Linear algebra application
The properties listed here for
inner products in quantum me-
chanics are just standard rules
for inner products in linear al-
gebra.
rather than the more familiar property of the Euclidean dot product
u·v=v·u.Inner product
Wavefunctions come equipped with an inner product that has the
properties described above.If we’re dealing with wavefunctions that are expressed as func-
tions of position, then it’s pretty clear how to define an appropriate
inner product: 〈u|v〉=∫
u∗vdx. The inner product axiom stated
above then requires that this (possibly improper) integral converge
in all cases, which means, for example, that we have to exclude
infinite plane waves from consideration. However, because it’s so
convenient sometimes to talk about plane waves, we may break this
rule when nobody is looking. Note the similarity between the ex-
pression∫
u∗vdxand the expressionuxvx+uyvy+uzvzfor a dot
product: the integral is a continuous sum, and the dot product is a
discrete sum.
Two wavefunctions have a zero inner product if and only if they
are completely distinguishable from each other by the measurement
of some observable. By analogy with vectors in Euclidean space,
we say that the two wavefunctions are orthogonal. For example,
〈 | 〉= 0, as can be verified from the integral∫π
0 sinxsin 2xdx=- These states are also distinguishable by measuring either their
momentum or their energy.
Let’s consider more carefully the general justification for this as-
sertion that perfect distinguishability is logically equivalent to a zero
inner product. We have described valid observables in quantum me-
chanics as being represented by operators that have real eigenvalues.
An alternative description of such an operatorO, called a hermitian
operator^2 after Charles Hermite, is that it is one such that for anyu
andv, the equation〈Ou|v〉=〈u|Ov〉holds.^3 Being hermitian is, for
an operator, analogous to being real for a number. (Cf. problem 8,
p. 1009.) Just as a randomly chosen complex number is unlikely to
be real, a randomly chosen linear operator will almost never be her-
mitian. Like love, patriotism, or beauty, a nonhermitian operator
fails to translate into anything a physicist can measure.
Using this alternative characterization of what makes a valid
(^2) The mathematician’s standard definition of a hermitian operator adds an
additional technical condition, which is that all of the operator’s eigenvalues
should have magnitudes below a certain fixed bound. This is much too restrictive
for our purposes, since, for example, an alpha particle in free space can have an
arbitrarily large kinetic energy. In fact, nothing really bad happens if we relax
our requirement for quantum-mechanical operators to be that they merely need
a property called beingnormal.
(^3) Proof that a hermitian operator has real eigenvalues: Letebe an eigenvalue,
Ou=euforu 6 = 0. Then〈Ou|u〉=〈u|Ou〉, so〈eu|u〉=〈u|eu〉, ande∗〈u|u〉=
e〈u|u〉, soe∗=e, meaning thateis real.
982 Chapter 14 Additional Topics in Quantum Physics