# Quick Dirac Notation question

1. Jan 3, 2012

### Ant_of_Coloni

1. The problem statement, all variables and given/known data
Find <lz> using $\Psi$, where $\Psi$=(Y11+cY1-1)/(1+c^2)).

Ylm are spherical harmonics, and <lz> is the angular momentum operator in the z direction.

2. Relevant equations

<lz> Ylm = [STRIKE]h[/STRIKE]mYlm

3. The attempt at a solution

The brackets around <lz> are throwing me off. This isn't defined in my book, but am I just supposed to apply the above equation to $\Psi$?

So <lz> = [STRIKE]h[/STRIKE]m(Y11+cY1-1)/(1+c^2)?

Also what would m be?

2. Jan 3, 2012

### bbbeard

The m is the eigenvalue of the lz operator for the spherical harmonic in question, i.e. the m in Ylm, i.e. +1 or -1 in your problem.

The <> notation surrounding an operator is implicitly the expectation value with respect to some given wavefunction:

<lz> = <$\Psi$| lz |$\Psi$>

lz Ylm = $\hbar$m Ylm
i.e. the operator is "naked" when it acts on the spherical harmonic. However, the equation you wrote is not exactly incorrect... it's just that <lz> is simply a number, not an operator, and the number is just $\hbar$m provided $\Psi$ = Ylm. That's subtly different from the equation I wrote, which indicates that the operator acting the wavefunction gives you a multiple of the wavefunction. Does that make sense?