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Normalization of the Angular Momentum Ladder Operator

  1. Nov 28, 2016 #1
    1. The problem statement, all variables and given/known data

    Obtain the matrix representation of the ladder operators ##J_{\pm}##.

    2. Relevant equations

    Remark that ##J_{\pm} | jm \rangle = N_{\pm}| jm \pm 1 \rangle##

    3. The attempt at a solution

    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!
     
  2. jcsd
  3. Nov 28, 2016 #2

    stevendaryl

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    Staff Emeritus
    Science Advisor

    Here's the part that you are missing: For any matrix element of the form [itex]\langle A |O|B \rangle[/itex], we have:

    [itex]\langle A |O|B \rangle = \langle B |O^\dagger|A \rangle^* [/itex]

    So in particular,
    [itex]\langle jm | J_\mp | jm \pm 1 \rangle = \langle jm\pm 1 | J_\pm | jm \rangle^*[/itex]

    See if that helps.
     
  4. Nov 28, 2016 #3
    Thanks! I got it right away with that.
     
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