Let's consider the harmonic oscillator immersed in a heat bath of temperature \(T\). For any Hermitian operator \(\hat{A}\), we can define its thermal average denoted as \(\langle\langle\hat{A}angleangle\) through the Boltzmann factors and partition function:

\[\begin{equation*}\langle\langle\hat{A}angleangle=\frac{1}{Z} \sum_{n=0}^{\infty} e^{-\beta E_{n}}\left\langle\psi_{n}|\hat{A}| \psi_{n}\rightangle \tag{12.154}\end{equation*}\]

where \(E_{n}\) is the \(n\) th-energy eigenvalue and \(\left|\psi_{n}\rightangle\) is its corresponding eigenstate. In this problem, we'll consider how a thermal system interfaces with the Heisenberg uncertainty principle.

(a) First, determine the expectation value of the Hamiltonian \(\hat{H}\), the mean value of the energy of this thermal harmonic oscillator.

 You can use the result of Eq. (12.152) in Exercise 12.5

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(b) Show that the expectation values of the position and momentum in this thermal harmonic oscillator are \(0:\langle\langle\hat{x}angleangle=\langle\langle\hat{p}angleangle=0\).

(c) Now, determine the thermal averages of the squared position and momentum, \(\left\langle\left\langle\hat{x}^{2}\rightangle\rightangle,\left\langle\left\langle\hat{p}^{2}\rightangle\rightangle\). What is the Heisenberg uncertainty principle for the thermal harmonic oscillator?

(d) Consider a classical harmonic oscillator with the energy you found in part (a). What is the probability distribution for the position \(x\) in the well? What about for the momentum \(p\) ? What is the product of the classical variances of the position and momentum and how does it compare to the quantum mechanical result?