On how to extract information about the momentum of particles from the wave function.
In the last section, we understood the probabilistic interpretation
of the wave function for
measurements of position at time
. In this lecture we ask the
question: how does the wave function encode information about
measurements of momentum?
4.1
Momentum in Classical Mechanics
For motivation, we first recall the interpretation of momentum in
classical mechanics as the generator of translations in space.
Recall from last term’s lectures that functions of position and
momentum generate infinitesimal canonical transformations in Hamiltonian
mechanics. The infinitesimal canonical transformation with parameter
generated by a function
is where is
the Poisson bracket.
In particular, the canonical transformation generated by the function
is Here we have used that The important lesson
is that momentum generates an
infinitesimal translation of .
4.2
Momentum in Quantum Mechanics
We now use this idea of momentum to understand momentum in quantum
mechanics. Let us consider translating a wave function by a infinitesimal amount in the positive direction - as shown below. As in the
previous lecture, we work at some fixed moment in time.
Infinitesimal translation of a wave
function to the right,
(and note the sign!).
The translated wave function is of course . To first order in , the change in the wave function
is found by Taylor expanding, If we want momentum to generate translations,
this suggests we should identify momentum with the derivative with respect to
.
Let us therefore define a momentum
operator such that where is a constant of proportionality.
The additional factors of are
introduced for convenience; we will get back to them shortly. Some
comments are in order
Note that while the momentum has units of , has units of . This means must have units , or ‘energytime’.
Since the wave function is complex, we could imagine is a complex number. In a moment,
we will show that must be
real for momentum measurements to yield real results. This is why we
introduced the extra factor of
in the definition.
Planck's constant has to be present in quantum mechanics for dimensional reasons. It has dimension of ‘energytime' or ‘action'.
.
The constant is known as the
(reduced) Planck
constant, and pronounced ‘h-bar’.
Its value cannot be determined by mathematical arguments. It must be
determined by comparing to experimental data, for example atomic
spectra. In our universe, The smallness of this number, in units that are natural to
humans, is why we do not observe quantum mechanical effects in everyday
life. You will in the literature also find , which is usually called Planck’s constant
(without the ‘reduced’ prefix).
4.3 A
Quick Commutator
In order have position and momentum on an equal footing, we can
introduce a position operator that simply multiplies a wave function by . In summary, The ‘commutator’ of these operators is defined by
Here the operators should
always be understood to act on everything to the right. Acting with this
equation on a wave function , we have where the final line follows from the product
rule. Since this holds for any wave function , we can summarise this result by
This is known as the canonical
commutation relation.
The commutator is reminiscent of the Poisson bracket formula from classical mechanics.
In fact, the commutator in quantum mechanics is found by replacing This replacement rule is known as ‘canonical quantisation’.
We study it further in later lectures after introducing some more
mathematical machinery.
4.4
Momentum Expectation Values
Just like position, in quantum mechanics we can only compute the
probabilities of the outcomes of momentum measurements. For now, we
satisfy ourselves with computing expectation values of functions of
momentum.
First, recall from the last lecture that the expectation value can be written We now propose, similarly, that the expectation
value of momentum is As for the position expectation value, we
emphasise that is
interpreted as the average of momentum measurements on an ensemble of
particles with the same wave function .
Let us now return to explain why must be real. Since the outcomes of
momentum measurements are real numbers, we require . Let us
imagine for a second that is
complex and compute the complex conjugate of , In the passing second line, we have integrated by
parts. In passing to the third line, we discarded the boundary term
because must vanish
as if the wave
function is square normalisable. Therefore, the momentum expectation
value is real if and only if
is real.
In a similar way, we can compute more general expectation values
Of particular importance is the momentum
uncertainty which gives a measure of
the spread of momentum measurements around made on an ensemble of
particles with identical wave function .
Example: Gaussian Wave function
Let us again consider the wave function
with normalisation . In the last lecture, we showed that the
position expectation values are given by and , and
therefore the uncertainty in position is .
Gaussian wave function with width .
The action of the momentum operator on this wave function is Note that the result is always a polynomial in
times the original wave function.
This means we can recycle our results for position expectation values to
compute momentum expectation values. For example, and similarly The momentum uncertainty is therefore .
Note that the product of position and momentum uncertainties is
independent of , This means that if we attempt to localise the particle in
space by making smaller,
the the uncertainly in momentum necessarily increases, and vice verse.
4.5
Heisenberg’s Uncertainty Principle
This example illustrates an important result known as Heisenberg’s uncertainty
principle. This states that for any normalised wave function, We will prove this result later in the course. It shows that
there is a fundamental limit in quantum mechanics on the degree we can
simultaneously reduce the uncertainty in position and momentum. The
Gaussian wave function saturates this limit: it is a ‘minimal
uncertainty’ wave function.
Remember that is an
extremely small number in human units. So while we cannot arrange for
both and to vanish, both uncertainties
can be simultaneously small in human units. This goes some way to
explaining why in everyday life, objects appear to have a definite
position and momentum.
4.6Problems
PROBLEMS CLASS 1
Momentum expectation value The
expectation value of momentum measurements on a particle with wave
function is You may assume the wave function is normalised, .
Show that
is real.
Show that if is real
then .
Suppose a wave function has momentum expectation value
.
Compute the expectation value for
Solution ▶
We compute the conjugate In passing to the second line we have integrated
by parts. The boundary term vanishes provided the probability density
vanishes at . This is a necessary
condition for the wave function to be normalizable, .
Assuming the wave function is real , then which vanishes for the same reason as the
boundary term in part (a).
By direct computation where in the last step we used that the wave
function is normalized.
Gaussian wave function:
Reconsider the Gaussian wave function which you have seen in one of
the problems in the previous chapter.
Explain why the momentum expectation value is zero, .
Compute the momentum uncertainty .
Show that the wave function saturates Heisenberg’s uncertainty
principle,
What happens to
when the particle is localised in space?
What changes if you multiply the wave function by ?
Solution ▶
The wave function is real.
First compute the action of the momentum operator
on the wave function, Acting again, we find We can now re-use the position expectation values
and
computed in question 3 (together with the normalisation condition ) to compute the
momentum expectation values, If you are uncomfortable with the above
manipulations, you can first compute the momentum expectation values
using the integral definitions You will find that the computation is equivalent
to the one above! Finally, .
Recalling that , we have .
The uncertainty in momentum increases.
If you multiply the wave function by , the momentum
expectation value changes to .
TUTORIAL 1
Non-normalisable wave functions:
Consider the wave function for some constant and . The total integrated probability is
not defined, because the integral does not exist for any non-zero value of . However, we can make sense of it by
‘cutting off’ the integral so that (‘putting the particle in a box’), doing all
computations at finite , and then
taking the limit . If you use this
‘regularisation’, what is the expectation value when the system is described by the wave
function above? Also compute .
Solution ▶If we cut off the integrals, and the
expectation value of the momentum becomes This is independent of . However, manipulations like
these are on a shoddy mathematical ground (we e.g. did not discuss the
boundary conditions on the wave function), and we will see some examples
where naive reasoning like this goes flat on its face.
Momentum in a box:
Consider a particle in an infinite potential well, so that . If you want to show that is real (see above), you
will encounter boundary terms which need to vanish. Do they? Why?
Solution ▶The boundary term you encounter
when computing reads For generic wave
functions this would not
vanish. It will, however, vanish if the wave function vanishes at the
edge of the box, which is precisely the condition we have used in the
chapter where we first introduced the wave function.