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Virial Theorem 
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#1
Oct1207, 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. 


#2
Oct1207, 10:42 PM

Emeritus
Sci Advisor
PF Gold
P: 11,155

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. 


#3
Oct1207, 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. 


#4
Oct1207, 11:33 PM

Emeritus
Sci Advisor
PF Gold
P: 11,155

Virial Theorem
For an operator A, that is not explicitly timedependent, [itex](i\hbar) dA/dt[/itex] is equal to a commutator. Does that help jog your memory?



#5
Oct1307, 12:21 AM

P: 126

Thank you! I solved the problem.



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