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terp.asessed
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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?