t + δ ) ⟨ α ( t ) ∣ p ^ x ∣ α ( t ) ⟩ = − m ω ℏ 2 2 ∣ α ∣ sin ( ω t + δ ) \langle\alpha(t)|\hat{p}_x|\alpha(t)\rangle = -\sqrt{\frac{m\omega\hbar}{2}}\,2|\alpha|\sin(\omega t + \delta) ⟨ α ( t ) ∣ p ^ x ∣ α ( t )⟩ = − 2 mω ℏ 2∣ α ∣ sin ( ω t + δ ) ∣ α ( t ) ⟩ |\alpha(t)\rangle ∣ α ( t )⟩ α = ∣ α ∣ e − i δ \alpha = |\alpha|e^{-i\delta} α = ∣ α ∣ e − i δ d ⟨ x ⟩ d t = ⟨ p x ⟩ m \frac{d\langle x\rangle}{dt} = \frac{\langle p_x\rangle}{m} d t d ⟨ x ⟩ = m ⟨ p x ⟩ d ⟨ p x ⟩ d t = ⟨ − d V d x ⟩ \frac{d\langle p_x\rangle}{dt} = \left\langle -\frac{dV}{dx}\right\rangle d t d ⟨ p x ⟩ = ⟨ − d x d V ⟩ Browse the problem library