2.3: Function Spaces
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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
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
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
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
where, from the orthonormality of the basis set
giving an identity operator in the space of continuous functions vanishing at 0 and
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
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,
Exercise
Write out the identity operator for the electron in a box
Functions on the Infinite Line
What happens if we take the analysis of the previous section and let
with
For the electron in a box (Fourier series) above we wrote the corresponding equation in Dirac notation as
with
It’s tempting to write down the analogous equations for the infinite line case, by translating the Fourier transform equations into Dirac notation, and blindly writing
This looks good, but has a problem—in contrast to the Fourier series basis functions
Furthermore,
But we never measure
This means we might be ok with this continuum basis of states: we don’t want them to be normalized in the traditional fashion
From our earlier definition of the delta function, we can express orthogonality of these
and, since the
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
Note: some authors prefer to define the normalized plane wave states by
Further Note: some prefer to go to a huge, but not infinite box, so the basis momentum eigenstates wave functions are the discrete set
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 force field—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
Differential Operators: the Momentum Operator on
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 trick is to integrate by parts:
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, so
Now
we have established that
So
(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
Proof:
We shall make clear that in this context we regard
where
Example
Take your favorite definition of the delta function, and prove that it isn’t normalizable, as defined in
Solution
(It wouldn’t be physically reasonable anyway—to localize a particle to a point would take infinite energy.) But the set of all
From the earlier result
it follows immediately that
Therefore,
Taking the inner product of at the point
Exercise
Check that Equation
It follows from the Equation
From this,
and
These are possibly the least rigorous equations in this section—we’re expressing one set of states outside of
Exercise: show these equations are consistent by substituting
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
The Basic Rules of Quantum Mechanics
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
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
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
Next Basic Rule: A physical variable, or observable, corresponds to a Hermitian operator
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,
Rule for Relating Operators to Experiments: any measurement of the value of the physical variable
The expectation value of an observable
It is important to note that two measurements of the same observable
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
and the expectation value of
where we’ve put a little hat on the