# Express a wave function as a combination of spherical harmonics

1. Oct 9, 2012

### mat8845

1. The problem statement, all variables and given/known data

An electron in a hydrogen atom is in a state described by the wave function:

ψ(r,θ,φ)=R(r)[cos(θ)+e(1+cos(θ))]

What is the probability that measurement of L2 will give 6ℏ2 and measurement of Lz will give ℏ?

2. Relevant equations

The spherical harmonics

3. The attempt at a solution

I know that I need to express the part in (θ,φ) as a linear combination of spherical harmonics. Then I would normalize my wave function, and the coefficients would lead me to the probability I need.

My problem is to express ψ in terms of the spherical harmonics Ylm. Is there a general method to do so? Thanks.

2. Oct 12, 2012

### PhysicsGente

The way I would do it is by looking at $\mathbf{L}^2|l \mspace{8mu} m> = \hbar^2 l(l+1) |l\mspace{8mu} m>$ and $\mathbf{L}_z |l \mspace{8mu}m> = \hbar m |l\mspace{8mu} m>$. And looking at a table of spherical harmonics. There is no need to normalize them since they are already normalized.

3. Oct 12, 2012

### vela

Staff Emeritus
You could find the projection of the angular part onto the state $\vert l\ m \rangle$ by calculating $\langle l\ m \vert \psi\rangle$. Seems to be kind of a pain though. I'd just futz around with the spherical harmonics to find a linear combination that works. For example, the $e^{i\phi}$ part has to come from an m=1 state.