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I am studying the book Introductory Quantum Optics by Gerry & Knight at the moment and as a reader, I stumble upon their seemingly interchangable use of the tems "mode" and "state". As far as I understand it now, a mode is related to frequency, while states involve energies and particle numbers. Could anyone elaborate on the general difference between these terms in quantum mechanics?

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Imagine with that we have a standing electromagnetic wave inside a cavity, as on page 11 in the book by Gerry & Knight. This cavity supports electromagnetic field modes of many different frequencies, which satisfy the given boundary conditions.

Now suppose that we look at a specific frequency $\omega$ i.e. a specific standing wave which is called a mode of the field. The state that the single-mode field is in, is denoted by the number state $|n\rangle$. Where the number $n$ corresponds to the number of quanta or loosely speaking "photons" in the single-mode field.

More general, for each frequency or "mode" $\omega_k$ in the cavity we have a corresponding state vector $|n_k\rangle$, that corresponds to the state that the mode $\omega_k$ is in. And using the state vector $|n_k\rangle$, we can for example calculate the mean energy $\langle E_k\rangle=\langle n_k|\hat{H}|n_k\rangle$ for the mode $\omega_k$.

I hope that you see the difference now between state and mode and how they are related.

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  • $\begingroup$ Thanks! So as I get it now, there can in fact be different states (corresponding to e.g. different photon numbers and energies) for a single mode (characterised by a frequency or a wave vector). $\endgroup$ Apr 3, 2018 at 10:44
  • $\begingroup$ Yes, exactly, as you're saying. $\endgroup$ Apr 3, 2018 at 11:23

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