1
Chapter 6 Systems of Particles
Generaalized Eigenvalue Equaton
We write the multiparticle Hamiltonian
H
which operates on a multiparticle wave state
Ψ
yielding the total
energy
E
of the system.
H
(1,2,3,4,5,.
..
N
)
!
"
(1,2,3,4,.
..
N
)
=
E
!
(1,2,3,4,.
..
N
) 1
#
(
x
1
,
p
1
) .
......
N
#
(
x
N
,
p
N
)
This equation can only be solved on computers if at all for multiparticle states, molecules,etc.
Independent or Weakly interacting Particles
If the particles are weakly interacting we might consider the Hamiltonian as a sum of single particle
Hamiltonians where the potential is a mean field felt by all particles equally.
H
(1)
+
H
(2)
+
H
(3)
+
....
H
(
N
)
( )
!
(1,2,3,.
..
N
)
=
E
1
+
E
2
+
E
3
+
.....
+
E
N
( )
!
(1,2,3,.
..
N
)
Perfectly
Seperable
equation
so
choose
a
combination
wave
function
!
1,
2,
3,
4,.
..
N
=
#
1
(1)
1
(2)
1
(3)
1
(4).
...
1
(
N
)
where
we
solve
the
system
of
sin
gle
particle
equations
H
1
(1)
1
=
E
1
(1)
1
H
2
(2)
2
=
E
2
(2)
2
H
N
N
=
E
N
N
In principle we can solve one equation and have all the solutions.
Example:
Let a system of 4 particles be confined to an infinite square well of width
a
.
!
n
(
i
)
=
2
a
sin(
n
x
a
)
E
n
=
!
2
2
m
n
a
#
$
%
’
(
2
ground
state
)
(1,2,3,4)
1,1,1,1
=
n
=
1
(1)
n
=
1
(2)
n
=
1
(3)
n
=
1
(4)
E
=
4
E
1
1
2
3
4
6
5
N
n=1
n=2