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POSTLUDE: SYMMETRY AND THE CONCEPTUAL FOUNDATIONS OF THERMOSTATISTICS
Physics — Statistical Mechanics · variational-perturbation-harmonic-oscillator
Problem Derive equation (h) of Example 1, first showing that for a harmonic oscillator
⟨ n ⟩ = − 1 z ′ ∂ z ′ ∂ ( β ℏ ω 0 ) \langle n \rangle = -\frac{1}{z'} \frac{\partial z'}{\partial(\beta\hbar\omega_0)} ⟨ n ⟩ = − z ′ 1 ∂ ( β ℏ ω 0 ) ∂ z ′ ⟨ n 2 ⟩ = 1 z ′ ∂ 2 z ′ ∂ ( β ℏ ω 0 ) 2 \langle n^2 \rangle = \frac{1}{z'} \frac{\partial^2 z'}{\partial(\beta\hbar\omega_0)^2} ⟨ n 2 ⟩ = z ′ 1 ∂ ( β ℏ ω 0 ) 2 ∂ 2 z ′ z ′ = e β ℏ ω 0 / 2 z = ∑ n = 0 ∞ e − β ℏ ω 0 n z' = e^{\beta\hbar\omega_0/2} z = \sum_{n=0}^{\infty} e^{-\beta\hbar\omega_0 n} z ′ = e β ℏ ω 0 /2 z = n = 0 ∑ ∞ e − β ℏ ω 0 n Browse the problem library