Potential/Kinetic Energy of Particles in Harmonic Oscillator

[messier992]

Homework Statement

I'm trying to reconcile the answers to two questions regarding the average potential and kinetic energies in simple harmonic oscillator

Question 1:
The average potential energy of the vibrational motion in the ground state of a
diatomic molecule is 12 meV. The average kinetic energy of the vibrational motion
in that state is:
Answer: 12meV?

Question 2:
The ground state energy of a particle in a harmonic oscillator potential is E. The average kinetic energy of that particle in that state is

Homework Equations

Potential Energy = 1/2*k*x^2
Total Energy = Kinetic + Potential Energy

The Attempt at a Solution

To find the ratio of potential to kinetic energy:
the total potential energy is given by the integral of x^2 from -1 to 1 => 2/3
the total kinetic energy is given by the integral of 1-x^2 from -1 to 1 => 4/3

Therefore, the average kinetic energy should be twice the average potential energy

 

[PeroK]

The Attempt at a Solution

To find the ratio of potential to kinetic energy:
the total potential energy is given by the integral of x^2 from -1 to 1 => 2/3
the total kinetic energy is given by the integral of 1-x^2 from -1 to 1 => 4/3

Therefore, the average kinetic energy should be twice the average potential energy

Where are you getting this from?

 

[jbriggs444]

the total potential energy is given by the integral of x^2 from -1 to 1 => 2/3
the total kinetic energy is given by the integral of 1-x^2 from -1 to 1 => 4/3

If you are integrating for displacements ranging from -1 to +1 and expecting to compute an average over time you should first have reason to believe that the oscillator spends an equal amount of time near each possible displacement.

But it is clear that this is not the case. An oscillator spends more time (proportionately) at its extremes and less near the center. You need to integrate over something that changes smoothly with time. A phase angle, perhaps?