Welcome back everyone. I'm Charles Clark, and these next few
lectures we're going to looking at the problem solving the Schrödinger Equation.
Now in the recent Broadway stage musical, "Curtains", a character says Putting on a
musical must be the most fulfilling thing a person could ever do.
Well, here at exploring quantum physics, we think that solving the Schrodinger
equation is one of the most fulfilling things a person can ever do.
and today you will be that person. We are going to start by.
Actually solving the Schrodinger equation or guiding you to solve the Schrodinger
equation this is tailored to people who've never solved that equation in
their lives before. But we hope there will always be some
information that's useful to people with much experience.
Well let's see. Well here is the Schrodinger equation.
You've seen it before. I h.
Bar decided to use h psi. And let me emphasize again according to
quantum mechanics, all the information that we are able to use to describe the
system is contained in the wave function. And so, this The equation is first order
in time because it, it, tells us how to determine the state at some later time
from the state at a given time. So this equation should be thought of as
an initial value problem in which we are given psi of t at some initial time t And
then, we integrate this equation to find[UNKNOWN] of T plus[UNKNOWN].
Well, you might ask, what could possibly go wrong in such a simple scheme.
And here is a little in line video to test your understanding.
Well I guess you see from this in line video that
The mechanics of the Schrodinger equation is apparently quite straightforward.
If we have a wave function and a Hamiltonian operator we can just advance
the equation gradually in time by repeated application of that operator.
Well like much in life The difficulty really emerges with the details.
So here's a recent paper, I, showing you my own work cause I do know something
about it. an it's a calculation relevant to some
atomic clock research involving the Ytterbium atom.
you can go look at your periodic table. Ytterbium is at the end of the rare earth
period has 70 electrons. Now let's just say, supposing we were
doing a, a, you know, a calculation that took very explicit account of all the
electrons in the turban atom, there are 70 of those.
And if we represented them on a, a spacial grid that was just ten points on
a side, which is not a lot to represent the motion of a particle we get this
impossibly large number to deal with. Now, it can be reduced by quite a bit.
Due to symmetry, considerations. But it's still just totally infeasible.
So, Much of the, the practical issues of
solving the Schrodinger equation have to do with finding feasible schemes.
Now let me emphasize that the full Schrodinger equation is explicitly time
dependence. It's, the equation for solving the, for
the evolution of quantum system time. And, there are many cases in which.
You need to pursue it from that standpoint.
Here, here's an example. It's actually a solution of a, of a
non-linear Schrödinger equation that describes Bose-Einstein condensate, and
it shows the breakup of a soliton in that condensate into a series of vortex rings.
So to l-, to understand things like this You need to have a full, time-dependent
description. On the other hand, as we have seen over
an over again, there're a lot of interesting phenomena, such as the, you
know, my favorite, the Balmer's spectrum of the hydrogen atom, that seem to be
described by stationary quantum states. That is, these are states that have, a
very simple time evolution. And they're persistent.
They lead to, to sharp structures in, in spectra such as the transitions between
levels. so for the rest of this, these lectures
we're going to focus on the stationary state problem in which we take the time
dependent Schrodinger equation and then look at the special case when the
solution is an eigenfunction of the Hamiltonian operator.
Okay, so just to remember the terminology, this is, this is an
eigenvalue equation, or eigenfunction, and the The Hamiltonian operator is
given, this is sort of like input to the calculation.
Psi and E are unknown, Psi is a so called eigenfunction or eigenvector, we tend to
use those terms exchangeably. And E, the energy of the system is the
eigenvalue. Okay, so what does it mean to actually
solve this equation? Well, the Schrodinger equation is
implemented to describe a wide variety of physical systems.
so our lectures are just going to focus on things that are described in terms of
a, a potential that affects to motion of an electro-, a particle of mass m.
So the, the general form that we're going to consider, which is very rich in terms
of its applications Is an equation, it's a partial differential equation in three
dimensions. we'll occasionally, we will of course for
illustrative purposes consider the problems in lower dimensions as well.
And so it has the form of the kinetic operator here minus h[UNKNOWN] squared
over 2M times del squared. just so you remember, delta squared is d
squared, dx squared plus d squared, dy squared plus d squared, dz squared.
this is the Laplacian operator, widely used throughout physics,
electromagnetism, acoustics. fluid dynamics, and quantum mechanics.
V, is the potential soul[/g] for, for the hydrogen atom .
V would be equal to minus Z E squared over R.
Or for the harmonic oscillator 1/2 M mega squared R squared.
So, for the usual type of problem, we know what the potential is, and we're
kind of solved for the unknown wave function and energy.
But we're going to start here with a, a constructive approach.
Which is based on the idea. I mean, how many actual functions do you
know? of course, you can say, okay.
A function is X, or X plus X to the 5th over 7.
You can generate a large number of ad hoc functions.
but perhaps many of you have a fairly limited.
Vocabulary in terms of generality that would be expanded during this course so
lets just say we are going to start with functions that we recognize and then see
if we can find a potential and an energy For which those functions are solutions
of the Schrodinger equation. This is actually quite a straightforward
thing to do. So here's the Schrodinger equation, now
what we do is we take this term here and move it over to the right hand side.
Then we take the Common factors of, of the wave function and put them over on,
onto the far left here, and we get this very simple equation.
Which means, I mean all you need to do to solve this, all you need to do is be able
to take derivatives So you take the trial wave function, apply the Laplacian
operator to it, divide by the wave function, and see what you get.
So here's a little in-line quiz. To, help walk you through, to what must
be the simplest possible example. So I hope you found that fairly
straightforward. Now, the function that we c-, chose,
which was, s-, size equal to a, a constant, is, you might say, it's not the
most useful solution that you've ever seen.
It does in fact have an application in scattering theory, but it's hardly a
generic function. Our message for the moment is that any
function will provide a solution to a Schrodinger equation.
But not necessarily for physically interesting potential.
And we'll see that in the next part that for any given value of the energy and
whatever the potential may be, there are infinitely many solutions of this, this
Schrodinger equation. So we need more information.
Then is just, evident in the mathematical statement of the equation of this level
in order to perform the solutions that we need to find.
Having said that, however, we're going to conclude this part by construction of E
and V for I say one of the most widely used functions in quantum mechanics, I
think I'll argue in the next segment is the most widely used function in quantum
mechanics, and you've seen it before but let's, let's look at it from the
standpoint of the constructive approach. As act at this point switch to the online
video, and online quiz and then we'll conclude with the a few-, the last few
words I say, in a moment. So that was a demonstration of recovering
the harmonic oscillator potential. From the Gaussian wave function, which is
solution, and we're going to take that up in more detail in the next part.