Commutation and Eigenfunctions

In summary, the conversation discusses two questions related to operators and their properties. The first question explores the concept of operator commutation and whether there is a simple proof to show that any operator commutes with itself. The second question delves into the properties of eigenvalue problems and whether the equation \hat{Q}^{2}\psi=\lambda^{2}\psi always holds true. The conversation also mentions the existence of a more general statement involving non-commuting operators and their eigenfunctions. However, this statement only holds for commuting operators.
  • #1
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My first question is, does any operator commute with itself? If this is the case, is there a simple proof to show so? If not, what would be a counter-example or a "counter-proof"?

My second question has to do with the properties of an eigenvalue problem. If you have an operator Q such that [tex]\hat{Q}[/tex][tex]\psi[/tex]=[tex]\lambda\psi[/tex], then is the following always true?

[tex]\hat{Q}^{2}[/tex][tex]\psi[/tex]=[tex]\lambda^{2}\psi[/tex]

Since [tex]\hat{Q}^{2}[/tex][tex]\psi[/tex]=[tex]\hat{Q}\hat{Q}[/tex][tex]\psi[/tex]=[tex]\hat{Q}\lambda\psi[/tex]=[tex]\lambda\hat{Q}\psi[/tex]=[tex]\lambda\lambda\psi[/tex]=[tex]\lambda^{2}\psi[/tex].

And is there a more general way of saying this? Like,

[tex]\hat{Q}[/tex][tex]_{i}[/tex][tex]\hat{Q}[/tex][tex]_{j}[/tex][tex]\psi[/tex]=[tex]\hat{Q}[/tex][tex]_{i}[/tex][tex]\lambda[/tex][tex]_{j}[/tex][tex]\psi[/tex]=[tex]\lambda[/tex][tex]_{j}[/tex][tex]\hat{Q}[/tex][tex]_{i}[/tex][tex]\psi[/tex]=[tex]\lambda[/tex][tex]_{i}[/tex][tex]\lambda[/tex][tex]_{j}[/tex][tex]\psi[/tex]

Which should always be true even if the operators don't commute because the lambdas are just scalars right?
 
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  • #2


Of course an operator commutes with itself [Q,Q] is always 0 from the definition of the commutator

For the last part you wouldn't be able to write it like that if the operators don't commute. Commuting operators have simultaneous eigenfunctions, non-commuting ones do not
 

1. What is commutation in quantum mechanics?

Commutation, or commutativity, refers to the property of two operators in quantum mechanics where their order of operation does not affect the result. This means that the operators can be applied in any order and the outcome will be the same.

2. How is commutation related to Heisenberg's uncertainty principle?

The commutation of two operators is related to the uncertainty principle because the degree of commutativity between two operators is inversely proportional to their uncertainty. This means that the more two operators commute, the more precisely the corresponding observables can be measured.

3. What are eigenfunctions in quantum mechanics?

Eigenfunctions, also known as eigenvectors, are special functions in quantum mechanics that correspond to particular properties or observables of a quantum system. They represent the states in which a measurement of the corresponding observable will yield a definite value.

4. How are eigenfunctions related to energy levels?

In quantum mechanics, eigenfunctions are related to energy levels through the eigenvalue equation. This equation represents the relationship between the eigenfunctions and the corresponding energy values of a quantum system. The eigenfunctions with the lowest eigenvalues correspond to the lowest energy levels.

5. Can commutation and eigenfunctions be applied to classical systems?

No, commutation and eigenfunctions are concepts specific to quantum mechanics and cannot be applied to classical systems. In classical mechanics, the observables are always well-defined and the operators always commute, making the concept of eigenfunctions unnecessary.

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