Angular Momentum Operator in terms of ladder operators

[Plutoniummatt]

Homework Statement

http://img716.imageshack.us/i/captur2e.png/
http://img716.imageshack.us/i/captur2e.png/

Homework Equations

Stuck on the last part

The Attempt at a Solution

http://img689.imageshack.us/i/capturevz.png/
http://img689.imageshack.us/i/capturevz.png/

 

[Ben Niehoff]

You missed the

$$e^{\pm i \phi}$$

in the definition of $L_{\pm}$

 

[Plutoniummatt]

You missed the

$$e^{\pm i \phi}$$

in the definition of $L_{\pm}$

dont they multiply to give 1?

$$e^{ i \phi}e^{- i \phi} = 1$$

Edit:

you were right, i got it in the end taking into account the exponential factors...was a ***** of an algebra grindfest though