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Matrix Elements

  • Thread starter Nusc
  • Start date
  • #1
753
2

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

 

Answers and Replies

  • #2
753
2
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>
 
  • #3
gabbagabbahey
Homework Helper
Gold Member
5,002
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Well, I'd start by calculating [itex]S_{-}|j,m\rangle[/itex]...what do you get for that?
 
  • #4
753
2
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.
 
  • #5
gabbagabbahey
Homework Helper
Gold Member
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Well, [itex]\langle j,m|S^2_{-}|j,m\rangle=\langle j,m|(S^2_{-}|j,m\rangle)[/itex] and [itex]S^2_{-}|j,m\rangle=S_{-}(S_{-}|j,m\rangle)=[/itex]___?
 
  • #6
753
2
ok thx
 
  • #7
753
2
Well, [itex]\langle j,m|S^2_{-}|j,m\rangle=\langle j,m|(S^2_{-}|j,m\rangle)[/itex] and [itex]S^2_{-}|j,m\rangle=S_{-}(S_{-}|j,m\rangle)=[/itex]___?
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?
 

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