- #1
terp.asessed
- 127
- 3
Homework Statement
Since Hamiltonian operator is:
Ĥ = - (ħ2/(2m))(delta)2 - A/r
where r = (x2+y2+z2)
(delta)2 = ∂2/∂x2 + ∂2/∂y2 + ∂2/∂z2
A = a constant
from Ĥg(r) = Eg(r) form, where:
g(r) = D e-r/b(1-r/b)
with b, D as constants, is an EIGENFUNCTION of Ĥ, find the correct b and give the eigenvalue E.
Homework Equations
g(r) = D e-r/b(1-r/b)
Ĥ = - (ħ2/(2m))(delta)2 - A/r
where r = (x2+y2+z2)
(delta)2 = ∂2/∂x2 + ∂2/∂y2 + ∂2/∂z2
D = a constant
The Attempt at a Solution
g(r) = D e-r/b(1-r/b)
g'(r) = -De-r/b/b - De-r/b/b + Dre-r/b/b2
= -2De-r/b/b + Dre-r/b/b2
g''(r) = 3De-r/b/b2 - Dre-r/b/b3
[delta_g(r)]2 = 5De-r/b/b2 -Dre-r/b/b3 - 4De-r/b/(br)
Ĥg(r) = Eg(r)
- (ħ2/(2m))(5De-r/b/b2 -Dre-r/b/b3 - 4De-r/b/(br)) - A/r * (D e-r/b(1-r/b)) = E*D e-r/b (1-r/b)
..which is then reduced to:
-5ħ2/(2mb2) + ħ2r/(2mb3) + 4ħ2/(2mbr) - A/r + A/b = E (1-r/b)...I managed to cancel out constant D and e-r/b, but I am at loss how how to eliminate r?
Should I equivelate:
ħ2r/(2mb3) + 4ħ2/(2mbr) - A/r = -E(r/b) ? I tried this way, and still haven't managed to find a way to cancel out r...to make b an independent number...
With regard to E, if I am not wrong, I think Energy is that of an excited state, NOT ground state, except I haven't figured it out (for I have not gotten b constant), and have no idea which energy level it is...
Any hints or notices to my mistakes would be appreciated!