What is the <lz> Expectation Value for Given Wave Function?

[Ant_of_Coloni]

Homework Statement

Find <l_z> using $\Psi$, where $\Psi$=(Y₁₁+cY_1-1)/(1+c^2)).

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

Homework Equations

<l_z> Y_lm = [STRIKE]h[/STRIKE]mY_lm

The Attempt at a Solution

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

So <l_z> = [STRIKE]h[/STRIKE]m(Y₁₁+cY_1-1)/(1+c^2)?

Also what would m be?

 

[bbbeard]

Homework Statement

Find <l_z> using $\Psi$, where $\Psi$=(Y₁₁+cY_1-1)/(1+c^2)).

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

Homework Equations

<l_z> Y_lm = [STRIKE]h[/STRIKE]mY_lm

The Attempt at a Solution

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

So <l_z> = [STRIKE]h[/STRIKE]m(Y₁₁+cY_1-1)/(1+c^2)?

Also what would m be?

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

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

<l_z> = <$\Psi$| l_z |$\Psi$>

I think your "relevant equation" should perhaps read

l_z Y_lm = $\hbar$m Y_lm

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 <l_z> is simply a number, not an operator, and the number is just $\hbar$m provided $\Psi$ = Y_lm. 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?


This is a pretty straightforward problem once you get the notation. The point is that the wave function is just a linear superposition of two eigenfunctions of the angular momentum operator, so the expectation is a linear function of the eigenvalues. But you have to do the algebra to get the right answer. And by "do the algebra" I mean to write down the expectation value, in which the wave function ψ shows up in both the bra and the ket form, in integral form, and make sure you understand how the orthonormality of the spherical harmonics makes the resulting integral "simple"...