Question

Suppose N independent harmonic oscillators being equilibrium of the temperature T. The Hamiltonian and the eigenvalue of the single harmonic oscillator are given by \begin{equation} H = \frac{p^2}{2m} + \frac{1}{2}m\omega^2 x^2, \label{1} \end{equation} \begin{equation} E_n = \hbar\omega(n + \frac{1}{2}), \label{2} \end{equation} respectively, where n is an integer satisfying n ? 0. Answer the following questions. 1. Find the partition function of the N-oscillator system. 2. Find the energy expectation value of the N-oscillator system and make its graph as a function of temperature. 3. Find the heat capacity of the N-oscillator system and make its graph as a function of temperature. 4. Discuss the difference between the temperature dependences of the heat capacities of the harmonic oscillator system and the two-level system. 5. Find the energy expectation value and heat capacity of the N-oscillator system by using classical canonical distribution. 6. Discuss the difference between the results obtained by the quantum and classical approaches.

          Suppose N independent harmonic oscillators being equilibrium of the temperature T. The Hamiltonian and the eigenvalue of the single harmonic oscillator are given by
\begin{equation} 
H = \frac{p^2}{2m} + \frac{1}{2}m\omega^2 x^2, \label{1}
\end{equation}
\begin{equation} 
E_n = \hbar\omega(n + \frac{1}{2}), \label{2}
\end{equation}
respectively, where n is an integer satisfying n ? 0. Answer the following questions.
1. Find the partition function of the N-oscillator system.
2. Find the energy expectation value of the N-oscillator system and make its graph as a function of temperature.
3. Find the heat capacity of the N-oscillator system and make its graph as a function of temperature.
4. Discuss the difference between the temperature dependences of the heat capacities of the harmonic oscillator system and the two-level system.
5. Find the energy expectation value and heat capacity of the N-oscillator system by using classical canonical distribution.
6. Discuss the difference between the results obtained by the quantum and classical approaches.

Suppose N independent harmonic oscillators being equilibrium of the temperature T. The Hamiltonian and the eigenvalue of the single harmonic oscillator are given by

H = (p^2)/(2m) + (1)/(2)mω^2 x^2,

En = ħω(n + (1)/(2)),

respectively, where n is an integer satisfying n ? 0. Answer the following questions.
1. Find the partition function of the N-oscillator system.
2. Find the energy expectation value of the N-oscillator system and make its graph as a function of temperature.
3. Find the heat capacity of the N-oscillator system and make its graph as a function of temperature.
4. Discuss the difference between the temperature dependences of the heat capacities of the harmonic oscillator system and the two-level system.
5. Find the energy expectation value and heat capacity of the N-oscillator system by using classical canonical distribution.
6. Discuss the difference between the results obtained by the quantum and classical approaches.

Added by Robert M.

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