Oscillators, 3-dimensional harmonic energy expression

We have demonstrated here that for the one-dimensional harmonic oscillator the integral required in the Euler equation, involving the functional derivative 8t/8p, can be exactly expressed in terms of the total kinetic energy. Indeed, the relation, involving a factor of 3, is exactly that given by the TF statistical theory. This latter theory gives for the density in d dimensions... 

In both cases, in the limit q -> 1 (r -> 0) one gets the standard expression for the energy of the one-dimensional harmonic oscillator ... 

Assuming the adsorbed molecule behaves as a three-dimensional harmonic oscillator and that the internal partition function (for electronic and nuclear energies) remains unchanged upon adsorption, A° may be expressed as ... 

The quantum mechanical information that follows from a normal mode analysis must reveal the same mechanical equivalence to a set of disconnected oscillators as the classical analysis. Each such oscillator (normal mode of vibration) can exist in any of the states possible for a one-dimensional harmonic oscillator. Each has its own contribution to the energy of the system, and thus, the Hamiltonian in Equation 7.35 corresponds to a quantum mechanical energy level expression... 

Consider a general three dimensional harmonic oscillator with force constants /C , k, and k. Write down the total energy expression for the energy levels of this system. What is the ground state energy ... 

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