Question

Relative kinetic energy of an atom in anharmonic oscillator

Answer 1

Answer:

For the quantum mechanical oscillator, the oscillation frequency of a given normal mode is still controlled by the mass and the force constant (or, equivalently, by the associated potential energy function). However, the energy of the oscillator is limited to certain values. The allowed quantized energy levels are equally spaced and are related to the oscillator frequencies as given by Equation 5.4.1 and Figure 5.4.1 .

Ev=(v+12)ℏω=(v+12)hν(5.4.1)

with

v=0,1,2,3,⋯∞(5.4.2)

In a quantum mechanical oscillator, we cannot specify the position of the oscillator (the exact displacement from the equilibrium position) or its velocity as a function of time; we can only talk about the probability of the oscillator being displaced from equilibrium by a certain amount. This probability is given by

PQ→Q+dQ=∫Q+dQQψ∗v(Q)ψv(Q)dQ(5.4.3)

We can, however, calculate the average displacement and the mean square displacement of the atoms relative to their equilibrium positions. This average is just ⟨Q⟩ , the expectation value for Q , and the mean square displacement is ⟨Q2⟩ , the expectation value for Q2 . Similarly we can calculate the average momentum ⟨PQ⟩ , and the mean square momentum ⟨P2Q⟩ , but we cannot specify the momentum as a function of time.

Physically what do we expect to find for the average displacement and the average momentum? Since the potential energy function is symmetric around Q=0 , we expect values of Q>0 to be equally as likely as Q<0 . The average value of Q therefore should be zero.

These results for the average displacement and average momentum do not mean that the harmonic oscillator is sitting still. As for the particle-in-a-box case, we can imagine the quantum mechanical harmonic oscillator as moving back and forth and therefore having an average momentum of zero. Since the lowest allowed harmonic oscillator energy, E0 , is ℏω2 and not 0, the atoms in a molecule must be moving even in the lowest vibrational energy state. This phenomenon is called the zero-point energy or the zero-point motion, and it stands in direct contrast to the classical picture of a vibrating molecule. Classically, the lowest energy available to an oscillator is zero, which means the momentum also is zero, and the oscillator is not moving.