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I have seen the Hamiltonian for a heat bath written as:

$$ H_B = \hbar \int_0^\infty \omega b(\omega)^\dagger b(\omega) d\omega $$

I was hoping to understand this equation better. This suggests that the heat bath is written as a sum of harmonic oscillators with raising/lowering operators $b^\dagger(\omega)$ and $b(\omega)$. I understand that is that this is just the continuous limit of the quantised electromagnetic field.

Please could someone explain the following:

1) Does the raising operator $b^\dagger(\omega)$ correspond directly to a mode in the heat bath with frequency $\omega$? ie does $b^\dagger(\omega)b(\omega)$ give us the number operator for the number of excitations of frequency $\omega$? or does the argument $\omega$ mean something else?

2) How does one alter this to change the temperature of the heat bath? My current interpretation is that this equation is for the vacuum/zero temperature case, so do we just add a Boltzmann factor inside the integral, to change the temperature or is there more subtlety required?

3) If you have any other intuitions/tips/tricks for thinking about this equation, I would appreciate them!

Many thanks!

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  • $\begingroup$ 1) Yes, $b(\omega)$ corresponds to a mode of frequency $\omega$. 2) The Hamiltonian is always temperature independant. It goes into the Boltzmann factor $\exp(-\beta H_B)$, not the other way round. $\endgroup$ – By Symmetry Feb 12 at 14:06
  • $\begingroup$ @BySymmetry That looks like an answer... $\endgroup$ – Mark Mitchison Feb 12 at 15:01
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  1. Yes the raising operator $b(\omega)^\dagger$ corresponds to a mode of frequency omega and $n(\omega) = b(\omega)^\dagger b(\omega)$ count the number of excitations. You can see that the mode has a frequency of $\omega$ because that is the prefactor of the mode in the Hamiltonian, so one excitation of that mode increases the energy by $\omega$.
  2. The Hamiltonian, quite generally, does not depend on temperature. The Hamiltonian tells you the possible energy levels of the system (and their degeneracy). Temperature (through the Boltzmann distribution) is related to how likely you are to find the system in a given energy level. You need to have the Hamiltonian first before you start talking about distributions or temperatures. Consequently the Hamiltonian should always be deperature independant.
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