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Harmonic oscillator at finite temperature: taking expectation values of operators

I have the Hamiltonian of an harmonic oscillator (with $\hbar=1$) $$ H = \omega \left(a^\dagger a + \dfrac{1}{2} \right) \;, $$ and the associated (canonical) partition function $$ Z = \text{Tr}\left[...
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Role of thermal fluctuations in restoring the symmetry in finite systems

A symmetry is spontaneously broken in a system with infinite number of degrees of freedom (DOF), when the system finds itself in the ground state that breaks the symmetry of the Hamiltonian. For ...
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What's the true reason behind thermal expansion?

Thermal expansion is a normal concept everyday. There are 2 explanations: 1, thermal expansion result in stress, then result in deformation 2, thermal expansion result in deformation, then result in ...