Commutation relation of operators involving momentum and position

[Kooshy]

Homework Statement

The problem is number 11, the problem statement would be in the first picture in the spoiler.
Basically, I'm trying to find if two operators commute. They're not supposed to, since they involve momentum and position, but my work has been suggesting otherwise, so I'm doing something wrong.

Homework Equations

Are in problem 10 and written next to it. (2nd picture in spoiler.)
x^ = x
p^ = -iħ d/dx

P^= p^/√(mωλ)
Q^=x^ * (√(mω/ħ))

The Attempt at a Solution

Also in the picture.
I think I'm messing up where the operators operate on each other and new terms are created, and I'm not sure where or how to fix it.

Spoiler

 

[grzz]

In (10) you have already proved that [P,Q] = -i[itex]\sqrt{hbar/\lambda}[/itex]
Now re (11), do not substitute for the operators P and Q but leave them as P and Q. Hence the resulting expression will be in terms of P and Q. Then use [P,Q] = -i[itex]\sqrt{hbar/\lambda}[/itex]

 

[grzz]

But be very careful since P and Q do not commute!

 

[Kooshy]

Thank you grzz, that worked out much easier than what I was trying.

 

[paranormal]

I can understand your frustration with trying to determine the commutation relation between operators involving momentum and position. This is a fundamental concept in quantum mechanics and it is important to get it right in order to accurately describe the behavior of particles at a microscopic level.

Firstly, it is important to understand that operators do not commute in quantum mechanics. This means that the order in which they are applied matters and can affect the outcome of calculations. In the case of momentum and position operators, their commutation relation is given by [x^, p^] = iħ, where the brackets represent the commutator operation.

As for your specific problem, it seems that you have correctly identified the operators x^ and p^, but you have not correctly applied them to each other. The correct way to do this is to use the definition of the operators given in the problem statement. For example, when applying p^ to x^, you should get [x^, p^] = x^ * (√(mω/ħ)) * (-iħ d/dx) - (-iħ d/dx) * x^ * (√(mω/ħ)). You can then simplify this expression to get the commutator relation.

It is also important to note that in quantum mechanics, operators can only operate on wavefunctions, not on each other. So when you have an expression like p^/√(mωλ), you need to first apply the operator p^ to the wavefunction and then divide by √(mωλ).

I hope this helps you in your understanding of commutation relations involving momentum and position operators. Remember to always carefully apply the definition of operators and keep track of the order in which they are applied. Good luck with your problem!