2.3. Theory QuestionsΒΆ
A quantum harmonic oscillator has energy levels:
\[ E_n = \left(n + \frac{1}{2}\right)\hbar \omega. \]Write down the corresponding canonical partition function \(Z(N,V,T)\). Note that in this case, the partition function forms an infinite geometric series, and can be rewritten in terms of the \(n \rightarrow \infty\) limit of the series. From the result you obtain, derive the expectation value of the energy. Use the limit of the geometric series for \(Z\), rather than the sum-based form.
Derive the Boltzmann distribution, (2.24), from (2.10), using the expectation value of the particle number in state \(s\), \(N_s\).
Bonus: Show that, based on (2.17), the canonical partition function (2.18) is obtained from (2.13) if the Hamiltonian admits an orthonormal eigenbasis \(\left\{\ket{\Psi_i}\right\}\) and the commutator \(\left[\hat{\mathrm{H}},\hat{\mathrm{O}}\right]\) vanishes. (There is no need to use the commutator itself, applying conditions that follow from vanishing commutators is sufficient. If you wish to provide an extended derivation in a general basis, recall that the time-independent case applies.)
Bonus: Show from (2.12) and (2.13) that the \(p_i\) are indeed equal to \(\frac{1}{Z}e^{-\beta E_i}\) for pure states, given that there exists a common eigenbasis \(\left\{\ket{\Psi_i}\right\}\) to the total Hamiltonian and explain the origin of this restriction. Explain why the density operator and the expression for the expectation value of an observable assume a general form (2.13) and (2.17), rather than being defined directly in terms of (2.15) and (2.18). (You may link your answer to the assumptions made in 3.).