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Virial Theorem

by ultimateguy
Tags: solved, theorem, virial
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ultimateguy
#1
Oct12-07, 07:47 PM
P: 126
1. The problem statement, all variables and given/known data
A particle has a potential [tex]\lambda X^n[/tex] and Hamiltonian [tex]H = \frac{P^2}{2m} + V(x)[/tex]

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

[tex]2<T>=n<V>[/tex]


2. Relevant equations



3. 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.
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Gokul43201
#2
Oct12-07, 10:42 PM
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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
#3
Oct12-07, 11:11 PM
P: 126
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
#4
Oct12-07, 11:33 PM
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Virial Theorem

For an operator A, that is not explicitly time-dependent, [itex](i\hbar) dA/dt[/itex] is equal to a commutator. Does that help jog your memory?
ultimateguy
#5
Oct13-07, 12:21 AM
P: 126
Thank you! I solved the problem.


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