Function Spaces
Michael Fowler 9/19/07
Linear Algebra in Infinite Dimensions
The motivation for our review of linear algebra was the observation that the set of solutions to Schrödinger’s equation satisfies some of the basic requirements of a vector space, in that linear combinations of solutions give another solution to the equation. Furthermore, Schrödinger’s equation itself, as a differential operator acting on a function, suggests that the concept of a matrix operator acting on vectors in an n-dimensional vector space can be extended to more general operators, such as differential operators, acting on functions in an infinite-dimensional space.
Our analysis of linear vector spaces began by defining an inner product, which was used to establish an orthonormal basis for the space. Constructing a well-defined basis for the space of all functions on the real axis sounds impossible, and probably is. Fortunately, we don’t need to be so all-encompassing. For one thing, we are not interested in functions with discontinuities, because in quantum mechanics that would be a wavefunction corresponding to infinite energy. (We can allow discontinuities in slope, although, as discussed in the Electron in a Box lecture, that occurs only where the potential is infinite. Infinite potentials are of course unphysical, but are convenient approximations in some cases, so we’ll keep that option open.) Another important restriction arises from the requirement that the wavefunction describe a single particle—it must be normalizable, that is to say the norm
and in fact 
must be scaled so that
this integral is equal to unity for actual computation of probabilities. Note
that 
means 
but the norm turns out
to be time-independent, as it must be, for the case of a single particle.
Building on the analogy with n-dimensional vector spaces, the requirement of finite norm suggests a definition for the inner product in function space:
This definition satisfies Dirac’s requirement that 
gives a positive norm,
and is linear in f, g. The
space of functions with this inner product, and with finite norm 
is written 
or just L2
. The functions are said to be “square
integrable”.
Notice that this inner product resembles the linear algebraic bra-ket product if we imagine every point on the line as an independent basis vector—mathematically meaningless, of course, but a hint of where we’re going.
Electron in a Box Again
As a preliminary to discussing functions on the infinite line, it is worth considering those restricted to the finite interval (0, L) and vanishing at the two ends. These are precisely the conditions satisfied by the electron-in-a-box wavefunctions (see the earlier lecture):
Recall from the Fourier Series lecture that any function
without discontinuities can be represented as a sum over Fourier
components. For the present case of
functions equal to zero at the two ends (as any physical wavefunction in a box must
be) the sine kets 
above form a complete
set, that is to say, at t = 0, any satisfying the
boundary condition can be written:
where, from the orthonormality of the basis set , the Fourier coefficients 
so (making explicit that is in fact a ket in this vector space)
giving an identity operator in the space of continuous functions vanishing at 0 and L:
exactly analogous to that in finite-dimensional vector spaces. The inner product of two functions
defined as in the preceding section by
is equivalently, in terms of Fourier coefficients,
and the normalization
So for the electron-in-a-box wavefunctions, the orthonormal basis of sine functions gives a well-defined infinite-dimensional vector space.
We have previously stated that the standard interpretation
of the wavefunction is that 
is the probability of
finding the particle in a small interval dx near x, and on
integrating over all x the total probability of finding the particle is
one. But we could also look for the
particle in a particular state, rather than in a particular small
interval dx. In this case, |an|2
is the probability of finding the particle in the nth
state. This is consistent with the
previous interpretation, and is parallel to our earlier analysis of the
probability of a particle having a particular momentum. The state coefficient an is
called the amplitude, or sometimes the probability amplitude.
You might be wondering how we would measure that a particle
is in a particular state. The answer is
to wait for it to jump out. If an atom
is excited (for example by a short burst of radiation) it will be excited to a state which is a
linear superposition of different energy eigenstates, 
, rather than to a single eigenstate. It will usually return to the ground state by
emitting one or a series of photons, and the frequency of an emitted photon
reveals the energy difference between the atomic states involved. For a collection of atoms excited in the same
way, the relative intensities of different spectral lines give the relative
probabilities of different states. Of
course, a long almost monochromatic wave packet of incoming radiation will tend
to put all the excited atoms into the same state.
Exercise: Write out the identity operator for the
electron in a box using the explicit
form 
Prove this is equivalent
to the delta function when operating on other functions within the box. What is the
behavior of this function outside the
box?
Functions on the Infinite Line
What happens if we take the analysis of the previous section
and let L go to infinity? This is
parallel to the analysis (two lectures back) of going from Fourier series to
the Fourier transform, the sum over a series of plane waves satisfying a
boundary condition becoming an integral over the continuum of all
plane waves. In that lecture, we saw
that as L went to infinity, the amplitude of the normalized eigenstates went to zero as
, and therefore so did the individual coefficients 
However, the density
of these eigenstates in momentum space increased as L, so overall the
factors of L cancelled and the sum
tended to a finite integral, specifically
For the electron in a box (Fourier series) above we wrote the corresponding equation in Dirac notation as
It’s tempting to write down the analogous equations for the infinite line case, by translating the Fourier transform equations into Dirac notation:
This looks good, but has a problem—in contrast to the
Fourier series basis functions 
these Fourier
transform “basis states” 
are infinitely long
plane wave states 
and therefore not
normalizable in the sense we’ve used that term so far. They’re not even in the space we’re supposed
to be working in!
Furthermore, 
is not the probability that a measurement
of the momentum of the electron will yield precisely the value 
The correct
probabilistic interpretation for a continuum of k-values is exactly parallel to
the continuum of x-values in ordinary space: 
is the probability
that a measurement of momentum would find the k-value to be in a small interval of width dk near k. The probability goes to zero with the width
of the interval, and so is vanishingly
small if we demand an exact value
of k.
But we never measure k with infinite precision anyway—that would take an infinitely large apparatus. The physically significant quantity is the probability of finding k in a small interval dk—in practice, with real detectors, we are always integrating over some (small) range in k.
This means we might be ok with this continuum basis of
states: we don’t want them to be normalized in the traditional fashion 
because that would
correspond to a finite probability of the particle having a mathematically
precise value of k, which makes no
physical sense—in fact it’s nonsense. The normalization we need is one that
makes sense in the context of an integral over a small interval in k—but still of course over a continuous
infinity of basis states!
From our earlier definition of the delta function, we can
express orthogonality of these states:
and, since the 
-function is normalized in the sense that it has total weight
one in an integral, we take this equation as the definition of the
normalization of the functions . That is to say, we
take the state to have wave function 
with A = 1.
Now the delta function is only meaningful inside an integral, therefore so is our normalization, and the formalism, a continuum basis of plane wave states with delta function orthogonality, although perhaps leaving something to be desired from a strict mathematical perspective, turns out to be a consistent and reliable way of formulating quantum mechanics.
Exercise: from the expression for the identity
operator above, 
Substitute 
and check that this
makes sense.
Note: some authors prefer to define the normalized
plane wave states by 
in which case 
and the 
appearing in the above
integral for the identity operator becomes simply dk. With our convention, dk always appears with a 
in the denominator.
Further Note:
some prefer to go to a huge but not infinite box, so the basis momentum
eigenstates wave functions are the discrete set 
, V being the volume. For this huge box, it is safe to replace the
sum over discrete momentum states by an integral, bearing in mind that the
density of states in phase space being proportional to L gives 
in three
dimensions. The L or V factors finally
cancel in computations, as we shall discover later.
Schrödinger’s Equation as an Operator on a Vector Space
As we recounted at the beginning of this course, when Schrödinger was challenged to find a wave equation for the electron wave, he constructed one parallel to the electromagnetic “photon wave equation”, that is to say, he took the energy-momentum equation and wrote
He discovered that the three-dimensional version of the differential equation constructed in this way could be solved by standard analytic methods for an electron in an inverse-square potential—the hydrogen atom. The standing wave solutions yielded the right set of energy levels—those Bohr had found earlier with his simplistic model. This confirmed that indeed the wave equation describing the propagation of the electron waves had been discovered, and it was
with E, p the differential operators given
above. Since the operator in brackets is linear,
the solutions 
form a linear vector space.
Differential Operators: the Momentum Operator on L2
Our task now is to recast this old approach of differential operators acting on wave functions in the equivalent Dirac language. Let’s begin with the simplest, the momentum operator. First, we need to show that it is Hermitian:
The important point here is that the last term, the contribution from the infinite endpoints of the integration, must be zero because square-integrable functions must go to zero at infinity. Then it becomes apparent that
with the appropriate sign change i going to -i on
taking the adjoint of 
We have established that 
so, taking the
adjoint:
and since 
, we have established that 
between any two states
in the space: so this is an operator identity, and 
is Hermitian. (The i is important: the differential
operator d/dx alone is not Hermitian, it’s anti Hermitian
in L2!
So px is a Hermitian operator, and therefore has real eigenvalues, which it must have since momentum is a physical quantity. But what are its eigenvectors? We already know, of course, that they are the plane wave states—this is the whole reason this particular operator was chosen in building the wave equation in the first place. Strictly speaking, though, as we’ve already discussed, these plane wave states are not in L2. Nevertheless, any smooth function in L2 can be expressed as an integral over these states, so they do form a complete basis for the functions relevant to physics.
(It is true that later, in scattering theory and some other places, we may talk about plane waves without always doing an integral: such loose talk should be understood as referring to a very long but finite wave packet, well approximated by a plane wave during the scattering event.)
The Position Operator and Its Eigenstates
The “position” is just the co-ordinate x, manifestly always real, and a Hermitian operator.
Proof: 
We shall make clear that in this context we regard x as an operator by writing it with a
little hat, 
It is equally clear
that the eigenstates of 
states in which the
particle has probability equal to one of being at a particular position, must
be delta functions corresponding to that position: that’s the only function
with zero probability of finding the particle anywhere else. So if 
is an eigenstate of x
with eigenvalue a,
where C is a constant. But, whatever value we choose
for C, this wave function, like the momentum eigenstate, isn’t
normalizable—so, in fact, itself could never be
the wave function of a particle!
Exercise: Take your favorite definition of the delta function, and prove that it isn’t normalizable, as defined in L2.
(It wouldn’t be physically reasonable anyway—to localize a
particle to a point would take infinite energy.) But the set of all ’s is certainly complete,
and therein lies its value: it is a
basis for the space. The convention is
to “normalize” these kets, or rather to construct an “orthonormal set”, by
analogy with the orthonormalization convention for the plane-wave momentum
states, that is, to take
From the earlier result
it follows immediately that C = 1.
Therefore,
Taking the inner product of 
with the bra 
just gives the value
of at the point x.
Consequently any function in L2
can be written:
Exercise: check that this is true by finding 
It follows from the above equation that the identity operator
in L2 can be written in terms of the eigenstates of 
From this, can be written
,
and
These are possibly the least rigorous equations in this
section—we’re expressing one set of states outside of L2 in terms of another such set, using both sets as
bases in L2! Obviously, this is only meaningful with an 
state defined as a
zero-width limit of narrowing Gaussians (say) and a state as a limit of
longer and longer wavepackets, tending to a single k-value. Yet despite the
lack of rigor in the above presentation, these states, used with care, are in
fact reliable and efficient tools for analyzing quantum mechanical
problems. We shall use them often.
Exercise: show these equations are consistent by
substituting from the first into
the right-hand side of the second, to give 
.
The Hamiltonian Operator
The Hamiltonian operator gives the time development of the wavefunction. It corresponds to the total energy. If the wavefunction corresponds to a definite energy, the time dependence can be factored out, and the spatial wave function is a solution of Schrödinger’s time independent equation:
Since we only
consider the space L2 of wave
functions on which both p and x are Hermitian, H must be Hermitian, and therefore has
real eigenvalues.
The Basic Rules of Quantum Mechanics
(You can also find these in Shankar Chapter 4, where my “rules” are called “postulates” or Griffiths Chapter 3 “Generalized Statistical Interpretation”, or in fact, in any Quantum text.)
Any quantum mechanical wave function must be normalizable, because the norm represents the total probability of finding the particle (or, more generally, the system) somewhere in its phase space, so
First Basic Rule: any state of the particle is a ket , symbolizing a function in L2.
Mathematicians use the term Hilbert space to refer to inner-product spaces of normalizable functions such that any convergent sequence in the space has a limit in the space (a property that, for example, the rational numbers don’t have, but the real numbers do) . Our functions above for the electron in the box do form such a space, with the sine waves an orthonormal basis. However, on going to the infinite line, although we still have normalizable wave functions, the two bases we have discussed above, the plane waves (momentum basis) and the delta functions (position basis) are not themselves in the space—by which we mean they are not normalized as defined in L2.
But these bases are both complete, meaning any wave function can be expressed in terms of a (continuous) sum over the elements of either of them.
Constructing these complete but not conventionally normalized bases was Dirac’s doing, and is extremely convenient in describing quantum mechanics. But it upset the mathematicians. Fortunately, they later justified it by inventing the theory of distributions, which are generalized functions, and include delta functions.
Bottom Line: we shall follow the other physicists in using the term “Hilbert space” more loosely than mathematicians do, to refer to L2, extended to include these non-normalizable bases.
Next Basic Rule: A physical variable, or observable, corresponds to a Hermitian operator A acting on L2.
We shall assume that the eigenkets of any such variable span the space: this is always true for a finite dimensional space, as previously discussed, but not for a general Hermitian operator in a Hilbert space, so this is a nontrivial assumption.
For an operator with a discrete
set of eigenvalues, 
, any wave function can be written
Rule for Relating
Operators to Experiments: any
measurement of the value of the physical variable A will yield one of the
eigenvalues 
of the operator A, and
the probability of finding the particular value is equal to 
The expectation value of an observable A is the average value of a series of measurements on identical quantum systems,
It is important to note that two measurements of the same
observable A on the same system, one
measurement being made immediately
after the other, must yield the same result.
That is to say, if the first measurement reads , the second must be with 100%
probability. But this can only happen if the wave function after the first
measurement is , which in general it wasn’t before the first
measurement. The jargon description of
this is that the act of measurement “collapses the wave function” into one of
the eigenstates of the variable being measured.
Measuring a Continuum Variable: For variables like position and momentum having continuum sets of eigenvectors, the statistical interpretation is in terms of finding the particle within some small range—the probability of finding it between x and x + dx is
and the expectation value of x is
where we’ve put a little hat on the x to remind us that it is an operator,
with eigenkets .