Sakurai's Formula for <j m | S^2_+ |j m >

[Nusc]

Homework Statement

Does anyone know what the formula is for the following:

<j m | S^2_+ |j m > = ?

Reference to equations in Sakurai would be helpful in deriving the relation. I would suspect 3.5.37 but what about the delta_j j' ?

Homework Equations

The Attempt at a Solution

 

[Nusc]

3.5.35 a

<j',m|J^2 |j,m> = j(j+1)hbar^2 delta_j j' delta m m'

3.5.37 is

J_+ |j,m> = c_jm^+ |j,m+1>

 

[gabbagabbahey]

Well, I'd start by calculating S_{-}|j,m\rangle...what do you get for that?

 

[Nusc]

J_- |j,m> = c_{j,m}^+ |j,m-1>

The formula that I'm interested in is :
<j m | S^2_+ |j m > = delta_m',m+2 c_jm^+ c_j m+1 ^+

But I don't understand how that is constructed, it's not in sakurai.

 

[gabbagabbahey]

Well, \langle j,m|S^2_{-}|j,m\rangle=\langle j,m|(S^2_{-}|j,m\rangle) and S^2_{-}|j,m\rangle=S_{-}(S_{-}|j,m\rangle)=___?

 

[Nusc]

ok thx

 

[Nusc]

Well, \langle j,m|S^2_{-}|j,m\rangle=\langle j,m|(S^2_{-}|j,m\rangle) and S^2_{-}|j,m\rangle=S_{-}(S_{-}|j,m\rangle)=___?

Normally J_+*J_- is substituted as J^2 - J_z^2+hbarJ_z

As you described above, J_+(J_-J_+), wouldn't the content in parenthesese just cancel out?