1Wavefunctions and the Schrodinger equation
IB Quantum Mechanics
1.2 Operators
We know that the square of the wavefunction gives the probability distribution of
the position of the particle. How about other information such as the momentum
and energy? It turns out that all the information about the particle is contained
in the wavefunction (which is why we call it the “state” of the particle).
We call each property of the particle which we can measure an observable.
Each observable is represented by an operator acting on
ψ
(
x
). For example, the
position is represented by the operator
ˆx
=
x
. This means that (
ˆxψ
)(
x
) =
xψ
(
x
).
We can list a few other operators:
position ˆx = x ˆxψ = xψ(x)
momentum ˆp = −i~
∂
∂x
ˆpψ = −i~ψ
0
(x)
energy H =
ˆp
2
2m
+ V (ˆx) Hψ = −
~
2
2m
ψ
00
(x) + V (x)ψ(x)
The final
H
is called the Hamiltonian, where
m
is the mass and
V
is the potential.
We see that the Hamiltonian is just the kinetic energy
p
2
2m
and the potential
energy
V
. There will be more insight into why the operators are defined like
this in IIC Classical Dynamics and IID Principles of Quantum Mechanics.
Note that we put hats on
ˆx
and
ˆp
to make it explicit that these are operators,
as opposed to the classical quantities position and momentum. Otherwise, the
definition ˆx = x would look silly.
How do these operators relate to the actual physical properties? In general,
when we measure an observable, the result is not certain. They are randomly
distributed according to some probability distribution, which we will go into full
details later.
However, a definite result is obtained if and only if
ψ
is an eigenstate, or
eigenfunction, of the operator. In this case, results of the measurements are the
eigenvalue associated. For example, we have
ˆpψ = pψ
if and only if ψ is a state with definite momentum p. Similarly,
Hψ = Eψ
if and only if ψ has definite energy E.
Here we are starting to see why quantization occurs in quantum mechanics.
Since the only possible values of
E
and
p
are the eigenvalues, if the operators
have a discrete set of eigenvalues, then we can only have discrete values of
p
and
E.
Example. Let
ψ(x) = Ce
ikx
.
This has a wavelength of
λ
= 2
π/k
. This is a momentum eigenstate, since we
have
ˆpψ = −~ψ
0
= (~k)ψ.
So we know that the momentum eigenvalue is p = ~k. This looks encouraging!
Note that if there is no potential, i.e. V = 0, then
Hψ =
ˆp
2
2m
ψ = −
~
2
2m
ψ
00
=
~
2
k
2
2m
ψ.
So the energy eigenvalue is
E =
~
2
k
2
2m
.
Note, however, that our wavefunction has
|ψ
(
x
)
|
2
=
|C|
2
, which is a constant.
So this wavefunction is not normalizable on the whole line. However, if we
restrict ourselves to some finite domain
−
`
2
≤ x ≤
`
2
, then we can normalize by
picking C =
1
√
`
.
Example. Consider the Gaussian distribution
ψ(x) = C exp
−
x
2
2α
.
We get
ˆpψ(x) = −i~ψ
0
(x) 6= pψ(x)
for any number p. So this is not an eigenfunction of the momentum.
However, if we consider the harmonic oscillator with potential
V (x) =
1
2
Kx
2
,
then this
ψ
(
x
) is an eigenfunction of the Hamiltonian operator, provided we
picked the right α. We have
Hψ = −
~
2
2m
ψ
00
+
1
2
Kx
2
ψ = Eψ
when
α
2
=
~
2
Km
. Then the energy is
E
=
~
2
q
K
m
. This is to be verified on the
example sheet.
Despite being a simple system, the harmonic oscillator is incredibly useful in
theoretical physics. We will hence solve this completely later.
Definition
(Time-independent Schr¨odinger equation)
.
The time-independent
Schr¨odinger equation is the energy eigenvalue equation
Hψ = Eψ,
or
−
~
2
2m
ψ
00
+ V (x)ψ = Eψ.
This is in general what determines what the system behaves. In particular,
the eigenvalues E are precisely the allowed energy values.