Welcome to another video on the i get
chem channel, where we help you learn
chemistry the easy way by showing you
how to solve problems. This is a video in
our Fundamentals of Quantum Mechanics
series. In today's video we will talk
about eigenfunction eigenvalue problems
in quantum mechanics. If you've watched
some of the other videos on our channel,
you know our focus is on solving
chemistry problems. But this video will
be a little different, because instead of
solving a problem,
this video will try to answer a simple
question: What is an eigenfunction
eigenvalue problem anyway?
But before we start, please be sure to
subscribe to our channel if you want to
see more contents on quantum chemistry. OK,
so let's take a look at today's question.
A key equation in quantum mechanics is
the Schrodinger's equation, or more
precisely, the time-independent
schrodinger equation: H psi equals to E psi.
In mathematics this equation is called
an eigenfunction eigenvalue problem.
Let's take a closer look at what it
means. Let's say our system is one
dimensional, in this case the wave
function psi is a function of X.
Representing this explicitly, we have H
psi of x equals to E times psi of x. On the
left side, H is an operator. An operator
operates on the function following it,
transforming psi of x to some output
function, say f of x. The right side
states that the output f of x is just
equal to the original psi of x multiplied
by a number E, not including 0. So to
distinguish between operators and
numbers, we sometimes put a special
upside-down v symbol above H. So what are
some examples of operators? Well, here is
one: D by dX. D by dX, when applied to the
purple psi of x
function morphs it into the green f of x
function. In this example, psi x is a
cosine function, so f of X is a sine.
Even though psi and f are both periodic, the
purple psi of x lags the green f of x by
a quarter of a period. So you see
there's no way to multiply psi by just a
number and make it come out equal to f.
So we see in general some operator
operated on some function psi will
usually not give back the same function.
So what does it mean to solve the
Schrodinger equation? It turns out that
for each operator H that satisfies a
certain mathematical requirement called
hermitian, you can always find a special
set of functions psi that when operated on
by H would give you back the same
function multiplied by a number. These
special functions are called eigen-
functions of the operator H. The number E
corresponding to each eigenfunction is
then called is eigenvalue. Let's take a
look at an example. It turns out that the
operator D squared dx squared is a
Hermitian operator. The cosine functions
are eigenfunctions of d squared dx
squared. This is clear since the
derivative of cosine is negative sine,
and the derivatives of negative sine is
negative cosine. This is shown as the
green function f of x. We see that we can
simply multiply psi by -2 and turn it
in to f of x, so the purple psi function
is an eigenfunction of d squared dx
squared, and correspondingly there is an
eigenvalue E, and this one is equal to
minus 2. So the sine function solves this
eigenfunction eigenvalue equation. In
fact, you can easily see that
not only this, but all sine functions are
eigenfunctions of d squared dx squared.
Let's look at another example.
This operator H equals to negative d
squared dx squared plus one-half x
squared is also Hermitian. The graph
shows the first three eigenfunctions of
this H. There are actually an infinite set
of eigenfunctions for this operator. To
illustrate that these are eigenfunctions
we can take the first one psi 1 and
calculate one-half x squared times psi 1.
This is shown in yellow. We can also
calculate minus d squared psi dx squared.
This is shown in red. And if we add the
yellow and the red curves, we get
back psi 1. In quantum chemistry, this H
operator is associated with molecular
vibrations and is called the harmonic
oscillator. In this next example, we will
return to the operator we looked at
earlier: minus d squared dx squared, but
this time we will restrict the solution
to be zero everywhere outside of x
equals plus minus 0.5. This requirement
is represented by the orange shaded
regions in the graph. The eigenfunctions
must be 0 in these forbidden regions.
The eigenfunctions are again sinusoidal
functions, but now to satisfy the special
boundary condition, they are cosine
functions of half integer periods, that
is 1 half, 3 halves
5 halves periods, etc. that fit within the
allowed region, and at the same time sine
functions of integer period, that is 1, 2
3 periods, etc. that fit within the
allowed region. Again there are an
infinite number of eigen functions for
this eigenfunction eigenvalue problem
that also meets the special boundary
conditions here. In quantum chemistry,
this is called the particle in a box
problem, and this is the simplest
example of quantum confinement. In
general, the solutions to an
eigenfunction eigenvalue problem will
depend on three things: 1. The
eigenfunctions are different for
different Hermitian operators. 2. The
eigenfunctions will depend on any
boundary conditions imposed on the
system. For example, while the blue
function psi 3 is a solution to the
particle in a box problem, this red
function whose second derivative is
indeed equal to itself is not a proper
solution, because it does not meet the
boundary condition that psi must be 0
in the forbidden regions. 3. Finally, the
eigenfunctions will depend on
dimensionality of the problem. You have
seen that these are the eigenfunctions
of the particle in a box problem in 1d,
but here are some of the eigenfunctions
in 2d when the particle is trapped
inside a circular disc. I hope this video
has helped you understand in a more
visual way what an eigenfunction
eigenvalue problem is in quantum
mechanics. If you want to see some of the
mathematics worked out in detail in some
eigenfunction eigenvalue problem
examples, please check out some of our
other videos on the channel. I will make
a link to a couple of them at the end of
this video. And if you have enjoyed this
video, please like this video and
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upcoming series Making Sense of the
Quantum Realm.
