A mathematical detour, in which we start viewing wave functions as vectors in an infinite-dimensional complex vector space: Hilbert space.
We now begin to develop the mathematical structures underlying
quantum mechanics more systematically. In this lecture, we introduce the
idea of wave functions as elements of a complex vector space with
Hermitian inner product. With a few additional assumptions, this is
known as a ‘Hilbert
space’.
6.1
Linear Algebra
Consider a finite-dimensional complex vector space . The most important
property is that of taking linear combinations: for any and complex numbers .
A Hermitian inner product on
is a map
that obeys
with equality iff .
It is often convenient to introduce an orthonormal basis such that and
any vector can be expressed Computing the inner product with
, the components of the vector
are . The
Hermitian inner product can then be expressed in component form as In particular, the squared norm of a vector is
.
6.2
Wave functions Revisited
At a fixed time , a wave
function is a continuous function In order for the wave function to have a
probabilistic interpretation, we require it to be square-normalisable,
This means that by multiplying by a constant,
we can ensure that the probability to find the particle anywhere is
.
The set of continuous square-integrable wave functions forms a
complex vector space. That is to say, for any square-integrable , and complex numbers , the wave
function is
square-integrable. This can be shown as follows. It is immediate that if
is square-integrable then
is square integrable for any
complex number . We can therefore focus on the sum . At each point we have where we used the properties of complex numbers,
and . We also have the
elementary inequality This implies and therefore This makes
it clear that if the wave functions and are square-integrable then the sum
is
square-integrable.
6.3
Inner Product
Let us define We claim this is a
Hermitian inner product:
with equality iff .
The first three properties follow immediately from the definition. To
prove the final property, note that and that the integrand is everywhere non-negative,
. This
immediately implies . Now suppose . Then vanishes everywhere except a
set of measure zero. However, since is a continuous function, we
must have
everywhere and therefore .
The Hermitian inner product obeys another property known as
‘completeness’. We will not need the definition in this course.
Including this property, the vector space of wave functions together
with the Hermitian inner product form a ‘Hilbert
space’.
6.4
Orthonormal Bases
It is frequently useful to introduce an orthonormal basis of wave
functions. In a later lecture, we will explain that there are natural
orthonormal bases that are ‘continuous’ or ‘discrete’ in nature and
arise from eigenfunctions of operators associated to observables such as
position, momentum and energy.
For now we define an orthonormal basis to be a discrete set of wave
functions such that
and any continuous square-integrable wave function
can be uniquely expressed where The Hermitian inner
product can be expressed in terms of the coefficients, while the the squared norm
becomes .
6.5
Example: Particle in a Box
Let us consider an infinite potential well in the region . We therefore restrict to
continuous square-integrable wave functions that vanish for and . In this case, we may replace
everywhere
The first three basis functions for a
particle in a box.
Let us define These wave functions are orthogonal
with respect to the inner product where in passing to the final line, we dropped
the second contribution because is impossible for .
The fact that any continuous square-integrable wave function has a
unique expansion of the form is the content of Fourier’s
theorem. The Fourier coefficients are found by taking the Hermitian
inner product with , The norm squared of the wave function
is which is
precisely the statement of Parseval’s
theorem.
The ‘pyramid’ wave function for a
particle in a box.
As an example, consider the ‘pyramid’ wave function displayed above.
With the correct normalisation, this wave function is The Fourier coefficients are computed as follows, In passing to the final line, the summands are
non-zero only when is odd, so we
introduced . As a
consistency check, so the wave
function is indeed correctly normalised.
6.6Problems
TUTORIAL 1
Properties of the inner product The
inner product on continuous square-integrable functions is
Show that
implies
for any constants .
Solution ▶
We have
We have
This follows from parts (a) and (b).
We have
since for all
.
Suppose . Then
can be non-zero at most
on a set of measure zero in . But since the wavefunction is
continuous,
everywhere and therefore .
PROBLEMS CLASS 2
Inconsistent? Consider the canonical
commutation relation We have argued that wave functions are vectors in
Hilbert space, and operators are ‘matrices’ acting on those vectors. In
this case, the operators
and are then both matrices,
and the right hand side should be read as ‘ times the unit matrix’. All fine
so far.
Now let use take the trace of the above equation, The left-hand side vanishes by virtue of
cyclicity of the trace. The right-hand side clearly does not vanish, as
it is the trace over the unit operator. How can this be consistent?
Solution ▶This argument shows that Hilbert
space cannot be finite-dimensional, as the trace of a commutator of
finite-dimensional matrices definitely vanishes, and the trace of a
finite-dimensional unit matrix does not.
If you write this in terms of wave functions, you can see the
kind of trouble we get into. The left-hand side would be an integral or
infinite sum over all normalisable wave functions, where the derivatives act on everything to the right,
and the ‘sum’ of course needs careful definition. However, the key point
is that e.g. in the second term, the derivative acts on , which may not be a
normalisable function!
For more details on this and related tricky points, see e.g.
F. Gieres.
Dirac's formalism and mathematical surprises in quantum mechanics.
Rept. Prog. Phys., 63:1893, 2000.
arXiv:quant-ph/9907069, doi:10.1088/0034-4885/63/12/201..