Question

Difference between a “mode” and a “state” in quantum mechanics?

Answer 1

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?

Answer 2

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 ωω 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⟩|n⟩. Where the number nn corresponds to the number of quanta or loosely speaking "photons" in the single-mode field.

More general, for each frequency or "mode" ωkωk in the cavity we have a corresponding state vector |nk⟩|nk⟩, that corresponds to the state that the mode ωkωkis in. And using the state vector |nk⟩|nk⟩, we can for example calculate the mean energy ⟨Ek⟩=⟨nk|H^|nk⟩⟨Ek⟩=⟨nk|H^|nk⟩ for the mode ωkωk.

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


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¶potter¶