Harmonic oscillators and commutators.

In summary, the commutation relation between two independent harmonic oscillators cannot be mixed, as the operators for each oscillator commute only with themselves and not with the operators of the other oscillator. This is due to the fact that the position and momentum operators for one oscillator do not affect the states of the other oscillator. Any other mix of these operators would also result in a commutator of 0.
  • #1
Beer-monster
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0

Homework Statement



If we have a harmonic oscillator with creation and annhilation operators [itex]a_{-} a_{+} [/itex], respectively. The commutation relation is well known:

[tex] [a_{+},a_{-}] = I [/tex]

However, if we have two independent oscillators with operators [itex]a'_{-} a'_{+} [/itex]

As the operators are the same function would they commute if mixed.

i..e is [tex] [a_{+},a'_{-}] = I [/tex]

Still valid?
 
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  • #2
When you say two operators A and B commute, it means AB=BA so that [A,B]=AB-BA=0.

As to your question about the commutation relation, the answer is no. The creation and annihilation operators for the one oscillator commute with the operators for the other oscillator.
 
Last edited:
  • #3
So [itex][a_{+},a'_{-}] = 0 [/itex]

Is there any reason for this beside the operators for one oscillator have no effect on the states of another?

Would assume that any other mix of these operators would commute?

e.g [a'_{-}a_{+},a'_{+},a_{+}] = 0
 
  • #4
The position and momentum operators, x1 and p1, for one oscillator commute with the position and momentum operators, x2 and p2 for the second oscillator:
\begin{align*}
[x_i, x_j] &= 0 \\
[p_i, p_j] &= 0 \\
[x_i, p_j] &= i\hbar\delta_{ij}
\end{align*}It follows from these that a primed and unprimed operator commute.
 
  • #5


I would say that the commutation relation between two different sets of operators is not necessarily the same as the commutation relation between two identical sets of operators. In the case of the harmonic oscillators, the commutation relation is valid because the operators are defined for the same oscillator. However, when we have two independent oscillators with different operators, the commutation relation may not hold true. It would depend on the specific definitions and properties of the operators and how they interact with each other. Therefore, it is important to carefully consider the context and definitions when determining the commutation relation between different operators.
 

1. What is a harmonic oscillator?

A harmonic oscillator is a physical system that follows a periodic motion, meaning it repeats the same motion over and over again. It is characterized by a restoring force that is directly proportional to the displacement from equilibrium, and the motion is governed by a simple harmonic motion equation.

2. What is a commutator in relation to harmonic oscillators?

A commutator is an operator used in quantum mechanics to describe the relationship between two observables. In the context of harmonic oscillators, the commutator is used to describe the relationship between position and momentum operators, which are used to determine the position and momentum of a harmonic oscillator at any given time.

3. How are harmonic oscillators and commutators related to quantum mechanics?

Harmonic oscillators and commutators are both important concepts in quantum mechanics. Harmonic oscillators are used to model the behavior of particles on a microscopic scale, while commutators are used to describe the relationship between observables in quantum mechanics.

4. What is the significance of studying harmonic oscillators and commutators?

Studying harmonic oscillators and commutators is important because it allows scientists to understand the behavior of particles on a quantum level. It also helps in the development of quantum technologies, such as quantum computers and quantum cryptography.

5. Can harmonic oscillators and commutators be used in other fields of science?

Yes, harmonic oscillators and commutators have applications in various fields of science, including physics, chemistry, and engineering. They are used to model and understand the behavior of systems that exhibit periodic motion, such as molecules, sound waves, and electrical circuits.

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