Major Project: Ground state of the He atom
PHZ 5156
This problem combines what we have learned about the technique of MonteCarlo
simulation with atomic physics. In particular, we will use MonteCarlo techniques to
estimate the groundstate wave function and energy of a helium atom. Remember
a He atom has two electrons, two protons, and two neutrons. While the nucleus we
can regard as a single particle for our purposes, the three body problem (nucleus and
two electrons) does not allow for an exact analytical solution.
The Hamiltonian for the problem in a set of units with

e

= 1 (i.e. the electron and
proton charge),
m
= 1 and
= 1 is,
ˆ
H
=

1
2
∇
2
r
1

1
2
∇
2
r
2

2
r
1

2
r
2
+
1
r
12
where
r
1
=

r
1

,
r
2
=

r
2

, and
r
12
=

r
1

r
2

. The groundstate solution satisfies
ˆ
H
Ψ
0
(
r
1
, r
2
) =
E
0
Ψ
0
(
r
1
, r
2
)
where
E
0
is the groundstate energy. For any trial wave function Φ
T
(
r
1
, r
2
) different
from the groundstate wave function Ψ
0
(
r
1
, r
2
), the energy expectation value given
by
E
=
...
Φ
*
T
(
r
1
, r
2
)
ˆ
H
Φ
T
(
r
1
, r
2
)
d
3
r
1
d
3
r
2
...
Φ
*
T
(
r
1
, r
2
)Φ
T
(
r
1
, r
2
)
d
3
r
1
d
3
r
2
satisfies
E > E
0
. We can approach the problem of obtaining the groundstate wave
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 Fall '08
 Johnson,M
 wave function, trial wave function, Coloumb

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