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Variation Method- Helium
If you guess a solution to the Schrödinger Equation, how do you find out how good
a guess you have made?
H
Ψ
= E
Ψ
can't be solved exactly:
but you have :
Ψ
(1)
guess
Ψ
(2)
better guess
Ψ
(3)
better yet
E
E
E
E exact
1
2
3
The Variation Theorem guarantees that the trial energy is always
greater than the
exact energy.
H
Ψ
i
= E
i
Ψ
i
∫
Ψ
i
*
H
Ψ
i
d
τ
=
∫
Ψ
i
*
E
i
Ψ
i
d
τ
solve for E
i
:
E
i
=
∫
Ψ
i
*
H
Ψ
i
d
τ
∫
Ψ
i
*
Ψ
i
d
τ
is exact
E
i
(1)
=
∫
Ψ
i
(1)*
H
Ψ
i
(1)
d
τ
∫
Ψ
i
(1)*
Ψ
i
(1)
d
τ
E
(1)
≥
E
by the variation theorem

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