Finding Eigenvalue for Ĥ: A Homework Statement

[terp.asessed]

Homework Statement

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.

Homework Equations

Given above

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?

 

[Orodruin]

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

 

[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...

 

[Orodruin]

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.

 

[terp.asessed]

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

 

[Orodruin]

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 ...

 

[terp.asessed]

Could you please clarify by what you mean by:

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

I thought -A/r was potential energy term?

 

[Orodruin]

- (ħ2/(2m))(delta)2[Ce^-r/a] - A/r = ECe^-r/a

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.

 

[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!

 

[Orodruin]

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

 

[terp.asessed]

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

 

[Orodruin]

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.

 

[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?