- #1
PatsyTy
- 30
- 1
Homework Statement
Obtain the matrix representation of the ladder operators ##J_{\pm}##.
Homework Equations
Remark that ##J_{\pm} | jm \rangle = N_{\pm}| jm \pm 1 \rangle##
The Attempt at a Solution
[/B]
The textbook states ##|N_{\pm}|^2=\langle jm | J_{\pm}^\dagger J_{\pm} | jm \rangle##, from here it is straightforward:
##\langle jm | J_{\mp} J_{\pm} | jm \rangle = \langle jm | (J^2 - J_z^2 \pm J_z | jm \rangle = \langle jm | \big( j(j+1)|jm\rangle-m^2\hbar^2 |jm\rangle \pm m \hbar^2 | jm \rangle\big)##
Giving us ##|N_\pm |^2=j(j+1)-\hbar^2 m (m\pm 1)##
What I do not understand is the very first step where we write out ##|N_{\pm}|^2=\langle jm | J_{\pm}^\dagger J_{\pm} | jm \rangle## from the given equation ##J_{\pm} | jm \rangle = N_{\pm}| jm \pm 1 \rangle##.
I have tried multiplying the first equation by the hermitian conjugate of ##J_\pm|jm\rangle## giving
##\langle jm | J_\mp J_\pm | jm \rangle = \langle jm | J_\mp N_{\pm}| jm \pm 1 \rangle = N_{\pm} \langle jm | J_\mp | jm \pm 1 \rangle ## however I don't think this is correct as I don't see how I can get a second ##N_\pm## from ##\langle jm | J_\mp | jm \pm 1 \rangle ##.
The text gives the explanation "Since both ##|jm\rangle## and ##|jm+1\rangle## are normalized to unity..." and that's the justification for this step. I don't fully trust this qualitative description so I am trying to write it out mathematically. Any help would be appreciated!