Virial Theorem

  1. [SOLVED] Virial Theorem

    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.
     
    Last edited: Oct 12, 2007
  2. jcsd
  3. Gokul43201

    Gokul43201 11,141
    Staff Emeritus
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    Gold Member

    Chegg
    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.
     
    Last edited: Oct 12, 2007
  4. 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.
     
  5. Gokul43201

    Gokul43201 11,141
    Staff Emeritus
    Science Advisor
    Gold Member

    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?
     
  6. Thank you! I solved the problem.
     
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