Welcome back, everybody. And this week we're going to go back to
more traditional quantum mechanics. That is quantum mechanics described in
terms of the Schrodinger equation. And today I'm going to solve the
Schrodinger equation on a number of very important examples.
Where we'll show you how quantum potentials can capture quantum particles
that results in simple balance states. So even though the solutions that we are
going to see, the examples that we are going to see are rather elementary from
the quantum mechanical point of view, well otherwise, they will require some
thinking. So, these examples are going to give us
insight into some rather complicated and fascinating problems.
Such as, for instance theory of superconductivities.
So, at the end of the lecture today, we're going to discuss how our solutions
are relevant to the key phenomenon that is responsible for super-conductivity
that is so called Cooper pairing. But before we get to this point, we need
to go over the basics. So let me start first, with the problem
with electron in a box, that we'll define in a second.
But before that, let me just talk about what kinds of problems we want to, we
can't possibly study with a Schrodinger equation.
So, here is the canonical Schrodinger equation.
we have seen it already, many times. It's a time-dependent Schrodinger
equation as we will see later today. So, actually, if the potential does not
depend on time, there is really no need to study the time-dependent Schrodinger
equation. So we'll see, it can be transformed into
an Eigenvalue problem that we will actually solve.
And in any case, so the first term in this in the single particle, again,
insinuates. So this first term in this Hamiltonian is
just kinetic energy. So there is essentially, it's a non
negotiable term. And the second term is something which is
sort of a problem, dependent, we should determine the potential in which our
particle, quantum particle, moves. So, amazingly the variety of all possible
problems In single particle quantum mechanics, are contained in this equation
in a compact way, and they differ simply by the choice of B of R.
But, this compactness of basically quantum mechanics, is a bit deceptive
because it can give you the impression that it can just solve all possible
quantum mechanics problems in one goals by solving this equation.
Well, this equation is rather each equation, and depending on the choice of
your far of the potential which is particle moves, so we can get completely
different physical situations. And understanding these physical
situations would require dramatically different mathematical approach is two.
However, very roughly, what we can do, we can classify our quantum potential by
putting it in one of two categories of either an attractive or a repulsive
potential, which in the context of quantum physics are called the former are
called potential wills and the latter are called potential barriers.
So here, let me present a, well, sort of simple illustration of what a typical
potential well looks like. So this is v of x, seen 1D quantum
mechanics. And this is a 1D caught, coordinate, and
a potential barrier would look sort of opposite to it, it would look like a
hill. And today we are going to focus our
attention on the physics of quantum potential wells.
So and basically to give you sort of the result right from the outset.
And maybe most of you probably already know this.
in potential wells we're going to see that the available energy levels for the
particle to occupy with the negative energy.
So they're going to be quantized, that is, the, the particle wouldn't be able to
have any arbitrary energy. It's just like in classical physics,
where we could assign any energy, we could, we could put let's say, a ball at
any level and let it oscillate between the turning points.
In quantum mechanics lets say if we put electron in a potential well with this
landscape we cannot assign it in arbitrary energy.
So the available energies are going to be quantized, and finding this quantized
energy levels is one of the sort of canonical problems that we're going to
solve. But this business about finding the
quantized energy levels in an arbitrary quantum well is in principle, rather
complicated technical. And the complexity of this, sort of,
exercise might depends on the particular form of the potential of what we're
studying. however to see, the appearance of this
quantum levels[/g] in general we can focus on the simplest of examples of the
effects. And that's what we're going to do now,
and perhaps the very simplest example is potential which has a so called hard wall
boundaries which implies the full length. So it basically means a potential where
beyond certain points, let's say x equals 0 and x equals l so the value of the
potential of the effects is equal to infinity.
So there is no way the particles can move beyond this point.
These are infinite walls. And in between these two, these two
points, the value of the potential is exactly equal to 0, so which means that
the particle is sort of free to move between these two walls but it cannot go
outside. And this is what is known as a problem of
an electron in the box. So as you can probably guess the solution
of this problem we're going to complete in the remainder of this segment is
going to involved certain energy levels which are going to form a discreet
series. So it turns out to understand the
phenomena of quantization one really doesn't need at quantum mechanics at all.
the quantization occurs in classical physics all over the place.
And so here, for instance, we have an example of a very familiar object a
guitar, which relies on quantization in some sense of the wavelengths and
frequencies available in its oscillating strings.
So here we have, strings sort of which are free to oscillate between two points
where they're pinned down. And that, those two points, in some sense
correspond to the hard world boundaries in the corresponding, quantum mechanical
problem that we're going to study a little later.
And the available wavelengths much depend on the distance between these points,
between these hard walls. Lets call it L.
And to corresponding frequencies are related through the available wavelengths
Y the speed of sound. Let us now think about, what is the
longest possible wavelength that we can induce in a finance strings.
So these are our basic, our hard walls. So this is our 0.
And this is our l. And the hard walls in this context
essentially mean that the the strings cannot oscillate here.
So the displacement let's call it u. from this horizontal axis is exactly
zero. It, x equals zero, and x equals l.
in other words, so we must have news, at these end points.
And, the longest possible wavelength that achieves that is, lambda equals 2l.
And this is sort of the fundamental oscillation that we can,
The longest possible wavelength that we can induce in this stream.
Because, if we try to make the wavelength even longer it would imply, essentially,
that we would have either no node in this point, or no node in this point.
And, this will violate our boundary condition.
But there of course many more wavelengths that are possible, that would satisfy the
proper boundary conditions. And we can achieve we can sort of find
these additional wavelengths, or higher harmonics, as they are called, by find,
by putting additional notes in between these endpoints.
And so for instance this example Gives us a wavelength which is exactly equal to
the L, to the distance between the inputs.
And we can continue this proceedure and generate even more wavelength.
And this wavelength, I'm going to follow this quantization rule, if you want.
with the corresponding. The solution while if you look at the
snapshot of oscillating string. at a certain time.
So the solution is simply going to be given by this sign.
Well some amplicude which is not really important.
Sign of two pie X or along the sub N. So and of course if X is equal to zero.
We're going to have the bond and we're going to set aside the boundary
condition. X equals zero.
And if x is going to be equal to L this quantization rule is going to enforce the
other boundary conditions namely that the displacement vanishes at x equals L so we
have a note here. Now if we look at the corresponding
quantum mechanical problem, so it is essential is very similar to the problem
with this oscillating string. With the only difference being that
instead of putting in the less extreme in between the two points, we put an
electron wave in between these two points.
But the wavelengths that are available for the electron, so the electron cannot
really move beyond this hard wall. Falls right where the probability of
finding the electron area is equal to zero.
Therefore we must demand that, well the probability of, side square of x equals
zero and l is identically equal to zero. Okay.
So or another words side itself is equal to zero and this is much similar to
having a node. At the, at the end point of this string.
So if we solve this problem for the electron, well which involve, which will
involves in this case actually solving the Schrodinger equation, we're going to
find exactly the same wavelength. And the wave function is actually going
to look exactly the same as the, of the displacement of the strings.
So there's actually no difference. I just copied pasted exactly the same
equations. So we're only replacing use of M which is
a displacement by side of M which is the wave function of electron.
So this shows you that there's actually there is a lot of analogy in quantization
of electric wavelength. Wavelength and quantization of wavelength
of classical objects, classical objects. So the only difference, and the important
difference here, would be in how the frequency or the energy in the case of
electron scale with the wavelength. So here we discuss that the frequency of
the oscillation, which by the way determines the sound you actually hear
scales linearly with the wavelength. On the other hand we know that the energy
of a quantum particle of just well, kinetic energy, essentially a free
particle is going to be p squared over 2 m.
And if we recall the, deployed relation between the momentum
and the wavelength, which is 2 pi h bar over lambda is equal to p.
So then we can readily combine these two results, the quantization of the
wavelength. And the scaling of the energy to get the
quantization of energy. So if you put everything together, the
quantization rule and the debroyal relation, we're going to get the
quantized energy levels. e sub n, where n is a positive integer
equal to pi squared. H squared, n squared divided by the l
squared, the distance here. This corresponds to the momentum squared
times 1 over twice the mass. So and notice that interestingly we
solved the Schrodinger equation without writing it down, the only thing we used
is Basically was the boundary conditions and
the some analogy with classical physics along with the, broad relation.
But of course one can solve it formally and we're going to show, later how it
works. So, sadly one cannot always solve the
Schrodinger equation in such a simplistic way without writing it down.
So and to understand. Why it happens, we can again, use this
guitar sort of string analogy. So, if we push down on the string, so we
effectively shorten the distance between these end points and the effective end
points that we enforce here. And by doing so, we, of course, change
the fundamental wavelength. And the quantization of the wave length
and the frequencies, this sort of results in different sounds that we produce this
way. But, if we push, not very hard, but if we
just touch the string here, so this creates instead of an infinite wall, sort
of, the, the impenetrable wall for the string to propagate beyond this point.
It creates a finite barrier. And to solve, the wave equation for this
string, the classical wave equation in the presence of such a small
perturbation, is actually more complicated than solving, this equation
for 2 hard walls. And likewise, we're going to see,
actually, that, the, problem of a quantum mechanical particle.
When, instead of the hard walls. We have, let's see, finite barrier here,
is slightly more complicated, well more complicated and it requires, actually a
serious mathematical calculations in solving the actual Schrodinger equation,
which is a differential equation. We're going to discuss it in the
following segment.