Exploring Coherent States in the Quantum Oscillator

[nolanp2]

i've just encountered coherent states while studying the quantum oscillator, and I'm trying to understand some of the semiclassical properties of them. can someone give me a brief description of what they represent in the system and of how they vary in time?

 

[strangerep]

i've just encountered coherent states while studying the quantum oscillator, and I'm trying to understand some of the semiclassical properties of them. can someone give me a brief description of what they represent in the system

In this case, coherent states can be described in three equivalent ways.

1) They saturate the Heisenberg uncertainty relation (i.e., minimize the simultaneous
uncertainty in position and momentum). One therefore says that they're "as classical
as possible".

2) They are eigenstates of the annihilation operator.

3) They can be generated by applying a certain operator from the Heisenberg
group to the vacuum state.

and of how they vary in time?

In simple cases, it often happens that coherent states evolve into
coherent states.

For a pedestrian amusing introduction to such things, try the old spr conversation
between Michael Weiss and John Baez on "Photons, Schmotons". It's available
in edited form at: http://math.ucr.edu/home/baez/photon/schmoton.htm

 

[nolanp2]

so a coherent state under the hamiltonian of a harmonic oscillator will be a coherent state for all t? are all coherent states identical?

 

[strangerep]

so a coherent state under the hamiltonian of a harmonic
oscillator will be a coherent state for all t? are all coherent states identical?

They are not identical. The set of coherent states forms an (overcomplete) basis for the
Hilbert space of states of the oscillator. (I.e., any state in the Hilbert space can be
expressed as an integral over the coherent states. "Over"-complete means they are
not mutually orthogonal.)

Try Wikipedia for a bit more info.

If you have access to a University library, try the book by Mandel & Wolf
"Optical Coherence & Quantum Optics". Their section on coherent states
explains quite a lot of interesting stuff.