In the lecture notes on the Einstein model there is the sentence:
“This oscillator is a so-called bosonic mode: when its wave function is in the n-th excited state, we say that it contains n bosonic excitations.”
How should I imagine the energy states of these bosons? The HO is in the En energy state and thus contains n bosonic excitations, therefore the energy of one boson is not En right? But the energy of all bosons together is En? Do all bosons just have 1 energy state available to them? With energy h_bar\omega_0?
I think I understand it now.
The bosons only have 2 energy levels available to them, 0 and \hbar \omega_0
\langle n \rangle is the average occupation of each energy level.
To get the average total energy of all bosons, you take the sum of the product of \langle n \rangle and the energy state of the boson:
\sum_j \langle n \rangle \varepsilon_j
This gives
\infty \cdot 0 + \frac{1}{e^{\hbar\omega_0 / k_B T} - 1} \hbar\omega_0
Then to find the average total energy of HO you can add the ZP energy to this since it’s a constant and you find E(T) as explained in the lecture notes.
Pretty much, but I would say it slightly differently: each boson occupying a mode with frequency \omega_0 has energy \hbar \omega_0, and there is an average number of n_B bosons occupying each mode.
Yes and in the case of the HO there is only 1 non-zero energy mode that bosons can occupy, the one with frequency \omega_0. If this is what you mean to say then I agree. Let me know if I understood it right.
Indeed, that’s exactly right.
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