Hello everyone, welcome back to exploring
quantum physics.
I'm Charles Clark.
In this lecture, we're going to use the
variational method
to make estimates of the properties of
quantum mechanical systems.
Now, there's been a lot of material that
you've been exposed to in this course.
And I just like to make one
recommendation, and that is, if
you can remember one thing, one thing to
sort of commit to
memory, which, I'll going to give you some
encouragement to do so by
the practice in this part of the lecture,
it's this simple Gaussian function.
its its shape is just
a, it's a little lump with a width that's
proportional to d.
And it's easy to integrate
numerically it's a very practical thing to
use, it's easy to calculate.
And it happens to be here, here we're, d
is just a parameter, it has the units of
length.
it expresses the
characteristic length scale of the
function and it enters the normalization
coefficient here.
So, as you see the wave function has
the units of 1 over the square root of
length.
and, now although d is just a parameter,
in this wave
function, when we interpret this as the
ground state of
a harmonic oscillator, then d, the
characteristic
length scale, is the square root of h bar,
Planck's constant, divided by the mass,
and the frequency
of the oscillator.
So in other words, you can use this
function, you
can remember it as the ground state of the
harmonic oscillator.
But then you can use it in a very
arbitrary way to represent wave functions
of complicated systems.
So now, we're going to do a calculation,
and it's something we're just going to do
it once, because when you do it once, I'm
going to give you a mnemonic.
I hope you, it will be useful for you to
remember the result.
and that is to calculate the expectation
value of
the kinetic energy operator applied to
this wave function,
which is in the the one-dimensional
kinetic energy operator,
-h power squared over 2m, d squared dx
squared.
Now, this is a straightforward integral
if you just differentiate this function
twice.
But I want you to think about
how you can do it without actually doing
the explicit integral,
and be able to remember how the procedure
works, and to reproduce it when necessary.
And the trick is the Virial Theorem which
you encountered in a
previous homework, and I think it was
mentioned in an earlier lecture.
And that is, for the harmonic oscillator,
here's the Hamiltonian, with
the usual form, the kinetic energy and the
potential energy, as indicated there.
The Virial Theorem states that the
expectation value of the kinetic energy
over the wave function is equal to the
expectation value of the potential energy.
So, these two, these two terms make on the
average
an equal contribution to the total energy.
So now, I'd like you
to think about what that implies for their
actual values.
So I hope that you remembered how to get
that result.
the Virial theorem states that the two
contributions are equal.
Well then the we must have t
plus v equal e, the total energy on the
average.
So in other words, the expectation value
of the kinetic energy operator is half
that of the the total energy,
which is equal to one half times one half,
h bar omega equal, in the ground state.
Now I j-, want to make two comments about
this.
the Virial Theorem, very powerful.
So first of all, it actually applies to
any state.
E, the most arbitrary state you can make
of the harmonic oscillator including a
time dependent wavepacket.
Now, in that case, this expectation value
has to, has to be generalized,
meaning it's time averaged.
but that makes it very useful.
And the next, the last point to make on
this
for the moment, it's valid in any number
of the dimensions.
So, if you have an n-dimensional harmonic
oscillator, then you can just count
the contributions to the expectation value
of
the kinetic energy from each individual
coordinate.
Last comment is that Virial Theorems exist
for
other potentials, but this case of the
harmonic oscillator
is very special when we have the so-called
equipartition of energy between the
kinetic and potential terms.
as we'll see later, the Virial theorem for
the coulomb problem
and hydrogen atom, also very useful, is an
entirely different form.
So to recapitulate, if we take this
Gaussian as our trial function, and
just compute the expectation value of the
kinetic energy, it's half the total
energy.
And it takes this form.
I've written it this way just to emphasize
there's a factor of
a half out front, which is due to the, the
Virial Theorem.
And this is the ground state energy of the
harmonic oscillator, but written in terms
of the characteristic length.
So, you'll see expressions of this
frequently in quantum mechanics.
h bar squared over the square
Planck's constant, divided by mass, and
the square of distance.
That's the, this is the energy of
localized function.
In other words, if you have a function
that's localized on a
length scale d, then the characteristic
motion, the uncertainty principle based
motion,
the average value of the kinetic energy of
that wave function is
of the order of h bar squared over 2 m d
squared.
So if you if you remember the
correspondents of the Gaussian with
the harmonic oscillator ground state,
which
I've just been emphasizing over and over
again, this part, you don't have to do
this integral again.
Having done it once, you can use it
forever.
And keep in mind that it always has a
contribution
that goes, the inverse square of the
characteristic length scale.
That means that as you, you try to
compress a wave packet, you raise its
kinetic energy.
It's a manifestation of the uncertainty
principle again.
So now let's see how this works in a
simple application.
Now, in one of the homework problems for
last week there's
a variational estimate of the at, at, at
attractive Dirac Delta function potential.
And I, and I can see from responses on the
student forums that, you know, some of
you have a lack of familiarity with the,
the Dirac Delta function, which is fine.
Now, I think there's a a
DLMF chapter.
A DLMF chapter,
D L M F, section 1.17
for the Dirac Delta.
I think that's very clearly and
excessively written.
and the delta function is a limit of a
sequence.
you can think of a sequence of functions
let's say localized on the line.
Let's take this square well function on
the line of width a and depth v not.
And, what you do is take a sequence of
such functions where a gets smaller
and smaller, and v not gets larger and
larger, so that the product is preserved.
So, in fact, you can, if you
are having trouble with the concept of the
Dirac Delta function, just
think of it in terms of a very specific
implementation
like this, and think of this as an
approximation of the delta function.
And if we use this as the approximation of
the delta function, then by its
definition, this potential here is equal
to minus v 08 delta of x because
as you can see when we perform this
integral of the potential
what we get is the only contribution is
the product v 08 with a negative
sign over that finite interval.
Now, we use the Delta function, the Delta
function is handy for sample
applications, but it's widely used in
practice
as sort of a pseudo-potential for
representing
information about complex interactions
between particle in a,
in a, in terms of a single parameter.
And this is useful when the deBroglie
wavelength of the
system, which is the characteristic
wavelength of the quantum mechanic
wave function, h over p, is much, much
greater than the
range of the range of interactions that
affect the function.
Let me give you just a simple concrete
example of that, before proceeding.
Here we see, a a schematic representation
of a
wave function for ultra-cold atom system,
two, two atoms colliding.
and here is the, here's the quantum
mechanical wave function
for two cases abound in a, in a free
state.
And then this is the, this is the inner
molecular potential.
And you see in this system, there's a lot
going on in the region
of close approach of the two atoms
and many, many, many wiggles in the
wavefunction.
But in the application, that's relevant
here
the application that's important in the
discussion in
this paper, is the behavior of the wave
function at large distances from the atom.
And so, really, the details of what
happens
in this inner region are only important in
so far, as it effects the, the long range
behavior of the wave function out here.
And so, though that, that behavior can be
built in by the use of a delta function
in an internal region, that just sets the
scale
for the evolution of the wave function
further out.
So let's go and complete the problem.
we're going
to we're going to calculate the
expectation value of h
as a function of d,
using our trial function.
Okay, so we have, we have this expression
for the kinetic energy.
It's h bar squared over 2 m d squared.
So, and now, now, let's, let's just,
recall what it must be for the
contribution for the potential energy.
Well, all we need to do there is when we,
when we integrate over
the delta function, we just get minus V 0
a, times
psi squared, evaluated x equal 0.
Right, because that that, the delta
function just
takes a measure of the weight on top of
it at the origin and so psi squared,
that's
equal to zero, is just d over the square
of pi.
So, this gives us, that small d, a very, a
repulsive.
In fact, let's just plot, plot it out
schematically.
This would be T.
Now let me see if I can change the color
here.
yeah.
So now, what about V?
Well, V is minus 1 over d.
So, that is negative here, and it falls
off inversely
proportional to the potential.
So this is V.
So now let's look at the sum of the two.
So, the sum of the two it's, it's, it's,
it's positive
at, at short distances, but then it's got
to become negative at, at long distances.
So, it turns over.
And, so this this divides a point of
minimum.
So you just have to to find the minimum as
a function of d, find the minimum value of
this the sum of these two terms.
And that, that gives you the best choice
of the the scale
parameter, to estimate the energy of the
attractive delta function.
I would like to say that this is, this is
the
same minimization that one encounters for
the hydrogen atom in three dimensions.
That is,
first of all, obviously, the kinetic
energy, the
contours of the kinetic energy is always
the same.
And then if you think of it, well the, the
hydrogen atom has, you know, v of r,
those as one over r, so it stands to
reason that you'd have an inverse
length in the expectation value for the
potential energy.
And we'll see in the subsequent part that,
that's the case.