Does \(A e^{iA} = e^{iA} A\)?

russdot · 2009-10-26T23:55:34+00:00

If A is an operator, is it correct/allowed to say: Ae^{iA} = e^{iA}A Thanks

Thread starter russdot
Start date Oct 26, 2009
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Operators

In summary, commuting operators refer to two operators in quantum mechanics that can be applied in any order without changing the outcome. This is important because it allows for simplified calculations and the ability to measure observables simultaneously. The significance of commuting operators lies in their role as physical observables that can be measured simultaneously. To determine if two operators commute, we can use the commutator, and non-commuting operators can still be used in quantum mechanics but with some limitations.

Oct 26, 2009

russdot

If A is an operator, is it correct/allowed to say:
[tex]Ae^{iA} = e^{iA}A[/tex]

Thanks

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Oct 26, 2009

Ben Niehoff

Science Advisor

Gold Member

Think of how [itex]e^{A}[/itex] is defined.

Oct 26, 2009

russdot

Ah yes, so since [tex]e^{A}[/tex] is defined with a power series [tex]e^{A} = 1 + A + \frac{A^{2}}{2!} + \frac{A^{3}}{3!} + ...[/tex] and A commutes with itself then A would commute with [tex]e^{A}[/tex]
Thanks!

FAQ: Does \(A e^{iA} = e^{iA} A\)?

What is a commuting operator?

A commuting operator refers to two operators that can be applied in any order without changing the outcome. In other words, the order in which the operators are applied does not matter, and they "commute" with each other.

Why is it important for operators to commute?

Commuting operators are important in quantum mechanics because they allow us to simplify complex mathematical calculations. By knowing that two operators commute, we can rearrange them in any order, making the calculations more manageable and efficient.

What is the significance of commuting operators in quantum mechanics?

Commuting operators play a crucial role in quantum mechanics as they represent physical observables that can be measured simultaneously. This means that if two operators commute, we can measure both observables at the same time without affecting the outcome.

How can we determine if two operators commute?

To determine if two operators commute, we can use the commutator, which is a mathematical operation that tells us if the operators commute or not. If the commutator of two operators is equal to zero, then they commute, and if it is not equal to zero, then they do not commute.

Can non-commuting operators still be used in quantum mechanics?

Yes, non-commuting operators can still be used in quantum mechanics. However, in these cases, we cannot measure the observables simultaneously, and the order in which the operators are applied will affect the outcome. Non-commuting operators are often used to describe systems with uncertainty, such as position and momentum in the Heisenberg uncertainty principle.