1Wavefunctions and the Schrodinger equation
IB Quantum Mechanics
1.1 Particle state and probability
Classically, a point particle in 1 dimension has a definitive position
x
(and
momentum
p
) at each time. To completely specify a particle, it suffices to write
down these two numbers. In quantum mechanics, this is much more complicated.
Instead, a particle has a state at each time, specified by a complex-valued
wavefunction ψ(x).
The physical content of the wavefunction is as follows: if
ψ
is appropriately
normalized, then when we measure the position of a particle, we get a result
x
with probability density function
|ψ
(
x
)
|
2
, i.e. the probability that the position
is found in [
x, x
+
δx
] (for small
δx
) is given by
|ψ
(
x
)
|
2
δx
. Alternatively, the
probability of finding it in an interval [a, b] is given by
P(particle position in [a, b]) =
Z
b
a
|ψ(x)|
2
dx.
What do we mean by “appropriately normalized”? From our equation above, we
see that we require
Z
∞
−∞
|ψ(x)|
2
dx = 1,
since this is the total probability of finding the particle anywhere at all. This
the normalization condition required.
Example (Gaussian wavefunction). We define
ψ(x) = Ce
−
(x−c)
2
2α
,
where c is real and C could be complex.
c
We have
Z
∞
−∞
|ψ(x)|
2
dx = |C|
2
Z
∞
−∞
e
−
(x−c)
2
α
dx = |C|
2
(απ)
1
2
= 1.
So for normalization, we need to pick C = (1/απ)
1/4
(up to a multiple of e
iθ
).
If
α
is small, then we have a sharp peak around
x
=
c
. If
α
is large, it is
more spread out.
While it is nice to have a normalized wavefunction, it is often inconvenient
to deal exclusively with normalized wavefunctions, or else we will have a lot of
ugly-looking constants floating all around. As a matter of fact, we can always
restore normalization at the end of the calculation. So we often don’t bother.
If we do not care about normalization, then for any (non-zero)
λ
,
ψ
(
x
) and
λψ
(
x
) represent the same quantum state (since they give the same probabilities).
In practice, we usually refer to either of these as “the state”. If we like fancy
words, we can thus think of the states as equivalence classes of wavefunctions
under the equivalence relation ψ ∼ φ if φ = λψ for some non-zero λ.
What we do require, then, is not that the wavefunction is normalized, but
normalizable, i.e.
Z
∞
−∞
|ψ(x)|
2
dx < ∞.
We will very soon encounter wavefunctions that are not normalizable. Mathe-
matically, these are useful things to have, but we have to be more careful when
interpreting these things physically.
A characteristic property of quantum mechanics is that if
ψ
1
(
x
) and
ψ
2
(
x
) are
wavefunctions for a particle, then
ψ
1
(
x
) +
ψ
2
(
x
) is also a possible particle state
(ignoring normalization), provided the result is non-zero. This is the principle of
superposition, and arises from the fact that the equations of quantum mechanics
are linear.
Example (Superposition of Gaussian wavefunctions). Take
ψ(x) = B
exp
−(x −c)
2
2α
+ exp
−
x
2
2β
.
Then the resultant distribution would be something like
We choose
B
so that
ψ
in a normalized wavefunction for a single particle. Note
that this is not two particles at two different positions. It is one particle that is
“spread out” at two different positions.
It is possible that in some cases, the particles in the configuration space
may be restricted. For example, we might require
−
`
2
≤ x ≤
`
2
with some
boundary conditions at the edges. Then the normalization condition would not
be integrating over (−∞, ∞), but [−
`
2
,
`
2
].