Express a wave function as a combination of spherical harmonics

[mat8845]

Homework Statement

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

ψ(r,θ,φ)=R(r)[cos(θ)+e^iφ(1+cos(θ))]

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

Homework Equations

The spherical harmonics

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 Y_l^m. Is there a general method to do so? Thanks.

 

[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.

 

[vela]

Homework Statement

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

ψ(r,θ,φ)=R(r)[cos(θ)+e^iφ(1+cos(θ))]

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

Homework Equations

The spherical harmonics

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 Y_l^m. Is there a general method to do so? Thanks.

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.