Z-operator acting on an angular momentum quantum state

[QuantumKyle]

Homework Statement

I need to show that < n l m | z | n l m > = 0 for all states | n l m>

2. Relevent Equations:

L^2 = Lx^2 + Ly^2 + Lz ^2
Lx = yp(z) - zp(y)
Ly = zp(x) - xp(z)
Lz = xp(y) - yp(x)
L+/- = Lx +/- iLy

The Attempt at a Solution

I really don't know where to begin because z is not an eigenfuntion of | n l m> (and if it was this equation would not be 0 anyways). My intuition tells me that I need to somehow represent z as a function of the operators L^2, Lz, and maybe L+/-. But I can't seem to isolate z. Maybe I'm looking at this problem the wrong way. Is there some fundamental theorem that would show that this equation is true?

 

[susskind_leon]

There is certainly a way to use the algebra, but it's actually eaasier if you look at the wavefunctions in position space.
The wavefunction is basically a Laguerre Polynomial with respect to r times a spherical harmonics.
The spherical harmonics is basically a complex exponential of phi times an associated Legendre polynomial with respect to $\cos \theta$ which is $\hat{z}$.
Now, check the first recurrence relation here: http://en.wikipedia.org/wiki/Associated_Legendre_polynomials#Recurrence_formula
and you will see that multiplying with $\hat{z}=\cos \theta$, which is the argument of the Legendre polynomial, gives you $P_{l+1}^m(\cos \theta)$ and $P_{l-1}^m(\cos \theta)$ (with awkward coefficients), so effectively you'll get |n l+1 m> and |n l-1 m>.

 

[QuantumKyle]

Thanks, that was a big help

 

[susskind_leon]

My pleasure ;)

 

[paranormal]

I would approach this problem by first reviewing the relevant equations and understanding how the Z-operator relates to the angular momentum operators Lx, Ly, and Lz. From the given equations, it is clear that the Z-operator cannot be expressed solely in terms of the angular momentum operators. Therefore, I would consider other relevant theorems or principles that may help in solving this problem. One possible approach could be to use the commutation relations between the angular momentum operators and the Z-operator to simplify the equation and show that it is equal to 0. Another approach could be to use the fact that the Z-operator is Hermitian and therefore has real eigenvalues, and use this information to manipulate the equation and prove that it is equal to 0. Additionally, I would consult with colleagues or reference materials to gather more information and insights on the problem. Ultimately, the key to solving this problem would be to carefully apply the relevant equations and principles in a logical and systematic manner.