Angular Momentum squared operator (L^2) eigenvalues?

[terp.asessed]

Homework Statement

Find the eigenvalues of the angular-momentum-squared operator (L²) for hydrogen 2s and 2p_x orbitals...

Homework Equations

Ψ_2s = A (2-r/a₀)e^-r/(2a₀)
Ψ_2px = B (r/a₀)e^-r/(2a₀)

The Attempt at a Solution

If I am not wrong, is the use of L² in eigenfunction L²Ψ = ħ² l(l+1) Ψ?

...so I wonder if eigenvalues are l(l+1) from this part, ħ² l(l+1) Ψ, as in:

for Ψ_2s, because for s orbital, l = 0--> l(l+1) = 0(0+1) = 0
for Ψ_2px, because for p orbital, l = 1--> l(l+1) = 1(1+1) = 2 ?

 

[DrClaude]

Homework Equations

Ψ_2s = A (2-r/a₀)e^-r/(2a₀)
Ψ_2px = B (r/a₀)e^-r/(2a₀)

That's only the radial part. You are missing the angular part, which is essential ;)

The Attempt at a Solution

If I am not wrong, is the use of L² in eigenfunction L²Ψ = ħ² l(l+1) Ψ?

...so I wonder if eigenvalues are l(l+1) from this part, ħ² l(l+1) Ψ, as in:

for Ψ_2s, because for s orbital, l = 0--> l(l+1) = 0(0+1) = 0
for Ψ_2px, because for p orbital, l = 1--> l(l+1) = 1(1+1) = 2 ?

How is an eigenvalue defined?

 

[terp.asessed]

That's only the radial part. You are missing the angular part, which is essential

OMG, I was like, why does the equation look funny----the correct one for Ψ_2px would be B (r/a₀)e^-r/(2a₀)sinθcosφ, right?

How is an eigenvalue defined?

Also, as how an eigenvalue is defined...well, from one previous eigenfunction example I solved, the one involving Hamiltonian operator, HΨ = EΨ, E is the eigenvalue?
So, is the eigenvalue for Ψ_2px 2ħ²?

 

[DrClaude]

OMG, I was like, why does the equation look funny----the correct one for Ψ_2px would be B (r/a₀)e^-r/(2a₀)sinθcosφ, right?

Right.

Also, as how an eigenvalue is defined...well, from one previous eigenfunction example I solved, the one involving Hamiltonian operator, HΨ = EΨ, E is the eigenvalue?

Yes. For an operator $\hat{A}$, $\hat{A} \psi = a \psi$ where $a$ is a scalar means that $\psi$ is an eigenfunction of $\hat{A}$ and $a$ its corresponding eigenvalue.

So, is the eigenvalue for Ψ_2px 2ħ²?

Yes.

I don't know what the level of the question is, so I don't know how simple the answer can be. If this is an advanced problem, you probably are expected to actually derive the eigenvalue.

 

[terp.asessed]

Thank you! I am quite surprised myself at how simple the answer turns out to be...but still thank you.