Matrix Mechanics Homework: Calculate [p,x] = i h-bar

[ehrenfest]

Homework Statement

My book says that we can express

$$p_{nm} = -i \sqrt{M \omega \hbar} \left( \delta_{n,m-1}\sqrt{m} - \delta_{n,m+1} \sqrt{m+1}\right)$$

and

$$x_{nm} = \sqrt{\hbar/2M\omega} \left( \delta_{n,m-1} \sqrt{m} +\delta_{n,m+1}\sqrt{m+1}\right)$$

for the simple harmonic oscillator potential.

I want to calculate [p,x] = i h-bar.

Homework Equations

The Attempt at a Solution

What I am saying is that when I calculate

$$\sum_{k}x_{nk}p_{km} - \sum_{k}p_{nk}x_{km}$$

I get 0. Can someone check that? If I need to I can post more work.

 

[2Tesla]

$$p_{nm} = -i \sqrt{M \omega \hbar} \left( \delta_{n,m-1}\sqrt{m} - \delta_{n,m+1} \sqrt{m+1}\right)$$

and

$$x_{nm} = \sqrt{\hbar/2M\omega} \left( \delta_{n,m-1} \sqrt{m} +\delta_{n,m+1}\sqrt{m+1}\right)$$

Shouldn't there be some creation and annihilation operators in here? Usually they're written as "a" with or without a little dagger.

 

[ehrenfest]

No, I think you are thinking of something else. x_nm is defined as <x_n|x|x_m> for example. |x_n> is nth eigenstate of the SHO Hamiltonian.

 

[2Tesla]

Ah, right. I remember them now.

These are not the x and p operators, they are expectation values. [p,x] = i h-bar is for the operators. The expectation values are just numbers, and the commutator of two numbers is always zero.

 

[ehrenfest]

I am saying that $$\sum_{k}x_{nk}p_{km} - \sum_{k}p_{nk}x_{km}$$ is equal to 0 for each n and each m. So the entire matrix is equal to zero.

The problem in my book says, "show that [x,p] = ihbar holds as a matrix equation."

 

[robphy]

What do you get for
$$\sum_{k}x_{nk}p_{km}$$?

 

[ehrenfest]

$$(1/2)i\hbar\delta_{n,m}$$

and I get the same for the

$$\sum_{k}p_{nk}x_{km}$$

All would be well if I got $$\sum_{k}p_{nk}x_{km}$$ equal to minus that, but I checked my algebra several times and I just don't know what is going on.

 

[malawi_glenn]

[p,x] = i h-bar

Just insert what p and x are in the linear combination of annihilation and creation operators, and use their commuting algeras.

 

[2Tesla]

OK, I'm over my denseness now. You're calculating $$<x_n|[p,x]|x_m>$$, right?

So, for $$\sum_{k}p_{nk}x_{km}$$ you should get that the $$\delta_{n,m}$$ terms actually cancel and the remaining terms have $$\delta_{n+2,m}$$ and $$\delta_{n-2,m}$$, and that for $$\sum_{k}x_{nk}p_{km}$$ you get those same terms, minus the final (correct) answer, proportional to $$\delta_{n,m}$$ of course. Can you show some work?

 

[malawi_glenn]

ehrenfest : can you please tell us exacly what you want to calculate?

 

[nrqed]

ehrenfest : can you please tell us exacly what you want to calculate?

Malawi, I think it's pretty clear what he is calculating. He has the matrix elements of p and x and wants to calculate the commutator using the matrix representation of those oeprators.

 

[malawi_glenn]

Ok, 2Tesla also asked. If this is a exercise in dealing with the matrix representation, or if he just wants [p,x]

Using the template is good. Saying exactly what the problem are. Now one can inteprent his problem to be:

I want to calculate [p,x] = i h-bar for the simple harmonic oscillator potential.

 

[ehrenfest]

OK, I'm over my denseness now. You're calculating $$<x_n|[p,x]|x_m>$$, right?

So, for $$\sum_{k}p_{nk}x_{km}$$ you should get that the $$\delta_{n,m}$$ terms actually cancel and the remaining terms have $$\delta_{n+2,m}$$ and $$\delta_{n-2,m}$$, and that for $$\sum_{k}x_{nk}p_{km}$$ you get those same terms, minus the final (correct) answer, proportional to $$\delta_{n,m}$$ of course. Can you show some work?

I see where I messed up. I just totally botched the replacement of n and m's with k's using the Kronecker deltas. Thanks.