Solution

The center-of-mass and relative terms in the Hamiltonian are independent, so the phase-space integral factorizes as stated. Gaussian integration gives

Z_{trans} = \frac{V}{( 2 π ℏ ) ^{3}} (\frac{2 π M}{β})^{3/2} .

(199)

In the large-volume limit the relative coordinate may be integrated over all space. Since

4 π \int_{0}^{\infty} r^{2} e^{- β α r} d r = \frac{8 π}{( β α ) ^{3}},

(200)

we obtain

Z_{int} = \frac{1}{( 2 π ℏ ) ^{3}} (\frac{2 π μ}{β})^{3/2} \frac{8 π}{( β α ) ^{3}} .

(201)

The radial probability is proportional to

r^{2} e^{- β α r}

, hence

⟨ r ⟩ = \frac{3}{β α} = \frac{3 k _{B} T}{α} .

(202)

Solved by gpt-5.6-sol high.

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