Query regarding Commuting operators

[legend]

I am having a problem with a couple of problems involving commutating operators.

Homework Statement

1. How do i find the commutation operators of x and ∂/∂x
2. If the angular momenta about 3 rotational axes in a central potential commute then how many quantum numbers we would get? And why will it be a problem?

Homework Equations

The Attempt at a Solution

1. For x i think it would be any F(x) is that so?
and for ∂/∂x is it d/dx ?
2. I think we would get 4 quantum numbers , right?

 

[gabbagabbahey]

1. How do i find the commutation operators of x and ∂/∂x

I'm not entirely sure what this question is asking...is it asking you to find the commutator of $x$ and $\partial/\partial x$, or something else?

2. I think we would get 4 quantum numbers , right?

Why?

 

[legend]

Thanks for your reply.

Yes, i guess it is asking for the commutator of x and also the commutator of $\partial/\partial x$

For the number of quantum numbers, we would have the principal quantum number and the three quantum numbers associated with the three components of the angular momentum and the azimuthal quantum number. So , i guess it should be 5 (sorry not 4), is it?

 

[gabbagabbahey]

Yes, i guess it is asking for the commutator of x and also the commutator of $\partial/\partial x$

Okay, so what is the definition of "commutator" between two operators? Use that definition, and show your calculations!

For the number of quantum numbers, we would have the principal quantum number and the three quantum numbers associated with the three components of the angular momentum and the azimuthal quantum number.

Why do you say this? How are these quantum numbers derived in the first place? Does that derivation hold if $[L_i,L_j]=0$?

 

[legend]

Thanks for your reply.

Okay, so what is the definition of "commutator" between two operators? Use that definition, and show your calculations!

As far as i know, commutation operator for a function would be such that FG - GF = 0, right? Based on that, i figured for $<br /> \partial/\partial x<br />$
it would be $<br /> \partial/\partial y<br />$
since $<br /> \partial/\partial x<br />$($<br /> \partial/\partial y<br />$) = $<br /> \partial/\partial y<br />$($<br /> \partial/\partial x<br />$)
And similarly for x and f(x), am i totally of the track here?

Why do you say this? How are these quantum numbers derived in the first place? Does that derivation hold if $[L_i,L_j]=0$?

I don't have much idea about this question actually.

 

[gabbagabbahey]

am i totally of the track here?

Way off track. Study your textbook/notes!

The commutator of two operators, $F$ and $G$ is defined as $[F,G]=FG-GF$. So, the commutator of $x$ and $\partial/\partial x$ is given by

$$[x,\partial/\partial x]=x\frac{\partial}{\partial x}-\frac{\partial}{\partial x} x$$

You can simplify this further by applying this commutator to a sample wavefunction and using the product rule to calculate the derivatives involved.

This is an incredibly basic task in operator algebra, so if you can't do it you need to study!