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===Quantum Harmonic Oscillator===
The most common approximation to the vibrational partition function uses a model in which the [[vibrational eigenmodes]] or [[normal mode|vibrational normal modes]] of the system are considered to be a set of uncoupled quantum harmonic oscillators. It is a first order approximation to the partition function which allows one to calculate the contribution of the vibrational degree of freedom of molecules towards thermodyanmic variables. A quantum harmonic oscillator has an energy spectrum characterized by:
<math>E_{j,i}=\hbar\omega_j(i+\frac{1}{2})</math>
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