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Hello everyone, welcome back to exploring
quantum physics.

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I'm Charles Clark.

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In this lecture, we're going to use the
variational method

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to make estimates of the properties of
quantum mechanical systems.

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Now, there's been a lot of material that
you've been exposed to in this course.

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And I just like to make one
recommendation, and that is, if

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you can remember one thing, one thing to
sort of commit to

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memory, which, I'll going to give you some
encouragement to do so by

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the practice in this part of the lecture,
it's this simple Gaussian function.

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its its shape is just

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a, it's a little lump with a width that's
proportional to d.

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And it's easy to integrate

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numerically it's a very practical thing to
use, it's easy to calculate.

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And it happens to be here, here we're, d

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is just a parameter, it has the units of
length.

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it expresses the

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characteristic length scale of the
function and it enters the normalization

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coefficient here.
So, as you see the wave function has

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the units of 1 over the square root of
length.

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and, now although d is just a parameter,
in this wave

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function, when we interpret this as the
ground state of

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a harmonic oscillator, then d, the
characteristic

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length scale, is the square root of h bar,
Planck's constant, divided by the mass,

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and the frequency

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of the oscillator.

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So in other words, you can use this
function, you

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can remember it as the ground state of the
harmonic oscillator.

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But then you can use it in a very

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arbitrary way to represent wave functions
of complicated systems.

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So now, we're going to do a calculation,
and it's something we're just going to do

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it once, because when you do it once, I'm
going to give you a mnemonic.

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I hope you, it will be useful for you to
remember the result.

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and that is to calculate the expectation
value of

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the kinetic energy operator applied to
this wave function,

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which is in the the one-dimensional
kinetic energy operator,

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-h power squared over 2m, d squared dx
squared.

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Now, this is a straightforward integral

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if you just differentiate this function
twice.

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But I want you to think about

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how you can do it without actually doing
the explicit integral,

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and be able to remember how the procedure
works, and to reproduce it when necessary.

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And the trick is the Virial Theorem which
you encountered in a

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previous homework, and I think it was
mentioned in an earlier lecture.

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And that is, for the harmonic oscillator,
here's the Hamiltonian, with

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the usual form, the kinetic energy and the
potential energy, as indicated there.

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The Virial Theorem states that the
expectation value of the kinetic energy

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over the wave function is equal to the
expectation value of the potential energy.

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So, these two, these two terms make on the
average

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an equal contribution to the total energy.
So now, I'd like you

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to think about what that implies for their
actual values.

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So I hope that you remembered how to get
that result.

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the Virial theorem states that the two
contributions are equal.

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Well then the we must have t

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plus v equal e, the total energy on the
average.

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So in other words, the expectation value
of the kinetic energy operator is half

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that of the the total energy,

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which is equal to one half times one half,
h bar omega equal, in the ground state.

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Now I j-, want to make two comments about
this.

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the Virial Theorem, very powerful.

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So first of all, it actually applies to
any state.

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E, the most arbitrary state you can make

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of the harmonic oscillator including a
time dependent wavepacket.

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Now, in that case, this expectation value
has to, has to be generalized,

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meaning it's time averaged.

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but that makes it very useful.

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And the next, the last point to make on
this

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for the moment, it's valid in any number
of the dimensions.

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So, if you have an n-dimensional harmonic
oscillator, then you can just count

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the contributions to the expectation value
of

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the kinetic energy from each individual
coordinate.

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Last comment is that Virial Theorems exist
for

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other potentials, but this case of the
harmonic oscillator

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is very special when we have the so-called

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equipartition of energy between the
kinetic and potential terms.

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as we'll see later, the Virial theorem for
the coulomb problem

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and hydrogen atom, also very useful, is an
entirely different form.

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So to recapitulate, if we take this
Gaussian as our trial function, and

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just compute the expectation value of the

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kinetic energy, it's half the total
energy.

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And it takes this form.

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I've written it this way just to emphasize
there's a factor of

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a half out front, which is due to the, the
Virial Theorem.

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And this is the ground state energy of the
harmonic oscillator, but written in terms

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of the characteristic length.
So, you'll see expressions of this

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frequently in quantum mechanics.
h bar squared over the square

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Planck's constant, divided by mass, and
the square of distance.

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That's the, this is the energy of

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localized function.

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In other words, if you have a function
that's localized on a

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length scale d, then the characteristic

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motion, the uncertainty principle based
motion,

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the average value of the kinetic energy of
that wave function is

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of the order of h bar squared over 2 m d
squared.

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So if you if you remember the
correspondents of the Gaussian with

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the harmonic oscillator ground state,
which

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I've just been emphasizing over and over

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again, this part, you don't have to do
this integral again.

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Having done it once, you can use it
forever.

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And keep in mind that it always has a
contribution

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that goes, the inverse square of the
characteristic length scale.

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That means that as you, you try to

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compress a wave packet, you raise its
kinetic energy.

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It's a manifestation of the uncertainty
principle again.

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So now let's see how this works in a
simple application.

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Now, in one of the homework problems for
last week there's

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a variational estimate of the at, at, at
attractive Dirac Delta function potential.

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And I, and I can see from responses on the
student forums that, you know, some of

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you have a lack of familiarity with the,
the Dirac Delta function, which is fine.

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Now, I think there's a a

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DLMF chapter.
A DLMF chapter,

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D L M F, section 1.17

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for the Dirac Delta.

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I think that's very clearly and
excessively written.

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and the delta function is a limit of a
sequence.

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you can think of a sequence of functions
let's say localized on the line.

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Let's take this square well function on
the line of width a and depth v not.

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And, what you do is take a sequence of
such functions where a gets smaller

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and smaller, and v not gets larger and
larger, so that the product is preserved.

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So, in fact, you can, if you

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are having trouble with the concept of the
Dirac Delta function, just

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think of it in terms of a very specific
implementation

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like this, and think of this as an
approximation of the delta function.

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And if we use this as the approximation of
the delta function, then by its

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definition, this potential here is equal
to minus v 08 delta of x because

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as you can see when we perform this
integral of the potential

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what we get is the only contribution is
the product v 08 with a negative

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sign over that finite interval.

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Now, we use the Delta function, the Delta
function is handy for sample

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applications, but it's widely used in
practice

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as sort of a pseudo-potential for
representing

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information about complex interactions
between particle in a,

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in a, in terms of a single parameter.

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And this is useful when the deBroglie
wavelength of the

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system, which is the characteristic
wavelength of the quantum mechanic

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wave function, h over p, is much, much
greater than the

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range of the range of interactions that
affect the function.

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Let me give you just a simple concrete
example of that, before proceeding.

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Here we see, a a schematic representation
of a

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wave function for ultra-cold atom system,
two, two atoms colliding.

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and here is the, here's the quantum
mechanical wave function

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for two cases abound in a, in a free
state.

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And then this is the, this is the inner
molecular potential.

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And you see in this system, there's a lot
going on in the region

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of close approach of the two atoms

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and many, many, many wiggles in the
wavefunction.

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But in the application, that's relevant
here

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the application that's important in the
discussion in

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this paper, is the behavior of the wave
function at large distances from the atom.

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And so, really, the details of what
happens

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in this inner region are only important in

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so far, as it effects the, the long range
behavior of the wave function out here.

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And so, though that, that behavior can be
built in by the use of a delta function

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in an internal region, that just sets the
scale

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for the evolution of the wave function
further out.

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So let's go and complete the problem.
we're going

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to we're going to calculate the
expectation value of h

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as a function of d,

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using our trial function.

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Okay, so we have, we have this expression
for the kinetic energy.

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It's h bar squared over 2 m d squared.

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So, and now, now, let's, let's just,

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recall what it must be for the
contribution for the potential energy.

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Well, all we need to do there is when we,
when we integrate over

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the delta function, we just get minus V 0
a, times

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psi squared, evaluated x equal 0.

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Right, because that that, the delta
function just

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takes a measure of the weight on top of

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it at the origin and so psi squared,
that's

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equal to zero, is just d over the square

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of pi.

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So, this gives us, that small d, a very, a
repulsive.

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In fact, let's just plot, plot it out
schematically.

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This would be T.

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Now let me see if I can change the color
here.

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yeah.
So now, what about V?

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Well, V is minus 1 over d.

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So, that is negative here, and it falls
off inversely

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proportional to the potential.
So this is V.

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So now let's look at the sum of the two.

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So, the sum of the two it's, it's, it's,
it's positive

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at, at short distances, but then it's got
to become negative at, at long distances.

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So, it turns over.

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And, so this this divides a point of
minimum.

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So you just have to to find the minimum as
a function of d, find the minimum value of

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this the sum of these two terms.

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And that, that gives you the best choice
of the the scale

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parameter, to estimate the energy of the
attractive delta function.

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I would like to say that this is, this is
the

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same minimization that one encounters for
the hydrogen atom in three dimensions.

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That is,

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first of all, obviously, the kinetic
energy, the

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contours of the kinetic energy is always
the same.

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And then if you think of it, well the, the
hydrogen atom has, you know, v of r,

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those as one over r, so it stands to
reason that you'd have an inverse

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length in the expectation value for the
potential energy.

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And we'll see in the subsequent part that,
that's the case.