Hello everyone, welcome back to Exploring
Quantum Physics.
I'm Charles Clark.
We're now going to start to apply some of
the greatest tools that mathematicians
have ever developed.
The special functions and apply them to
solving problems of Quantum mechanics of
simple systems.
Now special functions.
There are many good sources of information
on special functions available.
but it's nice, that they have a common
reference for, so
that everyone in this course will be using
the same one.
So we're using this the digital library
mathematical functions,
from my employer, the National Institute
of Standards and Technology.
It's a free App, freely available
worldwide.
And it contains information on,
on all the functions that we'll be using
in this course and in
many more, and, well, there are many other
things to be said about it.
But when I make a reference to something
like the Airy function, which
will be done later in this part, then what
I mean are the, the
specific representations that are provided
there, which
they're very much standard are the same
that you'll find in Wikipedia, but I
encourage you to look at this anyway.
Two several other resources for for you.
short crash course on mathematics.
that's something certainly worth looking
at because it
does provide a constructive by Zack Lains
and his
collaborators, and, gives you pretty good
idea of
the level of mathematical treatment that
started this course.
another one page guide to how, to the
solutions, known solutions of the
Schrodinger
equation in terms of special functions.
and what else, oh, a sample of a program
written in spreadsheet form for
solving the Schrodinger equation by
numerical integration.
So that's something we won't, I won't be
using
that in this part, but something worth
looking at.
There's a discussion discussion there.
And it'll give you some simple guides as
to
how to numerically integrate the
Schrodinger equation in cases
when you're dealing with a a complex
problem for which there is no known
solution.
So, who was the first person to use
this approach, solving a Schrodinger
equation using special functions?
no prizes for getting this right, it was,
in fact, Schrodinger.
And you might say that his first work,
first papers
on the, on the wave equation, were
influential for
several reasons.
But one, one lasting
development was that it, it, it cast the
equation, the Schrodinger equation of
Quantum mechanics, into a form that had
been well understood by in classical
mechanics.
And that was to re-, reduced
the Schrodinger equation, differential
equation that
had solutions in terms of functions that
were known at the time.
So what Schrodinger,
what Schrodinger did, of course, his, his
work,
the importance of the work stands on its
own.
He explained the spectrum of a hydrogen
atom, calculated the shifts of spectral
lines, he invented, invented perturbation
theory, all
in this series of four remarkable papers.
But certainly is very important for the,
the wider
implementation of his ideas, that he
developed a framework that mapped
on to well-developed problems of classical
mathematical physics,
and so we're following the same approach
here.
So in this part of the lecture, we're
going to look at
three, we're going to stick with
one-dimensional examples.
And we'll look at a constant potential, a
linear potential, and, I
guess, it'll be in the next part what are
this quadratic potential.
But you might say that these are the, the
three simplest possible potentials in
terms of development as, as polynomials.
And
the solutions of the Schrodinger Equation
for these cases are the exponential
function, for the linear, for the constant
potential, the Airy function for
the linear potential, so this is
exponential Airy,
and parabolic cylinder
for the quadratic.
Now, the harmonic oscillator's a special
case of that.
but we, we're going to go through in this,
in this
part we're going to go through these two
cases, and, and try
to infer from them some general principles
that are relevant both for the analytical
and numerical solution of one-dimensional
Schrodinger equations.
Right, so let's start with the linear
potential.
We just take V of x to be a constant.
And so here's the Schrodinger equation.
Now, the point is we want to find
solutions for any value of the energy.
We want to find a general, general
solution
valid for all, for all energies because
then
with, with such solutions in hand we can
find the particular solutions that we need
to satisfy
boundary conditions.
You'll see why this is important in a
moment.
But for example, if you just consider a
case where the, the potential is
is, is constant over different regions
but consider different values in different
regions.
And you want to find a wave function, a
solution to wave equation, that
is valid it, it'll be, it'll be a
sufficient to have this information.
So you can piece, you an stitch the
various pieces together.
So the approach we take is to transform
the Schrodinger equation up here into
a standard type.
in this case the standard
form that we see, that we see is here,
where rho is a
scaled variable, dimensionless variable,
and and
then the plus or minus sign has to do with
whether
energy, positive or negative.
We know that energy's can, can be any
number on the real line.
so we need to account for both of those
cases.
Okay, so we just take that system.
We
transform, we find that dimensionless
variable row is
kx, k is the wave vector and it's
depending on whether the energy is greater
than or less than
the potential energy, well this, the value
of k is, is
determined by that.
And so there's two classes of solutions.
So, when E is greater than V0, we have two
independent solutions.
So, you know, let me just say again, as a
second order differential equation,
any second order differential equation
like this
one, has at least two independent
solutions.
there are many different ways to choose
them but
here I've just, I've, I've given you two
independent
solutions sin a rho, and the cosine in a
rho.
we'll say a little bit more about the
choice momentarily.
And then in the region where E is
less than V0, the two inde-, the two
independent
solutions can be chosen of this form, E to
the rho and E to the minus rho.
So any other solution that exists in this
region can, is going to be a linear
combination of these two functions.
But that means the generic behavior is
divergence,
because if you just take you just take a
random choice of these two functions,
then the resulting solution is going to
diverge
as rho goes to infinity.
And in the exceptional case where you just
choose this function,
then it diverges going to and rho go to
the minus infinity.
Now that can be, you know that can be
changed by another boundary condition.
You have to be aware of that.
In this, for this particular circumsance,
the generic behavior is divergence.
So that, that's when that means that you,
you have difficult choices that have to be
made.
So I'll make two points about this,, this,
this, you know, subsets
trivial Schrodinger equation, which either
has oscillatory or exponential solutions.
And that is if you have a
general understanding of the solutions,
then you could
do a, a piecewise approximate solution, to
to
a real potential, get a real potential
that
has some arbitrary shape.
You can approximate it on a piecewise
basis, and then by knowing how to solve
the equation regions of constant
potential, you can
sort of, propagate solutions between
different such regions.
This is something that's used in modeling
semiconductor heterostructures and so on,
or the photonic band-gap materials are
well understood in terms of having
techniques for, to propagate a wave
function
across a constant region and then to
stitch
the, you know, stitch the transition
matrix
together, to see what the net result is.
So actually why don't we just now apply
that approach to the simplest possible
problem of crossing an interface, and
we're
going to look at an example that you,
you see every day.
if you look at a, look through a window
you see a, you see a reflection of
yourself.
I mean even if the glass is transparent
there is some reflection.
And so it turns out that this is a
phenomena that's an example
of this, the sort of the the potential
step function in Quantum mechanics.
So we have
V of x as a function of x.
this is air, this is glass.
So, the propagation of a light-wave as it,
enters a window is the same very
similar mathematically to the the problem
of a particle moving in a potential.
So the way that, that is framed is that
the
we have Y incident from a large distance
on the negative axis.
That wave function is a form of e to the
ikx.
Then there's a reflected component, which
is of the type R, the reflection
coefficient times e to the mi-, minus ikx,
and then the the
transmitted coefficient here.
Now, how do we,
how do we determine the effect of glass on
light?
let me refer you to an inline quiz.
So when light goes into glass, it's the,
the
refractive index of glass is about 1.5,
that means that its wave length decreases.
And so that is that means that the, the
wave, the, the wave
equation, spacial part of the wave
equation in the interior as this form.
And so now,
now it's very easy to it's very easy to
determine the reflection coefficient.
So what we, we have, we have, we have a
wave
function on two sides of the interface,
the air and the glass.
So we have to match the values of the wave
function to the boundary.
So for x equals 0, that's very easy, it's
1 plus R here.
It's T here, so we have to have 1 plus R
is equal to T.
And then, we match the derivatives,
that's is the equation of continuity
that's required in Quantum mechanics.
So, once again, it's easy to differentiate
this side and the answer's ik 1
minus R, and this side is just iknT.
And so now
what do we do?
Well we divide
we, we
we, we, we take this side,
and divide it by, by the other one.
So, you see, I'm just
taking, this divided by that, this divided
by that, equals that divided by that.
Get this simple equation for the
reflective's coefficient.
So, you see that, there, the k, the
wave vector disappears as does the
transmission coefficient.
And then it's easy to invert this
equation, to get this.
So R is equal to minus 0.5 over 2.5, which
is 1
fifth.
So the actual reflected intensity is R
squared.
Coefficient is 0.04, so it's 4%.
This is, this is about right.
When you put light through a window that
normal instance about 4% is reflected.
So on the next part of this lecture we're
going to explore the next
sequence, next in the sequence of
complexity, the linear potential.