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Particle in a finite two-dimensional harmonic trap

For

H = \frac{p _{x}^{2} + p _{y}^{2}}{2 m} + \frac{κ}{2} (x^{2} + y^{2}), x^{2} + y^{2} < R^{2},

put

a = κ R^{2} /2

. The normalized classical one-particle partition function is

Z_{1} = \frac{4 π ^{2} m}{h ^{2} β ^{2} κ} (1 - e^{- β a}) .

Its mean total and potential energies are

⟨ E ⟩ = \frac{2}{β} - \frac{a}{e ^{β a} - 1}, ⟨ V ⟩ = \frac{1}{β} - \frac{a}{e ^{β a} - 1} .

Table of contents 81 1
- Hard-wall correction to harmonic equipartition Particle in a finite two-dimensional harmonic trap 47

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