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The problem of a quantum oscillator coupled to an independent-oscillator model of a heat bath is discussed. The transformation to normal coordinates is explicitly constructed using the method of Ullersma. With this transformation an alternative derivation of an exact formula for the oscillator free energy is constructed. The various contributions to the oscillator energy are calculated, with the aim of further understanding this formula. Finally, the limitations of linear coupling models, such as that used by Ullersma, are discussed in the form of some critical remarks

Independent oscillator model of a heat bath: Exact diagonalization of the Hamiltonian