Welcome back everyone, to exploring
Quantum physics.
I'm Charles Clark.
And today you're going to follow the
bouncing ball.
So just to remind you we're working with
special functions and today,
this part we're going to especially be.
involve with the ari function.
So I encourage you to make this reference
available to you.
Though, everything I'll show you is
self-contained, but
this is a, a good source of additional
information.
It's also appropriate for me to remind you
the short course.
Crash course on mathematics
reference on the solution of Schrodinger
equation, known Schrodinger equations, and
access to numerical, simple numerical
integrator that runs on a spreadsheet.
In the previous part, we started out, we
treated the first of these three simple
one dimensional examples.
And we're now going to today this part
look at the linear of potential.
And I think, I think actually we'll just
we will have a
very brief discussion about the Quadratic
at the end of the linear potential.
Does sort of, is the best thing to focus
on, because it gives one
a very good framework for competence of
understanding the solution of one
dimensional Schrodinger equation.
Now, in the previous part we say for this
constant potential, and
we found that we could transform the
Schrodinger equation into a standard form.
[SOUND]
And the key point about that standard
form.
Well, there were two variants of it.
Difference in sides.
whether, according to weather E, what was
greater
than or less than the constant V naught.
but in both cases we can identify two
solutions.
Now I should have brought this point up in
the,
in the previous part, but here is really
good tool for helping you to
make decisions about choices of the
alternative pairs
of wave functions that you will always
face, in solving the Schrodinger equation.
This is the so-called Wronskian, the
Wronskian of two functions is given by
fg prime, right, so, so, g prime equals dg
dx.
And this is a very good comparison tool,
tell you decision making.
So, the in line quiz, I'm going to invite
you to discover
for yourself, the key property, the
Wronskian operant, why it's so useful.
So I guess you can see that in these two
cases you can calculate the Wronskian in
these two functions easily.
It's, that's plus or minus one depending
on how you do it.
And here it's plus or minus 2, I guess.
Depending on, on how you do it.
but perhaps, perhaps if you think about
it, you'll see
that if you have if you have,
if you have divergent behavior, In this,
in i, in this
case, of one function, then you should be
able to find another wane that converges.
Or, maybe it's a easier to say that if you
have oscilatory
behavior of one fuinction, in this region,
and you pick another
one, Then that's also a solution its also
going
to be isolating in order to keep the
Wronskian
constant so that's the whole real
important is that
these alternative pairs of solutions the
Wronskian must be constant.
Now the solution to the linear potential.
Schrodinger's Equation for a linear
potential.
Some very nice properties, so I suppose
that these
are, I suppose, the area functions must be
less
familiar to everybody, then must be less
familiar than
the exponential or the trigonometric
functions, functions to everybody.
but if you, this is your first exposure to
them, let me point out that this case the
linear
potential gives us two universal solutions
that combine both
the oscillatory behaviour of the
trigonometric functions.
And the exponential, converging, diverging
or
converging behavior of the exponential
function.
So, there's basically,
a kind of universal function for each type
that,
that covers behavior in the allowed
Versus forbidden
regions.
And that is if you have, if you have a
linear potential, then eventually
depending on the choice of v1,
Eventually, eventually the, the potential
eventually gets larger than the energy.
Or eventually gets smaller than the
energy, either way around.
So in this sense, linear potential.
very, is a very nice, tool for
understanding the behavior of quantum
mechanical wave functions.
So again, in this case, there's a
demarkation point.
Well, these, of course these are two
particular choices of function and
you can find their definition in this
chapter.
And they're both oscillate for, when the
arguments less than 0.
And then the AI converges for row, greater
than 0,
and VI diverges, and these two have
implementations in
physical systems.
The one we're going to look at in
detail here, is the quantum mechanical
bouncing ball.
but the second type of airy function
becomes important.
In modeling things like scanning time
telling microscope.
Now the bouncing ball problem, is, as
you'll see,
is important from a, a technical,
theoretical perspective and
in term of learning how to solve the short
new equation, but actually it's something
that has been.
recently realized a remarkable experiment
of the,
bouncing of a neutron confined in a, in a
trap and bouncing off a mirror.
and there's actually experiments that
measured
the quantum mechanical spectrum of the
Earth subject of the, of the neutron
subject to the Earth's graviational field.
There have also been some implentations in
ultra cooled, atom systems.
So we're actually going to, solve the
problem with reference
to parameters that are germane to this
recent experimental realization.
So, here, we're going to describe this
amazing neutron experiment.
The starting point is something from
freshman physics.
V is equal to MG V is equal to MGH, or V
is equal to MGX.
So the potential energy of the neutron, is
given by this form, where
X is the distance going vertically.
off the surface of the mirror.
G, acceleration due to the Earth's
gravity, we'll just
take a nominal value, 9.8 meters per
second squared.
M, the mass of the neutron, about the same
as the mass of
the proton, here's the nominal value, just
going to use a round number value.
And now here is the boundry condition.
for x equal to 0.
So in other words the wave function has to
vanish at the surface of the mirror.
That's an idealization that's largely
true.
And then it's got to, it's got to, the
[INAUDIBLE]
function has to oscilllate and reach a
point of whatever
the energy is The wave function must die
off eventually
in order for the neutron to be localized
and continue
to bounce back and forth in the way that
we expect.
Now, in order to cast this equation into
appropriate dimensional form, I just,
we're going to use.
We're going to define you know I, I've
worked this problem out in
advance so to define our dimensions
coordinate grow in terms of this
transformation and I like you to look at
that in the, in video quiz.
Right, well I hope that you found that you
could solve the quiz.
Solve the quiz question using dimensional
analysis only because if you just look at
e over mg, I mean that, that
tells you the that dimensional analysis of
this expression tells you the answer.
Oh yeah.
So in other words this parameter alpha,
has to have the units of energy.
I mean since row is dimensionless as well,
you can see that that's quite clear.
Now, in the next brief in-line quiz, just
take it a step
further, and this has to do with I, the,
relationship between rho and x.
So if
you were to draw a rho axis, where does it
go in this mirror, and what,
what you know for example, there's an
important,
important issue regarding what does rho
equal 0 means?
What is the physical interpretation of
the value that particular mentioned
coordinates.
So I hope it was obvious, I hope it's easy
for you to find that the
value of rho equals 0.
Describes the classical turning point.
That is, when I say the place where the
velocity changes sign.
It, it represents the maximum distance,
represents the maximum distance,
maximum value of x for
which is attained in classical
motion.
So that means that the function that we
are going to, that we want is Ai because
what it attains is maximum
the little bit before rho equal 0 but it's
the one that dies off,
as rho gets large.
As rho increases beyond, into the positive
region, which is the classically forbidden
region.
So we know that we, we must of the two
solutions that we are going to use, we
must use AI.
That is the only acceptable solution.
in the region rho greater than 0, because
the only other possible solution there
would involve an ad mixture of Bi, because
any solution in
this region can be written as a linear
combination of Ai and Bi,
the one unique solution, up to some
overall constant multiple factor, is Ai.
So if we take that, we now say, now how,
knowing that that is the solution.
How do we satisfy the boundary condition.
That psi of x equals 0
is equal to 0.
Well, we're going to
write, we write the wave function in terms
of an area function.
So we have an additional parameter avail,
available to us, which is the unconcern
offset in the solution.
And so that, that means that we basically,
we.
We take this area function and we sort of
start pulling it up, until we,
until the node lies at the relevant, at
the surface of the mirror.
So I think you can see that what we're
going to get is,
here's the lowest solution will be like
that.
Then the next lowest solution Oppss.
The first excited state has a, has a node
one node in the wave function and so one.
So basically, the a, the, this curve ai,
shows you a universal set of profile of
the
excited state of the bouncing ball and
there's
just obtained by displacing the surface of
the mirror.
To each successive node in the solution
ai.
That will conclude
actually really with just a very brief
description
of what goes on in the quadratic system.
We're not going to discuss it in a lot of
detail.
The most important case of this is
concerns the harmonic oscillate.
Later which we've solved elsewhere.
But there's sometimes
things like a harmonic oscillator with
dis, impurities and other disruptions,
and so it's, it's nice to have the ability
to solve the general second order
equation.
And, I'll point out that once again.
it turns out that the, the divergent
character of solutions is the generic one.
So for example,
in, in this system, because of the
quadratic dependence of x.
You know, eventually v is greater than e.
If you go, if you go sufficiently far
away, sufficient large
values of x, positive or negative, you're
in the forbidden region.
So,
the the, because you have propagation
that, because of the way the equation
in forbidden region, you always get a
divergent solution.
but there are special cases, and these are
you know, these are exactly the.
eigenvalues in the harmonic oscillator
here.
You can see this is a harmonic oscillator
looking equation.
that when
n is an integer.
you get, you get bound, eigenstates.
So this, this just shows
the, the characteristics of this so called
parabolic cylinder function.
And we, we're not going to make practical
use of that
in this, in this course but I thought you
should see it.
Again, the quadratic equations solutions
to the
shortage equation of quadratic potential
are part
of this, this systematic family special
functions
that's been developed to solve the
shortage equations
in general.
Okay, that concludes this lecture.
I'll hope to see you in the next one.