Matrix representation of ladder operators

[Ed Quanta]

Homework Statement

Find the matrices which represent the following ladder operators a+,a_, and a+a-
All of these operators are supposed to operate on Hilbert space, and be represented by m*n matrices.

Homework Equations

a+=1/square root(2hmw)*(-ip+mwx)
a_=1/square root(2hmw)*(ip+mwx)
a+a_= 1/hw*(H) - 1/2

The Attempt at a Solution

I recognize that once I find the matrix which represents a+, a_ will be represented by the conjugate of the transpose of that matrix. I also can then find a+a_ by matrix multiplication. Not sure about anything else though.

 

[dextercioby]

Do you know the action of the ladder operators onto the standard basis vectors in the state space ?

 

[Ed Quanta]

No clue. There is nothing on this in my Griffiths quantum mechanics text that I have seen.

 

[nrqed]

No clue. There is nothing on this in my Griffiths quantum mechanics text that I have seen.

Griffiths does give the equation. It is 4.120 and 4,121 (given in problem 4.18), Second edition.

$$L_{\pm} |l, m> = \hbar {\sqrt{l(l+1)-m(m \pm 1)}}|l,m\pm 1>$$

 

[Ed Quanta]

But we haven't gotten to chapter 4 yet, or done anything having to do with angular momentum. The ladder operators were introduced with respect to the harmonic oscillator.

Would I still just plug in values for l and m to find the elements of the matrix?

We were told that
(a+)mn=∫ψn(x)(a+)(ψm(x))dx

And same type of definition for (a_)mn only with a_ instead of a+.

Is there any way for me to come up with the equation in Griffiths 4.120 given this information?

I'm sorry if I am slow.

 

[nrqed]

But we haven't gotten to chapter 4 yet, or done anything having to do with angular momentum. The ladder operators were introduced with respect to the harmonic oscillator.

Would I still just plug in values for l and m to find the elements of the matrix?

We were told that
(a+)mn=∫ψn(x)(a+)(ψm(x))dx

And same type of definition for (a_)mn only with a_ instead of a+.

Is there any way for me to come up with the equation in Griffiths 4.120 given this information?

I'm sorry if I am slow.

My APOLOGIES! I was thinking about the ladder operators for angular momenta (I read replies to your posts but not your initial post in details...my bad! )

My sincerest apologies.

Ok, what you need then is his equation 2.66, page 48 (in the second edition):

$$a_+ \psi_n = {\sqrt{n+1}} \psi_{n+1} \,\,\,\,\,\, a_- \psi_n = {\sqrt{n}} \psi_{n-1}$$

If the $\psi_n$ are represented by explicit column vectors (usually the ones with 1 in one slot and zero and the others), it is easy to use that to find the matrix representations of the ladder operators (they will be matrices with entries below or above the diagonal only)

My apologies, again.

Patrick

 

[Ed Quanta]

Thanks agaom. Easier than I thought. That is what I got on wikipedia but I wasn't sure why until now.