This is easily seen to be zero without an explicit calculation, be-
cause when we take the antiderivative ofe−^2 iθ, we will get the
same type of exponential, whose values when we plug in the up-
per and lower limits of integration will cancel each other out.
Imaginary momentum? example 13
Here’s a paradox. If we take a wavefunctioner x, whereris a
constant, then applying the momentum operatorOp = −id/dx
(example 5, p. 976) givesOper x=−i r er x.For a state of definite momentum, we normally have in mind, as
in examples 5 and 12, an oscillating wave wherer=i kis purely
imaginary. But what ifris real, sayr= 1 (ignoring units)? Then
our wavefunction isex, and it’s a state of definite momentum
—imaginarymomentum. Oh no, what’s going on? Nice polite
observables like momentum aren’t supposed to have imaginary
eigenvalues.
The resolution to this paradox lies in the fundamental principles of
quantum mechanics that we’ve learned. Wavefunctions are sup-
posed to belong to a vector space in which we have a well-defined
inner product. A wavefunction likeΨ=exis ruled out by this re-
quirement, because〈Ψ|Ψ〉is infinite, and therefore undefined.
Of course we could raise the same objection to a wavefunction
likeΦ=ei k xdefined for all real values ofx. But when we work
with wavefunctions likeΦ, we usually just have in mind a compu-
tational shortcut, with the actual wavefunction being some kind of
wavepacket or wave train consisting of a finite number of wave-
lengths. (Or we could be talking about rotation, as in the quantum
moat of example 12. Note that in such an example, oscillating
functions can be made to join smoothly to themselves as they
wrap around, but this doesn’t work with functions likeex.)Averages
The average family lives down the street from me. Their family
income in 2014 was $72,641, and they have 2.5 kids. This joke
depends on the fact that you can’t superpose families to make a
single family — but wecan do this for wavefunctions. Suppose
that the particle-in-a-box wavefunction has a definite energy of
1 unit,OE = 1. This says that is a state of definite energy
1, so that when we act on it with the energy operatorOE, the result
is just to multiply the wave by 1 (the eigenvalue).
If this is true, then shortening the wavelength by a factor of 2
means increasing the momentum by a factor of 2, and increasing the
energy by a factor of 4. Therefore the wavefunction has 4 units
of energyOE = 4.
Now there is nothing wrong with mixing these together to get aSection 14.6 The underlying structure of quantum mechanics, part 2 985