Difference between a "mode" and a "state" in quantum mechanics? 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?  
 A: 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.
