Multiparticle Quantum Systems
We developed a photon model of electromagnetic radiation using a complex-
valued wavefunction that could predict and explain both the interference and
photon-counting phenomena we observed. A wavefunction
2
2
2
2
2
2
2
2
2
2
1
( , , , )
( , , , )
( , , , )
x y z t
x y z t
x y z t
c
t
x
y
z
∂
∂
∂
∂
Ψ
=
+
+
Ψ
≡ ∇ Ψ
∂
∂
∂
∂
Photon wavefunctions satisfy the wave equation,
(
)
( , , , )
i k r
t
x y z t
e
ϖ
⋅ -
Ψ
∝
a
a
represents a photon state with definite energy and momentum
and
E
=
ℏ
p
k
=
a
a
ℏ
so that their energy and momenta are properly related,
E
2
= p
2
c
2
.
To predict and explain interference effects in electron scattering and other
experiments involving massive particles we made an analogy to our photon
model and began using wavefunctions to represent particle states. To
impose the correct relationship between nonrelativistic particle kinetic energy
and momentum,
K = p
2
/2m
, we require that these wavefunctions satisfy the
free-particle Schrödinger equation,
2
2
2
2
2
2
2
( , , , )
( , , , )
2
i
x y z t
x y z t
t
m
x
y
z
∂
∂
∂
∂
Ψ
= -
+
+
Ψ
∂
∂
∂
∂
ℏ
ℏ