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Hello, everyone.
Welcome back to Exploring Quantum Physics.

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I'm Charles Clark, and we're now going to
start talking about how

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to solve the Schrödinger Equation, what
that means, how it's done in practice.

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And we're going to start with some simple

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constructive techniques to give you a few
tools

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that you can you use if you are

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if you're stuck in solving a
quanta-mechanics problem.

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Sometimes a few of these simple tricks
will help you see

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your way through, help you identify

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appropriate scaling relations and the
like.

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So, let's proceed.

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Okay, here's the Schrödinger Equation
which you've

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heard a lot about already in the course.

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size of wave function H, the Hamiltonian
operator.

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ih bar, d divided dt,

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psi.
And the precept of quantum mechanics is

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that all accessible information about the
state is contained in the wave function.

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Now, you know, this will, you, this is for
an isolated system.

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When we're talking about two interacting
systems or a system in

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a bath and there's, there's a lot more
nuance, but let's just,

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let's just stick with simple many body
systems and

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the ab, absence of noise or thermal

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bath because that, in itself, is an
interesting problem.

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So, in other words if we know the wave
function at a given time

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we can just, we can just sequentially
integrate this equation

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using some technique to find it at a later
time.

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So why in this era of high performance
computing and all these other

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wonderful things, is anyone really worried
about solving the Schrödinger Equation?

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The inline quiz contains some sort of an

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exercise on thinking about how you use the
Schrödinger

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Equation to propagate the wave function
forward in time.

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So, from a formal and even a practical
perspective for very simple systems

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solutions to the Schrödinger Equation are
fairly straight forward to find.

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But, in, in real life we're encountered
with

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problems that involve atoms like
ytterbium, 70 electrons.

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This is a system I happen to

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have worked on myself with wonderful
collaborators, Marianna

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Safronova, Sergey Porsev, during the past
year and

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it's, it's motivated by determining the
corrections to atomic

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transition frequency due to the presence
of a

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thermal background for application next,
next generation atomic clocks.

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Well a very simple, a very simple
consideration will show

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you that even a, a crude representation of
the 70 electron

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wave function on a course grid would just
be completely infeasible

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by today's and presumably just about any
future technology that we know of.

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So Physics-oriented approaches that take
ad, that take advantage of approximations,

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reasonable approximations are, I think are
always going to be essential.

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And so, learning how to solve the, the
simple systems gives one good

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guidance as to what you're going to have
to do to deal with more complicated ones.

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Now, there are many cases in which one

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needs to solve the full time-dependent
Schrödinger equation.

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say, studying non-equilibrium quantum
systems.

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Here, here's an example of equation
solution of a, actually

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of a nonlinear Schrödinger equation that
describes a Bose-Einstein condensate.

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This, these are, this is a spatial
representation of density

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of, of an atomic cloud at, at various
times and

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so it's, it's sometimes critical to have
information like this.

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However, there's quite enough to be
learned

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by understanding how to solve the
stationary states

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of the system which underpin a lot of the
important phenomena we see in nature.

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And so the rest of this, this lecture's

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going to focus on the solution of
stationary state problems.

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So, here is the, the time-independent
version of the Schrödinger Equation.

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The Hamiltonian operating on size and
eigenvalue, ei,

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eigenfunction of a Hamiltonian operator
with eigenva, value.

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E.

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What does it mean to solve this equation?
Well, the real-life

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problem that faces research physicists is
they have some material system

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with a atomic and molecular composition so
they have a good idea of the

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Hamiltonian, but then they have particles
moving subject to that, that Hamiltonian.

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And they need to find they need to find
the the eigenfunctions.

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Well, in other words, usually we know

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the form of interactions between the
particles and

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we want to find the the, the energy

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and the wave function describes a
stationary state.

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We're going to get to that.

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But, at first, I'm going to take

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a slightly different approach than is
conventional.

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And that is look at some well known
function

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psi and see which Schrödinger equations
they satisfy.

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Because if you have some experience doing
that,

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then you can often make your way forward
much

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more rapidly in solving a practical
problem.

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This constructive approach that we're

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discussing now, is actually extremely
simple.

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Here's, for example, let's take a, a
motion of

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a particle mass m and a potential in three
dimensions.

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here's the, here's the Schrödinger
equation for that and by

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tri, trivial mathematical transformation,
you can reduce it to this form.

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So, in other words, if you, if you know
how to differentiate psi

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you can find an energy and a potential for
which it is a solution.

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So,

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there's nothing better than solving a
Schrödinger equation.

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let's do it in the next inline example.

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So, I hope you found that was a fairly
easy problem to solve.

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Just involves taking a derivative.

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And now, I might say that, that, the
solution's not entirely a contrived one.

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It's one that does occur in the theory of
particle scattering.

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I mean it's the, the basis

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for an important limit in the description
of atomic collisions.

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Okay, so now back to this somewhat trivial
procedure.

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you put in any function and it will solve
a Schrödinger

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equation, but usually not, not an
interesting one.

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But, it's worthwhile, looking at the

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cases where the interesting solutions are
obtained.

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furthermore there are constraints on what
are

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allowable solutions of Schrödinger
Equation,

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for example in three dimensions, d cubed r
psi star

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psi has got to be equal to 1 for a
stationary state.

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The, the, the probability of the, of the,
the net probability of

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the particle being in the presence of the
potential, must always be unity.

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And so, there are, there are lots of lots
of choices of wave function here

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which cannot possibly be brought into this
form.

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But let's, I mean, let's go find some
useful ones.

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Okay, so the last inline quiz

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simply has you

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take one of the most useful functions, not
just in quantum mechanics but in, in

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practical mathematical analysis in just
about anything, and find

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out what sort of what type of Schrödinger
equation

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it is a solution of.
And that, that'll be it for this lecture.

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Then, we'll go and apply that function in
following lectures.