Kinetic and potential energies of a harmonic oscillator

[Von Neumann]

Problem:

In a harmonic oscillator

\left\langle V \right\rangle=\left\langle K \right\rangle=\frac{E_{0}}{2}

How does this result compare with the classical values of K and V?

Solution:

For a classical harmonic oscillator

V=1/2kx^2
K=1/2mv^2

I don't really know where to begin. Is it safe to say that quantum oscillator must depend on the average values of the kinetic and potential energies? Also, the values of the energies in the classical system are conserved in such a way to conserve total mechanical energy.

 

[Pagan Harpoon]

Is the problem statement telling you that the expectation values for the kinetic and potential energies are each E/2 in the quantum case?

If so then perhaps it wants you to calculate the average values in the classical case and show that they're the same.

 

[Von Neumann]

Is the problem statement telling you that the expectation values for the kinetic and potential energies are each E/2 in the quantum case?

If so then perhaps it wants you to calculate the average values in the classical case and show that they're the same.

Oh yes, I've already done that. I calculated <U> and <K> separately.

 

[Pagan Harpoon]

Then haven't you already solved the problem? If you have the expectation values for the kinetic and potential energies in both the classical and quantum cases, then you can compare them and see that they are the same.

Is the question asking you for something more?

 

[Von Neumann]

The question is simply asking to compare the expectation values <V> and <K> of the quantum harmonic oscillator (given above) with the classical values of V and K.

 

[Pagan Harpoon]

If I were faced with that question, I would calculate the average values for V and K in the classical case (which you have done) and then comment that they are the same as the expectation values in the quantum case - that's how they compare, they're the same.

What more do you want to do?