5Axioms for quantum mechanics
IB Quantum Mechanics
5.4 Discrete and continuous spectra
In stating the measurement axioms, we have assumed that our spectrum of
eigenvalues of
Q
was discrete, and we got nice results about measurements.
However, for certain systems, the spectra of
ˆp
and
H
may be continuous. This
is the same problem when we are faced with non-normalizable momentum
wavefunctions.
To solve this problem, we can make the spectra discrete by a technical device:
we put the system in a “box” of length
`
with suitable boundary conditions of
ψ
(
x
). We can then take
` → ∞
at the end of the calculation. We’ve discussed
this in some extent for momentum eigenstates. We shall revisit that scenario in
the more formal framework we’ve developed.
Example.
Consider
ψ
(
x
) with periodic boundary conditions
ψ
(
x
+
`
) =
ψ
(
x
).
So we can restrict to
−
`
2
≤ x ≤
`
2
.
We compare with the general axioms, where
Q = ˆp = −i~
d
dx
.
The eigenstates are
χ
n
(x) =
1
√
`
e
ik
n
x
, k
n
=
2πn
`
.
Now we have discrete eigenvalues as before given by
λ
n
= ~k
n
.
We know that the states are orthonormal on −
`
2
≤ x ≤
`
2
, i.e.
(χ
n
, χ
m
) =
Z
`
2
−
`
2
χ
m
(x)
∗
χ
n
(x) dx = δ
mn
.
We can thus expand our solution in terms of the eigenstates to get a complex
Fourier series
ψ(x) =
X
n
α
n
χ
n
(x),
where the amplitudes are given by
α
n
= (χ
n
, ψ).
When we take the limit as
n → ∞
, the Fourier series becomes a Fourier integral.
There is another approach to this problem (which is non-examinable). We
can also extend from discrete to continuous spectra as follows. We replace the
discrete label n with some continuous label ξ. Then we have the equation
Qχ
ξ
= λ
ξ
χ
ξ
.
These are eigenstates with an orthonormality conditions
(χ
ξ
, χ
η
) = δ(ξ −η),
where we replaced our old
δ
mn
with
δ
(
ξ − η
), the Dirac delta function. To
perform the expansion in eigenstates, the discrete sum becomes an integral. We
have
ψ =
Z
α
ξ
χ
ξ
dξ,
where the coefficients are
α
ξ
= (χ
ξ
, ψ).
In the discrete case,
|α
n
|
2
is the probability mass function. The obvious gener-
alization here would be to let
|α
ξ
|
2
be our probability density function. More
precisely,
Z
b
a
|α
ξ
|
2
dξ = probability that the result corresponds to a ≤ ξ ≤ b.
Example.
Consider the particle in one dimension with position as our operator.
We will see that this is just our previous interpretation of the wavefunction.
Let our operator be Q = ˆx. Then our eigenstates are
χ
ξ
(x) = δ(x − ξ).
The corresponding eigenvalue is
λ
ξ
= ξ.
This is true since
ˆxχ
ξ
(x) = xδ(x − ξ) = ξδ(x − ξ) = ξχ
ξ
(x),
since δ(x − ξ) is non-zero only when x = ξ.
With this eigenstates, we can expand this is
ψ(x) =
Z
α
ξ
χ
ξ
(x) dξ =
Z
α
ξ
δ(x − ξ) dξ = α
x
.
So our coefficients are given by α
ξ
= ψ(ξ). So
Z
b
a
|ψ(ξ)|
2
dξ
is indeed the probability of measuring the particle to be in
a ≤ ξ ≤ b
. So we
recover our original interpretation of the wavefunction.
We see that as long as we are happy to work with generalized functions like
delta functions, things become much nicer and clearer. Of course, we have to be
more careful and study distributions properly if we want to do this.