Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 2 / 37C / b / i

Table of contents
- Solution i

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} .

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}},

we obtain

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

The radial probability is proportional to

r^{2} e^{- β α r}

, hence

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

 Ancestors (11)

 View article source

 Discussion (0)

There are no discussions about this article yet.

 Articles by others on the same topic (0)

There are currently no matching articles.

  See all articles in the same topic Create my own version