3Expectation and uncertainty
IB Quantum Mechanics
3.1 Inner products and expectation values
Definitions
Definition
(Inner product)
.
Let
ψ
(
x
) and
φ
(
x
) be normalizable wavefunctions
at some fixed time (not necessarily stationary states). We define the complex
inner product by
(φ, ψ) =
Z
∞
−∞
φ(x)
∗
ψ(x) dx.
Note that for any complex number α, we have
(φ, αψ) = α(φ, ψ) = (α
∗
φ, ψ).
Also, we have
(φ, ψ) = (ψ, φ)
∗
.
These are just the usual properties of an inner product.
Definition
(Norm)
.
The norm of a wavefunction
ψ
, written,
kψk
is defined by
kψk
2
= (ψ, ψ) =
Z
∞
−∞
|ψ(x)|
2
dx.
This ensures the norm is real and positive.
Suppose we have a normalized state
ψ
, i.e.
kψk
= 1, we define the expectation
values of observables as
Definition
(Expectation value)
.
The expectation value of any observable
H
on
the state ψ is
hHi
ψ
= (ψ, Hψ).
For example, for the position, we have
hˆxi
ψ
= (ψ, ˆxψ) =
Z
∞
−∞
x|ψ(x)|
2
dx.
Similarly, for the momentum, we have
hˆpi
ψ
= (ψ, ˆpψ) =
Z
∞
−∞
ψ
∗
(−i~ψ
0
) dx.
How are we supposed to interpret this thing? So far, all we have said about
operators is that if you are an eigenstate, then measuring that property will give
a definite value. However, the point of quantum mechanics is that things are
waves. We can add them together to get superpositions. Then the sum of two
eigenstates will not be an eigenstate, and does not have definite, say, momentum.
This formula tells us what the average value of any state is.
This is our new assumption of quantum mechanics — the expectation value
is the mean or average of results obtained by measuring the observable many
times, with the system prepared in state ψ before each measurement.
Note that this is valid for any operator. In particular, we can take any
function of our existing operators. One important example is the uncertainty:
Definition
(Uncertainty)
.
The uncertainty in position (∆
x
)
ψ
and momentum
(∆p)
ψ
are defined by
(∆x)
2
ψ
= h(ˆx − hˆxi
ψ
)
2
i
ψ
= hˆx
2
i
ψ
− hˆxi
2
ψ
,
with exactly the same expression for momentum:
(∆p)
2
ψ
= h(ˆp − hˆpi
ψ
)
2
i
ψ
= hˆp
2
i
ψ
− hˆpi
2
ψ
,
We will later show that these quantities (∆
x
)
2
ψ
and (∆
y
)
2
ψ
are indeed real
and positive, so that this actually makes sense.
Hermitian operators
The expectation values defined can be shown to be real for
ˆx
and
ˆp
specifically,
by manually fiddling with stuff. We can generalize this result to a large class of
operators known as Hermitian operators.
Definition
(Hermitian operator)
.
An operator
Q
is Hermitian iff for all nor-
malizable φ, ψ, we have
(φ, Qψ) = (Qφ, ψ).
In other words, we have
Z
φ
∗
Qψ dx =
Z
(Qφ)
∗
ψ dx.
In particular, this implies that
(ψ, Qψ) = (Qψ, ψ) = (ψ, Qψ)
∗
.
So (ψ, Qψ) is real, i.e. hQi
ψ
is real.
Proposition.
The operators
ˆx
,
ˆp
and
H
are all Hermitian (for real potentials).
Proof.
We do
ˆx
first: we want to show (
φ, ˆxψ
) = (
ˆxφ, ψ
). This statement is
equivalent to
Z
∞
−∞
φ(x)
∗
xψ(x) dx =
Z
∞
−∞
(xφ(x))
∗
ψ(x) dx.
Since position is real, this is true.
To show that
ˆp
is Hermitian, we want to show (
φ, ˆpψ
) = (
ˆpφ, ψ
). This is
equivalent to saying
Z
∞
−∞
φ
∗
(−i~ψ
0
) dx =
Z
∞
−∞
(i~φ
0
)
∗
ψ dx.
This works by integrating by parts: the difference of the two terms is
−i~[φ
∗
ψ]
∞
−∞
= 0
since φ, ψ are normalizable.
To show that H is Hermitian, we want to show (φ, Hψ) = (Hφ, ψ), where
H = −
h
2
2m
d
2
dx
2
+ V (x).
To show this, it suffices to consider the kinetic and potential terms separately.
For the kinetic energy, we just need to show that (
φ, ψ
00
) = (
φ
00
, ψ
). This is true
since we can integrate by parts twice to obtain
(φ, ψ
00
) = −(φ
0
, ψ
0
) = (φ
00
, ψ).
For the potential term, we have
(φ, V (ˆx)ψ) = (φ, V (x)ψ) = (V (x)φ, ψ) = (V (ˆx)φ, ψ).
So H is Hermitian, as claimed.
Thus we know that hxi
ψ
, hˆpi
ψ
, hHi
ψ
are all real.
Furthermore, observe that
X = ˆx − α, P = ˆp − β
are (similarly) Hermitian for any real α, β. Hence
(ψ, X
2
ψ) = (ψ, X(Xψ)) = (Xψ, Xψ) = kXψk
2
≥ 0.
Similarly, we have
(ψ, P
2
ψ) = (ψ, P (P ψ)) = (P ψ, P ψ) = kP ψk
2
≥ 0.
If we choose
α
=
hˆxi
ψ
and
β
=
hˆpi
ψ
, the expressions above say that (∆
x
)
2
ψ
and
(∆p)
2
ψ
are indeed real and positive.
Cauchy-Schwarz inequality
We are going to end with a small section on a technical result that will come
handy later on.
Proposition
(Cauchy-Schwarz inequality)
.
If
ψ
and
φ
are any normalizable
states, then
kψkkφk ≥ |(ψ, φ)|.
Proof. Consider
kψ + λφk
2
= (ψ + λφ, ψ + λφ)
= (ψ, ψ) + λ(ψ, φ) + λ
∗
(φ, ψ) + |λ|
2
(φ, φ) ≥ 0.
This is true for any complex λ. Set
λ = −
(φ, ψ)
kφk
2
which is always well-defined since
φ
is normalizable, and then the above equation
becomes
kψk
2
−
|(ψ, φ)|
2
kφk
2
≥ 0.
So done.