and . Allow this number to be . Since the total number of normal modes is , the function is given by:

The integration is performed over all frequencies of the crystal. Then the internal energy will be given by:

In quantum mechanics

Bound states in quantum mechanics are analogous to modes. The waves in quantum systems are oscillations in probability amplitude rather than material displacement. The frequency of oscillation, , relates to the mode energy by where is the Planck constant.

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