Failed to validate the raising operator

Questions
,
1

I tried to validate the ladder operator, but the raising operator returns nothing, as shown below:

Selection_166.png
Selection_166.png1404×569 75.5 KB

Attached is the test notebook.

Any hints for this problem will be highly appreciated.

Regards,
HZ

Angular_momentum_operator.wl (4.59 KB)

2

You are applying the raising operator to the maximal state of angular momentum. Perhaps that is the reason you are not getting the eigenvalue.

In fact, applying the lRaising to the minimal state with the eigenvalue -l:

In[107]:=Solve[Simplify[Lz[(x - I y)^l] == eigenvalue (x - I y)^l],eigenvalue][[1]]
         Solve[Simplify[L2[(x - I y)^l] == eigenvalue (x - I y)^l],eigenvalue][[1]]

Out[107]= {eigenvalue -> -l \[HBar]}
Out[108]= {eigenvalue -> l (1 + l) \[HBar]^2}

does increase the eigenvalue to -l+1:

In[109]:=Solve[Simplify[Lz[lRaising[(x - I y)^l]] == eigenvalue lRaising[(x - I y)^l]],eigenvalue][[1]]
         Solve[Simplify[L2[lRaising[(x - I y)^l]] == eigenvalue lRaising[(x - I y)^l]],eigenvalue][[1]]

Out[109]= {eigenvalue -> -((-1 + l) \[HBar])}
Out[110]= {eigenvalue -> l (1 + l) \[HBar]^2}