The Hamiltonian (ignores vacuum energy), [tex] H = \hbar\omega_{p}a_{p}^+a_{p} [/tex], represents some cavity at temperature T. For simplicity assume the cavity only supports a single mode.(adsbygoogle = window.adsbygoogle || []).push({});

[tex] H = \hbar\omega_{m} a_{m}^+ a_{m} [/tex]

1) Given that in thermal equilibrium the probability of a system to have energy E is proportional to ~exp(-E/kT), give an expression for [tex]\rho_{m}(n) [/tex] where:

[tex] \hat{\rho}_{m} = \sum_{n}\rho_{m}(n) | n>_{m} _{m}<n| [/tex]

The constant is independent of n

2) Find this constant by imposing condition [tex] Tr( \rho_{m}) = 1 [/tex]

For the first part since the constant is independent of n and the Hamilitonian is governed by [tex] \hbar\omega a^+a[/tex] where [tex]a^+a[/tex] equals n so in order to isolate the n component from the Hamiltonian and write an expression [tex]\rho[/tex] I think we should start by evaluating <m|[tex]\rho[/tex]|n> where we will get a summation expression that needs to be evaluated. This is what I think we should do and I tried it but it is not taking me anywhere..I am afraid that I don't even know what I am doing. I just need a small hint to get started and then I will be all set...

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# Homework Help: Thermal States of Light

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