Thus far, the only operator
we've considered has been the
energy operator.
That is, the Hamiltonian.
As we mentioned, for every
measurable physical quantity
in quantum mechanics, there
should be an operator.
As we prepare here to look at
the uncertainty principle, a
relation that applies to the
measurement of position and
momentum, we will first formally
define the operators
associated with these measurable
quantities.
Of course, these will allow
us to define and calculate
expectation values of both
position and momentum.
We will start first by formally
introducing the
momentum operator.
So for momentum, we're going
to write an operator.
So we put a little hat over this
p, and we're simply going
to postulate that this momentum
operator can be
written as minus i times h bar
times the gradient operator.
And of course, the gradient
operator is just this set of
derivatives in each of the
coordinate directions, and
here, to avoid some other
confusions, we're defining x
nought and y nought, and z
nought as unit vectors in the
x, and y, and z directions.
So therefore, we're saying the
momentum operator is minus h
bar times the gradient, the
spatial derivative operator in
vector form.
So with this postulated form
here that our momentum
operator is minus ih bar times
a gradient operator, we find
that if we simply square this
operator and put it over 2m,
then what we have is minus
h bar squared over
2m times del squared.
And of course, we recognize
that from the part of
Schroedinger's equation.
And we see therefore that with
this choice which we have just
postulated, we're beginning to
see a correspondence between
the classical notion of the
energy, E, which is p squared
over 2m in classical mechanics
plus V, the potential, and the
corresponding Hamiltonian
operator of the Schroedinger
equation, so the Hamiltonian
operator with the hat being
minus h bar over 2m,
del squared plus V.
And we could rewrite that with
our new definition of the
momentum operator as the
momentum operator squared over
2m plus V. Now, note that if we
operate on the state e to
the ik dot r with the momentum
operator, that would be minus
ih bar times the gradient
of e to the ik dot r.
If we take the gradient of e
to the ik dot r using the
components in each of the
different vector directions
and all of those different
vector components of the
gradient, we're going to get i
times k, e to the ik dot r.
And that i times the minus i we
had here just gives us 1.
So the effect of minus i h bar
times the gradient on e of the
ik dot r is to give us h bar
k times e to the ik dot r.
And so far this is just a
mathematical operation saying
that if we operate with the
momentum operator on this kind
of plane wave spatial state,
we're going to get h bar k
times the same plane
wave spatial state.
Well, that means that these
spatial plane waves are
actually the eigenfunctions
of the operator p.
And they have eigenvalues
h bar k.
Why do we say that these plane
waves are the eigenfunctions
of this operator p, with
eigenvalues h bar k?
This is a vector eigenvalue,
but that's a detail.
The reason is that we operated
with this operator p on this
function, and found that we got
this value, in this case a
vector value times the
same function.
That means by definition that
these functions are the
eigenfunctions.
And these h bar k's here
are the eigenvalues.
Now as I said, these are vectors
in this case, but it's
still the same basic idea.
We can therefore say for these
eigenstates that the momentum
is h bar k.
That's the eigenvalue of the
momentum in this state.
It's a perfectly well defined
value, vector value when we
have this kind of state
and we operate with p.
Note that p here is a vector
with three components with
different scalar values.
And it's not an operator.
This p is not an operator, it's
just a vector, which we
can say is momentum if we
happen to be in this
particular kind of state,
e to the ik dot r.
Now, up to this point we've
only been talking about
eigenfunctions and eigenvalues
of energy.
Here we are saying that we
have a new operator, one
representing another physical
quantity, here, momentum.
And this operator also can
have eigenfunctions and
eigenvalues.
It's important to understand
that these are not necessarily
also functions of the energy
operator, that is, of the
Hamiltonian.
For the particular case where
the potential is constant
everywhere, the momentum
operator and the Hamiltonian
happen to have the same
eigenfunctions, but generally
they do not.
If we add a potential into the
Hamiltonian, we will get some
different set of eigenfunctions
for the
Hamiltonian, but the
eigenfunctions of the momentum
operator remain unchanged.
They are still these
plane waves.
At the very least,
we're making a
mathematical statement.
For the operators that we come
across that represent some
physical quantities such as
energy, or momentum, position
or angular momentum, each one
of these will have its own
eigenfunctions and
eigenvalues.
The energy eigenfunctions have
the special property that they
represent states that are stable
or constant, or what we
might call stationary in time,
as far as any measurable
quantity is concerned.
In general, the eigenfunctions
of other operators, like
momentum, do not represent
states that are stable in time
unless they also happen
to be eigenfunctions
of the energy operator.
Mathematically, the existence
of these eigenfunctions and
eigenvalues of these other
operators is quite important
as we handle quantum
mechanics.
But, as I said, they don't
necessarily have the
particular properties of the
energy eigenstates of
representing systems that are
stable or stationary in time.
To emphasize, for the momentum
operator, we just made the
observation that if we operated
with it on a plane
wave state, e to the ik dot r,
then we got a constant, here h
bar times the vector k
times the function.
By the definition of
eigenfunctions and
eigenvalues, that means, as
we've pointed out, that this
plane wave is necessarily an
eigenstate of this operator,
in this case with the vector
eigenvalue h bar
times the vector k.
Since such spatial plane wave
states generally can be used
to describe any function in
space, a fact we may already
be familiar with from Fourier
transforms, we're seeing that
this set of eigenfunctions
is also a complete set.
Again, we will see this property
of completeness
rather generally for the
eigenfunctions of quantum
mechanical operators
representing physical
quantities.
Now let's look briefly at
the positional operator.
For the position and the
positional operator, we simply
postulate that the operator,
which turns out to be almost
trivial when we're working with
functions of position, is
simply the position
vector r itself.
At least when we're working in
a representation that is in
terms of position, we therefore
typically do not
write r with a little hat on it,
though perhaps rigorously
we really should do that.
The operator for the z component
of position would,
for example, also simply
be z itself.
As I said, this is a consequence
of the fact that
we are working in what we call
the position representation.
We're still talking about
functions that are functions
of position.
We're essentially writing down
a list of values of the
function for each different
position as the way we're
thinking about the function.
And because of that reason,
we tend to think about the
position operator as just being
the position itself.
Now we're going to take
our first look at
the uncertainty principle.
A commonly quoted form is
to say that we cannot
simultaneously know both
the position and the
momentum of a particle.
Another form would say that
measuring a system necessarily
changes it.
Actually stating the uncertainty
principle these
ways, especially that second one
about changing the system
is really mixing two
different things.
One aspect is quantum mechanical
measurement.
There we say that measuring a
quantity collapses it into an
eigenstate.
Now, as we've discussed, this
has at least philosophical
problems, and arguably
we don't really
understand this process.
We could say that this issue
is not the uncertainty
principle, however.
This is the measurement
problem.
A second and quite separate
issue is the notion that it's
not meaningful to require
the state of a system to
simultaneously have well defined
values of two specific
quantities.
This is what we will be
referring to as an uncertainty
principle, and at least in our
minds, we're separating it out
from the measurement problem.
At the time when the uncertainty
principle was
first being discussed,
understandably these two
issues were mixed together.
And in popular discussions,
that mixing
continues to this day.
In our discussions, though, here
we are going to separate
these two problems.
The second problem of a physical
state not being able
simultaneously to have well
defined values of two specific
physical quantities is not
actually a problem in quantum
mechanics, and is something that
occurs in many situations
in the classical world without
causing any real confusion.
We already mentioned
this point briefly.
A good example is that a musical
note or tone must be
long enough if it's to
have a relatively
well defined frequency.
We can notice if two musical
instruments are slightly out
of tune if they play for
a long enough time.
If the note is long, however,
it does not have a well
defined time associated
with it.
A short musical note does not
have a well defined frequency,
but does have a relatively
well defined time.
This phenomenon of the inverse
relation between time and
frequency uncertainties is well
known in Fourier analysis.
Now, we will have a first look
at this uncertainty principle
for position and momentum.
The only real surprise here is
that before we started on our
quantum mechanical view here,
we just did not expect this
phenomenon for particles
with mass.
Now that we're describing them
as waves, however, this
uncertainty principle arises
naturally, just as it does for
other wave or oscillation
phenomena.
Now we're going to have our
first look at the uncertainty
principle, then.
And here we're going to
illustrate the position
momentum uncertainty principle
by an example.
Actually, we're going to take
the example of a Gaussian
wavepacket.
We've looked at Gaussian
wavepackets before, and we
know that we could write those
as a sum over waves of
different k values with
Gaussian weights, with
Gaussian amplitudes in front
of each of those waves with
different k values.
And perhaps slightly more
elegantly, we could take the
limit of that sum by using
an integration.
It's really still the same
thing as a sum, just it's
mathematically slightly more
elegant to write it this way.
So our Gaussian wave packet,
then, is a sum over several
different waves.
In fact, every wave with
different k in a particular
range with these Gaussian
factors weighting
them out the front.
So this is just a Gaussian
wavepacket.
Now, we could rewrite this
Gaussian wave packet here at
specifically the time t equals
0 just as this simpler
expression here.
In other words, if we put t
equals 0 in here, then the
expression simplifies down,
where this psi with a
subscript k of k is just
this expression here.
It's just Gaussian weights, or
Gaussian amplitudes for each
of these different waves, in
this case at time t equals 0.
So in this expression here with
these Gaussian weights,
we could say that psi with the
subscript k as a function of k
is the representation of this
wave function in k space.
This is the set of amplitudes
in k space of all of these
plane waves.
So this is just as good a
representation of the wave
function as a set of all of the
amplitudes in real space.
And with that statement that
these are the amplitudes in k
space, then we can interpret the
modulus squared of these
amplitudes as being the
probability p sub k, strictly
the probability density in k
space now, that if we measure
the momentum of the particle,
actually, the z component of
the momentum, because we're only
working in one direction
here at the moment, it would
be found to have
the value h bar k.
So, we are interpreting this
as a probability, or
probability density that the
particle would be found to
have momentum in the z direction
of value h bar k.
Now, with this expression here,
the set of amplitudes,
then this probability density of
finding a value h bar k for
the momentum when we do some
measurement would be, as we
said, this modulus squared of
this amplitude here, and that
would be proportional to
this exponential here.
Now, a slight difference.
The 2 that was inside the square
here, the 2 is now
outside the square here, because
in taking the square
here, we've multiplied
by 2 on the top line.
So that's canceled one of these
2's down here, leaving
just one of them.
And this Gaussian corresponds to
the way that a statistician
would write down a Gaussian
probability distribution, with
this parameter, delta k, being
what a statistician would call
the standard deviation.
So, this expression here is a
set of probabilities that
happens to be written in such a
way that they look like the
Gaussian distribution in
statistics with this standard
deviation, which is a standard
statistical term to express
the width of a statistical
distribution.
Note also that the wave function
expressed in space
here is written down like
this at time t equals 0.
We saw that before.
That is technically what is
known as the Fourier transform
of this function, and it's
well known from the
mathematics of Fourier
transforms that the Fourier
transform of a Gaussian
is also a Gaussian.
And that makes the Gaussian a
very convenient function to
work with here in our example.
Specifically here, then, the
Fourier transform of this
Gaussian function here, which
is formally this expression,
is this Gaussian now in
real space, now as a
function of z here.
And it's not quite written yet
in the form of what we would
regard as our statistical form
we're going to get to for the
modulus squared here.
So if we do rewrite this
modulus squared of this
function now of position, our
Gaussian in position, of
course the modulus squared
would look like
this, with a 2 in here.
If we want to put that into a
standard form for a Gaussian,
for a probability distribution
that would be in the standard
statistical form, which would
be in this form with some
width parameter in real space,
delta z down here, and our
factor of 2, then we would
now be looking at
the standard deviation.
This delta z would be the
standard deviation for the
probability distribution
for measuring
position z in real space.
Now, to make these two
expressions be the same thing,
of course, so in other words,
to find what this delta z is
in terms of the delta k, well,
that's algebraically quite
simple, just comparing these
two expressions.
What we would find is that for
this to be the standard
deviation in space when this
is a standard deviation in
momentum space, the product
of delta k times delta
z is equal to 1/2.
In other words, these two
expressions are the same thing
if delta k and delta
z multiplied
together, give us 1/2.
You can check that out by
comparing these two
expressions.
Substitute this in, we
can get from one
expression to the other.
From this expression here, then
we can multiply by h bar,
and that would get us for h bar
k, the standard deviation
we would measure in momentum,
we could call that standard
deviation in momentum
h bar times delta k.
We could call that delta p,
and then this expression,
delta k delta z equal to 1/2
turns into this one.
Delta p times delta z equal
to h bar over 2.
And what this is is for our
Gaussian wave packet, this is
the relation between the
standard deviations that we
would see in measurements of
position, so the statistical
variability of our measurement
of position, and our
measurements of momentum.
We measure position, we get some
variation in the answers
we get statistically.
We measure momentum, we get some
variation in the answers
we get statistically.
We can describe those variations
in terms of the
standard deviations, delta
p and delta z of those
statistical distributions, and
what this expression is
telling us is that the product
of these two standard
deviations, in this case, is
equal to a specific quantity
here, h bar over 2.
This relation that we have here
is as good as we can get
for a Gaussian.
For example, if we have that
Gaussian pulse and we let it
broaden in space as it
propagates, it's still got the
same range of k values
in it, but it's
become broader in space.
So that was the best that we
could do there was that
relation, and it also turns out
that this Gaussian shape
is the one with the minimum
possible product of these
standard deviations.
We're not going to prove that
here, but we will come back
later on to prove this
mathematical minimum, and the
Gaussian happens to be as
good as you can do.
So quite generally, therefore,
we can write that delta p
times delta z is greater than
or equal to h bar over two,
and the greater than or equal
to is for two reasons.
The first is that if we let
our Gaussian propagate, it
will get a little bit
wider anyways.
So the delta z will increase,
even if the delta p did not.
So we'd have to have a greater
than or equal to from our
Gaussian experiment,
as it were.
But as we said, the Gaussian
is also the
best possible shape.
Other shapes give us worse
products of delta p and delta
z, so they would also be greater
than h bar over 2.
And this, of course, is our
uncertainty principle for
position and momentum
in one direction.
What this uncertainty principle
implies is that if
we make our Gaussian
distribution here narrower in
space, it gets broader when we
look at the distribution of
momenta used to make
up the pulse.
And if we make our Gaussian
distribution broader in space,
it gets narrower when
we look at the
distribution of momentum.
It's this inverse relationship
between width in ordinary real
space and width in what we call
momentum space that the
uncertainty principle is
describing for us.
Though demonstrated here only
for a specific example, this
uncertainty principle
is quite general.
It expresses this non-classical
notion that if
we know the position of a
particle very accurately, we
cannot know its momentum
very accurately.
Our modern understanding of
quantum mechanics says that
it's not merely that we cannot
simultaneously measure these
two quantities, or that quantum
mechanics is only some
incomplete statistical theory
that does not tell us both
momentum and position
simultaneously , even though
somehow they both exist
to arbitrary accuracy.
Quantum mechanics is apparently
a complete theory,
not merely a statistical image
of some underlying
deterministic theory.
A particle simply does not
simultaneously have both a
well defined position and
a well defined momentum.
As we said, this notion is
quite common also in the
classical world.
We mentioned it in the relation
between uncertainty
and time, and uncertainty
and frequency.
We can look at that one
more explicitly.
Of course, this kind of
uncertainty principle is
already well known in the
mathematical subject of
Fourier analysis.
One cannot simultaneously have
both a well defined frequency
and a well defined time
for a signal.
If a signal is a short pulse, it
is necessarily made up out
of a range of frequencies,
some range of order delta
omega, for example.
And in that case we would by
similar mathematics, in fact
ultimately identical
mathematics, we'd be able to
prove that the product of these
two width parameters,
the width in frequency and the
width in time, when we're
talking about angular frequency
here, would be
greater than or equal to 1/2.
That's simply a mathematical
statement that will fall out
if we do the analysis, the
Fourier analysis of signals.
The shorter the pulse is,
the larger the range of
frequencies.
As I say, this is a normal
phenomenon in signals in time
and frequency.
Another good example is
diffraction, which we looked
at previously for waves
in general.
There we saw that if the
aperture was small, the beam
had a wide divergence angle.
And if the aperture was large,
the beam had a narrow
divergence angle.
This is an exactly similar
uncertainty principle.
Indeed, it's also possible to
interpret the width of the
pattern in the far field over
here as representing an
uncertainty in the momentum of
the particle in that vertical
direction, and the width
of the aperture as the
uncertainty in the position in
the vertical direction, in
which case we get back exactly
the same uncertainty principle
for position and momentum that
we've been discussing.
