Hello, everyone.
Welcome back to Exploring Quantum Physics.
I'm Charles Clark, and we're now going to
start talking about how
to solve the Schrödinger Equation, what
that means, how it's done in practice.
And we're going to start with some simple
constructive techniques to give you a few
tools
that you can you use if you are
if you're stuck in solving a
quanta-mechanics problem.
Sometimes a few of these simple tricks
will help you see
your way through, help you identify
appropriate scaling relations and the
like.
So, let's proceed.
Okay, here's the Schrödinger Equation
which you've
heard a lot about already in the course.
size of wave function H, the Hamiltonian
operator.
ih bar, d divided dt,
psi.
And the precept of quantum mechanics is
that all accessible information about the
state is contained in the wave function.
Now, you know, this will, you, this is for
an isolated system.
When we're talking about two interacting
systems or a system in
a bath and there's, there's a lot more
nuance, but let's just,
let's just stick with simple many body
systems and
the ab, absence of noise or thermal
bath because that, in itself, is an
interesting problem.
So, in other words if we know the wave
function at a given time
we can just, we can just sequentially
integrate this equation
using some technique to find it at a later
time.
So why in this era of high performance
computing and all these other
wonderful things, is anyone really worried
about solving the Schrödinger Equation?
The inline quiz contains some sort of an
exercise on thinking about how you use the
Schrödinger
Equation to propagate the wave function
forward in time.
So, from a formal and even a practical
perspective for very simple systems
solutions to the Schrödinger Equation are
fairly straight forward to find.
But, in, in real life we're encountered
with
problems that involve atoms like
ytterbium, 70 electrons.
This is a system I happen to
have worked on myself with wonderful
collaborators, Marianna
Safronova, Sergey Porsev, during the past
year and
it's, it's motivated by determining the
corrections to atomic
transition frequency due to the presence
of a
thermal background for application next,
next generation atomic clocks.
Well a very simple, a very simple
consideration will show
you that even a, a crude representation of
the 70 electron
wave function on a course grid would just
be completely infeasible
by today's and presumably just about any
future technology that we know of.
So Physics-oriented approaches that take
ad, that take advantage of approximations,
reasonable approximations are, I think are
always going to be essential.
And so, learning how to solve the, the
simple systems gives one good
guidance as to what you're going to have
to do to deal with more complicated ones.
Now, there are many cases in which one
needs to solve the full time-dependent
Schrödinger equation.
say, studying non-equilibrium quantum
systems.
Here, here's an example of equation
solution of a, actually
of a nonlinear Schrödinger equation that
describes a Bose-Einstein condensate.
This, these are, this is a spatial
representation of density
of, of an atomic cloud at, at various
times and
so it's, it's sometimes critical to have
information like this.
However, there's quite enough to be
learned
by understanding how to solve the
stationary states
of the system which underpin a lot of the
important phenomena we see in nature.
And so the rest of this, this lecture's
going to focus on the solution of
stationary state problems.
So, here is the, the time-independent
version of the Schrödinger Equation.
The Hamiltonian operating on size and
eigenvalue, ei,
eigenfunction of a Hamiltonian operator
with eigenva, value.
E.
What does it mean to solve this equation?
Well, the real-life
problem that faces research physicists is
they have some material system
with a atomic and molecular composition so
they have a good idea of the
Hamiltonian, but then they have particles
moving subject to that, that Hamiltonian.
And they need to find they need to find
the the eigenfunctions.
Well, in other words, usually we know
the form of interactions between the
particles and
we want to find the the, the energy
and the wave function describes a
stationary state.
We're going to get to that.
But, at first, I'm going to take
a slightly different approach than is
conventional.
And that is look at some well known
function
psi and see which Schrödinger equations
they satisfy.
Because if you have some experience doing
that,
then you can often make your way forward
much
more rapidly in solving a practical
problem.
This constructive approach that we're
discussing now, is actually extremely
simple.
Here's, for example, let's take a, a
motion of
a particle mass m and a potential in three
dimensions.
here's the, here's the Schrödinger
equation for that and by
tri, trivial mathematical transformation,
you can reduce it to this form.
So, in other words, if you, if you know
how to differentiate psi
you can find an energy and a potential for
which it is a solution.
So,
there's nothing better than solving a
Schrödinger equation.
let's do it in the next inline example.
So, I hope you found that was a fairly
easy problem to solve.
Just involves taking a derivative.
And now, I might say that, that, the
solution's not entirely a contrived one.
It's one that does occur in the theory of
particle scattering.
I mean it's the, the basis
for an important limit in the description
of atomic collisions.
Okay, so now back to this somewhat trivial
procedure.
you put in any function and it will solve
a Schrödinger
equation, but usually not, not an
interesting one.
But, it's worthwhile, looking at the
cases where the interesting solutions are
obtained.
furthermore there are constraints on what
are
allowable solutions of Schrödinger
Equation,
for example in three dimensions, d cubed r
psi star
psi has got to be equal to 1 for a
stationary state.
The, the, the probability of the, of the,
the net probability of
the particle being in the presence of the
potential, must always be unity.
And so, there are, there are lots of lots
of choices of wave function here
which cannot possibly be brought into this
form.
But let's, I mean, let's go find some
useful ones.
Okay, so the last inline quiz
simply has you
take one of the most useful functions, not
just in quantum mechanics but in, in
practical mathematical analysis in just
about anything, and find
out what sort of what type of Schrödinger
equation
it is a solution of.
And that, that'll be it for this lecture.
Then, we'll go and apply that function in
following lectures.