So let's look, then, at some
specific uncertainty
principles.
We can go back and look again
at the position momentum
uncertainty principle.
We can formally derive it now
without presuming Gaussians or
any other specific forms
of distributions.
We consider the commutator of
the momentum operator and x,
where we're treating x as the
operator for position in this
one direction, x.
And to be sure we're taking the
derivatives correctly, we
will have the commutator
operate on
an arbitrary function.
So we've got the commutator of
px and x operating on f, and
we have minus ih bar d by
dx from the px operator.
So this is px, x, and this one
comes from x, px, and we have
it operating on our arbitrary
function.
And when we do that, we can
explicitly multiply
these out like this.
And taking the derivative with
respect to x of this product
here gives us two terms, one
of which is just f, and the
other of which is
x d by dx of f.
But that cancels out with this
term, and so we're left with
minus ih bar times f--
the function f.
Since this function was
arbitrary, then we can drop it
on both sides here, and get
this operator equation
in this form here.
So the operator that corresponds
to the commutator
of px and x is just minus ih
bar, or technically, this
multiplied by the identity
operator.
And the commutation rest
operator, therefore, is just
this number minus h bar, or
technically, minus h bar times
the identity operator.
Hence, the average value-- the
expectation value of this
operator for any state
is just minus h bar.
And in this case, we have an
uncertainty principle that is
going to apply to
any conceivable
state of the system.
So from our general form of
the uncertainty principle,
which we wrote like this, then
for our position and momentum
uncertainty principle in the
direction x, we're left with
this expression.
We can also look at another
uncertainty principle, and
this one is between
time and energy.
This is not quite on such sound
theoretical ground as
some of the others, and we'll
see why in a minute.
But it does work.
The energy operator is the
Hamiltonian H, and from
Schrodinger's equation, we write
H operating on psi is
just the same thing as ih
bar d by dt of psi.
And so we identify that we can
write down the Hamiltonian as
this operator, d by dt.
Now, if we take the time
operator to be just t, we can
work through this here.
It's not quite clear that we
really have a time operator in
this quantum mechanic, so this
is a little less rigorous from
a mathematical and perhaps
physical point of view.
But nonetheless, we can derive
an uncertainty principle here,
at least by analogy.
Then, using essentially
identical algebra as we just
used for the momentum position
uncertainty principle, then we
can take the commutator
of H and t for time.
Exactly the same algebra,
as I said.
And we get ih bar, or
technically ih bar times an
identity operator.
And that enables us to propose
an uncertainty principle for
energy and time that is delta
E times delta t is greater
than or equal to h bar over 2.
And that turns out to be
a useful relation.
It certainly works in
the situations we
typically look at.
And this, of course, is very
closely related to a frequency
time uncertainty principle
that is quite rigorous in
Fourier analysis.
We would use, actually,
mathematically, an identical
form of proof here to get this
uncertainty principle in
Fourier analysis.
And we can get it by analogy, at
least from our energy time
uncertainty principle by noting
the energy is h bar
omega here.
So therefore, delta omega delta
t is greater than or
equal to half just by taking the
h bar out of both sides of
our energy time uncertainty
principle.
But quite rigorously, from a
mathematical point of view, we
conceptually derive this
uncertainty principle in
ordinary classical
Fourier analysis.
