1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Thermal average

  1. Nov 4, 2011 #1
    1. The problem statement, all variables and given/known data
    2. Relevant equations


    I read a book where a free real scalar field with Hamiltonian

    [tex] H = \int \dfrac{\mathrm{d}^{3}p}{(2 \pi)^{3}} \, E_{\vec{p}} a_{\vec{p}}^{\dagger} a_{\vec{p}} [/tex]

    is beeing discussed.

    Note that:

    [tex] E_{\vec{p}} = \sqrt{\vert \vec{p} \vert^{2} + m^{2}} [/tex]


    [tex] \left[ a_{\vec{p}}, a_{\vec{q}}^{\dagger} \right] = (2 \pi)^{3} \delta^{3}(\vec{p} - \vec{q}) [/tex].

    Now there is the statement that the thermal average of the operator
    [tex] a_{\vec{p}}^{\dagger} a_{\vec{q}} [/tex] is given by:

    [tex] \left \langle a_{\vec{p}}^{\dagger} a_{\vec{q}} \right \rangle = \dfrac{1}{Tr \left[ \mathrm{e}^{-\beta H} \right]} Tr\left[\mathrm{e}^{-\beta H} a_{\vec{p}}^{\dagger} a_{\vec{q}} \right] = \dfrac{1}{\mathrm{e}^{\beta E_{\vec{p}}} -1} (2 \pi)^{3} \delta^{3}(\vec{p} - \vec{q}) [/tex]

    I don't know how to obtain this average value.

    3. The attempt at a solution

    When I first start to compute [tex] Tr \left[ \mathrm{e}^{-\beta H} \right] [/tex] I end up with an ill-defined expression:

    [tex] Tr \left[ \mathrm{e}^{-\beta H} \right] = \int \dfrac{\mathrm{d}^{3} k}{(2 \pi)^{3}} \left \langle \vec{k} \right| \mathrm{e}^{-\beta H} \left| \vec{k} \right \rangle = \int \dfrac{\mathrm{d}^{3} k}{(2 \pi)^{3}} (2 \pi)^{3} \delta^{3}(0) [/tex]

    where I used

    [tex] H \left| \vec{k} \right \rangle = \int \dfrac{\mathrm{d}^{3} k}{(2 \pi)^{3}} E_{\vec{p}} a_{p}^{\dagger} a_{\vec{p}} a_{\vec{k}}^{\dagger} \vert 0 \rangle = \int \dfrac{\mathrm{d}^{3} k}{(2 \pi)^{3}}

    E_{\vec{p}} \delta^{3}(\vec{p} - \vec{k}) (2 \pi)^{3} \vert \vec{p} \rangle = E_{\vec{k}} \vert \vec{k} \rangle [/tex]

    I think the problem is that I used momentum states as eigenstates of the Hamiltonian, but they are not normalizable.

    Does anyone have an idea how to compute this thermal average?

    (If I have the simple Hamiltonian [tex] H = \omega a^{\dagger} a [/tex] and consider a complete set of eigenstates |n> it is very easy to compute this average. But in the case described above I don't know how to do that).

    I hope someone can help me.


    The same problem is also described here (page 5, eq. (3.10))
    http://www.actaphys.uj.edu.pl/vol38/pdf/v38p3661.pdf [Broken]
    Last edited by a moderator: May 5, 2017
  2. jcsd
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted