Calculating Average Values and Proving Inequality for Particle Potential - N^nX

[ultimateguy]

[SOLVED] Virial Theorem

Homework Statement

A particle has a potential $$\lambda X^n$$ and Hamiltonian $$H = \frac{P^2}{2m} + V(x)$$

Knowing that the commutator of H and XP is $$i\hbar(n\lambda X^n - \frac{P^2}{m})$$, find the average values <T> and <V> and verify that they satisfy:

$$2<T>=n<V>$$

Homework Equations

The Attempt at a Solution

The question asked to calculate the commutator and that is what I found, but I'm lost as to how to get the average values and proove the inequality.

 

[Gokul43201]

The question doesn't actually ask you to calculate the commutator; it gives you the value of the commutator (though with a sign error; it should read ...-P^2/m).

The next step is to recall what the commutator of any operator with the Hamiltonian gives you (Hint: Heisenberg EoM). After that you just have to take the time average on both sides, and take the limit of loooong times.

 

[ultimateguy]

The question asked to calculate the [H, XP] commutator, I just didn't write it because I already found it and wanted to save time.

I'm not sure I understand the hint.

 

[Gokul43201]

For an operator A, that is not explicitly time-dependent, $(i\hbar) dA/dt$ is equal to a commutator. Does that help jog your memory?

 

[ultimateguy]

Thank you! I solved the problem.