7
$\begingroup$

Given the differential operator: $$A=\exp(-\beta H)$$ where $$H=\frac{1}{2}\left( -\frac{d^2}{dx^2}+x^2 \right)$$ and $\beta\gt 0$ How can I get the trace of this operator? Thanks in advance.

$\endgroup$
  • 1
    $\begingroup$ In general, this Wikipedia entry may help; in the specific case, Jon's answer gives you how to compute. $\endgroup$ – Willie Wong May 24 '13 at 15:04
7
$\begingroup$

The trace of this operator is easily obtained in the following way: $$ Z={\rm Tr}\exp(-\beta H). $$ that is equivalent to $$ Z=\sum_n \langle n|\exp(-\beta H)|n\rangle. $$ Assuming $H|n\rangle=E_n|n\rangle$, this is just $$ Z=\sum_n\exp(-\beta E_n). $$ Your case is the harmonic oscillator $E_n=n+\frac{1}{2}$ and the sum is just a geometric series easy to perform.

$\endgroup$
  • 2
    $\begingroup$ In general, the trace of an operator is the sum over sandwiching it between eigenstates. Thus $\text{tr} f(A)=\sum f(\lambda_n)$. $\endgroup$ – Sharkos May 24 '13 at 15:06
  • $\begingroup$ @Sharkos: Correct. $\endgroup$ – Jon May 24 '13 at 15:08
  • $\begingroup$ Thanks, though I wasn't seeking confirmation, but merely emphasizing the underlying idea for the OP's benefit... $\endgroup$ – Sharkos May 24 '13 at 15:10

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.