Exploring Coherent States in the Quantum Oscillator

In summary, coherent states in the quantum oscillator system can be described as states that minimize the uncertainty in position and momentum, are eigenstates of the annihilation operator, and can be generated by applying a certain operator to the vacuum state. In simple cases, they evolve into coherent states and are not mutually identical. For more information, one can refer to the book "Optical Coherence & Quantum Optics" or Wikipedia.
  • #1
nolanp2
53
0
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?
 
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  • #2
nolanp2 said:
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
 
  • #3
so a coherent state under the hamiltonian of a harmonic oscillator will be a coherent state for all t? are all coherent states identical?
 
  • #4
nolanp2 said:
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.
 

FAQ: Exploring Coherent States in the Quantum Oscillator

1. What is a coherent state in a quantum oscillator?

A coherent state in a quantum oscillator is a quantum state that exhibits classical-like properties. It is described by a wave function that is a Gaussian distribution with a well-defined phase and amplitude. These states are also known as minimum uncertainty states, as they have the lowest possible uncertainty in position and momentum.

2. How are coherent states created in a quantum oscillator?

Coherent states can be created in a quantum oscillator by applying a coherent state operator to the vacuum state of the oscillator. This operator is a displacement operator that shifts the position and momentum of the vacuum state by a certain amount, creating a coherent state with a well-defined amplitude and phase.

3. What are the properties of coherent states in a quantum oscillator?

Coherent states in a quantum oscillator have several important properties. They are eigenstates of the annihilation operator, meaning they are stable over time. They also exhibit classical-like behavior, such as following a predictable trajectory in phase space. Additionally, they have the lowest possible uncertainty in position and momentum.

4. How do coherent states differ from other quantum states?

Coherent states differ from other quantum states in that they are eigenstates of the annihilation operator and exhibit classical-like behavior. They also have the lowest possible uncertainty in position and momentum, making them more stable and predictable compared to other quantum states.

5. What are the applications of exploring coherent states in a quantum oscillator?

Exploring coherent states in a quantum oscillator has many applications in quantum mechanics, such as in quantum computing and quantum communication. Coherent states are also used in quantum metrology, where they can improve the precision of measurements. Additionally, coherent states play a crucial role in understanding the dynamics and behavior of quantum systems.

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