# I Eigenfunction of L2 operator

1. Apr 6, 2016

### kenyanchemist

Ψ2Px= [Ψ2p+1 +Ψ2p-1]1/2 an eigen function of L2 or Lz?
if so, how is it?
and if so kindly explain the values of l and m
thanks

2. Apr 7, 2016

### Simon Bridge

Good question - did you apply the operators to find out?

$\psi_{2p,x} = \frac{1}{\sqrt{2}}\left(\psi_{2p,+1}+\psi_{2p,-1}\right)$
This says that the 2p state for, say, a hydrogen atom, is equally likely to involve the electron spin up or spin down.
The "2" is the energy eigenstate number, the "p" is from "s,p,d,f..." notation, and is the orbital angular momentum state number. You can look these up.

3. Apr 7, 2016

### kenyanchemist

By all means tell me more.... you are being a major help.
So how would I go about applying
And from the explanation you've just given above am guessing l and m are normal quantum number values

4. Apr 7, 2016

### kith

Let's say we have an operator $A$, an eigenvalue $a$ and the corresponding eigenfunction $\psi_a$. How does the eigenvalue equation look like?

What do you get if you replace $A$ by $H, L^2$ or $L_z$ and $\psi_a$ by $\psi_{n,l,m}$?

5. Apr 7, 2016

### kenyanchemist

Oh and the entire equation in the brackets are to the power of 1/2 (square root) not multiplied by the root of 1/2
Sorry I think there was a font issue there

6. Apr 10, 2016

### Simon Bridge

No - I was just assuming someone made a mistake.

What you wrote was:

Ψ2Px= [Ψ2p+1 +Ψ2p-1]1/2

Another interpretation would be: $\psi_{2px} = \left[ \psi_{2p+1}+\psi_{2p-1}\right]^{1/2}$
... but that does not make any sense, though it looks like it can be used to answer the questions.

... how would you normally go about applying an operator?
(By the time you get to see angular momentum in QM, you have met operators and how to apply them: go back through your notes.)

... have you looked up "spdf notation" yet?
https://en.wikipedia.org/wiki/Electron_configuration

The quantum numbers for an atomic state are n,l,m(,s) - so $H\psi_{nlm} = E_n\psi_{nlm}$
... for hydrogen, $E_n=-13.6\text{eV}/n^2$. The advantage of doing QM this way is that you don't have to do lots of differentiating and integrating.
Also see: http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/qangm.html