Biomedical Engineering Reference
In-Depth Information
U
(
x
,
t
)
Ψ (
x
,
t
)
h
2
8π
2
m
∂
2
∂
x
2
+
h
2π
∂Ψ (
x
,
t
)
∂
t
−
=
j
(11.1)
where
j
is the square root of
1). Ψ(
x
,
t
) is the
wavefunction
and in general it is
a complex valued quantity. The square of the modulus of Ψ can be found from
−
1(
j
2
=−
|
Ψ (
x
,
t
)
|
2
=
Ψ
∗
(
x
,
t
) Ψ (
x
,
t
)
where Ψ
∗
is the complex conjugate of Ψ . The fact that complex wavefunctions are
allowed and that the Schrödinger equation mixes up real and imaginary quantities caused
Schrödinger and his contemporaries serious difficulties in trying to give a physical meaning
to Ψ . According to Max Born, the physical interpretation attaching to the wavefunction
Ψ(
x
,
t
) is that Ψ
∗
(
x
,
t
)Ψ (
x
,
t
)d
x
gives the probability of finding the particle between
x
and
x
d
x
at time
t
. It is therefore the square of the modulus of the wavefunction rather than
the wavefunction itself that is related to a physical measurement.
Since Ψ
∗
(
x
,
t
)Ψ (
x
,
t
)d
x
is the probability of finding the particle (at time
t
) between
x
and
x
+
d
x
, it follows that the probability of finding the particle between any two values
a
and
b
is found by summing the probabilities
+
b
Ψ
∗
(
x
,
t
) Ψ (
x
,
t
) d
x
P
=
a
In three dimensions the equation is more complicated because both the potential
U
and the
wavefunction Ψ might depend on the three dimensions
x
,
y
and
z
(written as vector
r
)as
well as time:
∂
2
∂
x
2
+
U
(
r
,
t
)
Ψ (
r
,
t
)
h
2
8π
2
m
∂
2
∂
y
2
+
∂
2
∂
z
2
h
2π
∂Ψ (
r
,
t
)
∂
t
−
+
=
j
(11.2)
You are probably aware that the operator in the outer left-hand bracket is the
Hamiltonian
operator
and that we normally write the equation
h
2π
∂Ψ (
r
,
t
)
∂
t
H
Ψ (
r
,
t
)
=
j
(11.3)
The Born interpretation is then extended as follows:
Ψ
∗
(
r
,
t
)
Ψ
(
r
,
t
)
d
x
d
y
d
z
(11.4)
which gives the probability that the particle will be found at time
t
in the volume element
d
x
d
y
d
z
. Such volume elements are often written dτ .
I have not made any mention of time-dependent potentials in this text, but they certainly
exist. You will have met examples if you have studied electromagnetismwhere the
retarded
potentials
account for the generation of electromagnetic radiation.
For the applications we will meet in this topic, the potentials are time independent; in
such cases the wavefunction still depends on time, but in a straightforward way as I will
now demonstrate.
