Quantum Mechanics - Commuting Operators (very quick question)

In summary, if two operators commute, there exists a complete set of common eigenvectors. This is only applicable to Hermitian operators and is often discussed in terms of wave mechanics rather than matrix mechanics.
  • #1
Brewer
212
0
Just a quickie:

If two operators commute, what can be said about their eigenfunctions?

The only thing I can glem from the chapter in my textbook about this is that the eigenfunctions are equal? Is this right, or have I misread it?
 
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  • #2
If two (Hermitian) operators commute, then there exists a complete set of eigenvectors which is common to both operators.
 
  • #3
Not to interrupt, but if I were asking this question, I'd be doing so exclusively in terms of wave mechanics. It seems to be the 'mdern' way to ignore matrix mechanics when teaching QM - at least it is for my department.
 

FAQ: Quantum Mechanics - Commuting Operators (very quick question)

What are commuting operators in quantum mechanics?

Commuting operators in quantum mechanics refer to operators that can be applied in any order without changing the final outcome. In other words, the order in which these operators are applied does not matter.

What is the significance of commuting operators in quantum mechanics?

The significance of commuting operators lies in the fact that they represent physical observables that can be measured simultaneously with no uncertainty. This is a fundamental concept in quantum mechanics and plays a crucial role in understanding the behavior of quantum systems.

Can all operators in quantum mechanics commute with each other?

No, not all operators in quantum mechanics commute with each other. Only a specific set of operators that satisfy certain mathematical conditions can commute with each other. These conditions are known as the commutation relations.

How do commuting operators relate to the uncertainty principle?

Commuting operators are directly related to the uncertainty principle in quantum mechanics. The uncertainty principle states that certain physical quantities, such as position and momentum, cannot be known simultaneously with absolute precision. Commuting operators represent physical observables that can be measured simultaneously with no uncertainty, while non-commuting operators represent physical observables that have inherent uncertainty.

Can commuting operators be used to simplify calculations in quantum mechanics?

Yes, commuting operators are often used to simplify calculations in quantum mechanics. Since the order of application does not matter, we can rearrange the operators to make calculations easier. This is particularly useful in systems with multiple operators and can save time and effort in solving complex problems.

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