# Finding Eigenvalue

1. Oct 18, 2014

### terp.asessed

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

Ĥ = - (ħ2/(2m))(delta)2 - A/r
where r = (x2+y2+z2)
(delta)2 = ∂2/∂x2 + ∂2/∂y2 + ∂2/∂z2
A = a constant

Then, show that a function of the form,

f(r) = Ce-r/a

with a, C as constants, is an EIGENFUNCTION of Ĥ provided that the constant a is chosen correctly. Find the correct a and give the eigenvalue.

2. Relevant equations
Given above

3. The attempt at a solution
Because Ĥ is an energy (Hamiltonian) operator, I put E as an eigenvalue in the following equation: Ĥf(r) = Ef(r)
So....
- (ħ2/(2m))(delta)2[Ce-r/a] - A/r = ECe-r/a
- (ħ2/(2m)) [Ce-r/a/a2 - 2Ce-r/a/(ra)] - A/r = E*Ce-r/a
..was what I've been doing...so:

E = - {(ħ2/(2m)) [Ce-r/a/a2 - 2Ce-r/a/(ra)] - A/r}/(Ce-r/a)

..but I am at loss as to if it is the right E eigenvalue and as how to get the "a" value? Also, a question--is {-A/r} in the Hamiltonian operator a potential energy part?

2. Oct 18, 2014

### Orodruin

Staff Emeritus
Hint: The energy eigenvalue must be a constant independent of the coordinate r.

3. Oct 18, 2014

### terp.asessed

Independent of r? So, does it mean E should be some number based on a, C, and A?

I have been trying to eliminate "r", but I keep failing:

- (ħ2/(2m))(delta)2[Ce-r/a] - A/r = ECe-r/a
- (ħ2/(2m)) (C/a) [1/a - 2/r] - A/(re-r/a) = E*C
- Cħ2/(2ma) (1/a - 2/r) - A/(re-r/a) = EC

...I still haven't managed to eliminate r.......

Last edited: Oct 18, 2014
4. Oct 18, 2014

### Orodruin

Staff Emeritus
Well, C is a normalization constant and will not matter. You must determine a such that E is independent of r. Being a constant independent of the coordinates is the entire point of E being an eigenvalue to H.

5. Oct 18, 2014

### terp.asessed

Wait, I am getting slightly confuse--I thought a was supposed to be some value, without "r" in it?

6. Oct 18, 2014

### Orodruin

Staff Emeritus
Yes, a is also a constant independent of r.

Note that you have also forgotten the f(r) in the potential energy term in the original post ...

7. Oct 18, 2014

### terp.asessed

Could you please clarify by what you mean by:

I thought -A/r was potential energy term?

8. Oct 18, 2014

### Orodruin

Staff Emeritus
In this expression, you have included Ce-r/a in all terms except the -A/r term. This term is also a part of the Hamiltonian and must also act on the wave function. In the end, you should be able to divide out this term.

9. Oct 18, 2014

### terp.asessed

Ohhhhhhhh, so Hf(x) = - (ħ2/(2m))(delta)2[Ce-r/a] - A[Ce-r/a]/r ? Plus, -Af(x)/r is the potential term, then?

Thank you for your patience with me!

10. Oct 18, 2014

### Orodruin

Staff Emeritus
Yes. After performing the derivatives you should now be able to fix a such that E is a constant independent of r.

11. Oct 18, 2014

### terp.asessed

So, a = Am/ħ2 and E = -ħ2/(2ma2) = -ħ6/(2m3A2)?

12. Oct 18, 2014

### Orodruin

Staff Emeritus
I believe you have made an arithmetic error. You should obtain the inverse of your expression for a. Otherwise I think you now have the correct idea.

13. Oct 18, 2014

### terp.asessed

fixed my mistake and got a = ħ2/(Am), and E = -A2m/(2ħ2)

By the way, just on my own (aside from hw-related question previously), I sketched the function f(r) = Ce-r(Am/ħ2) as a function of r and I am curious if the graph does correspond to the ground state?

Last edited: Oct 18, 2014