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visit MIT OpenCourseWare at ocw.mit.edu. ROBERT FIELD: OK,
let's get started. So last time we talked about the
one-dimensional wave equation, which is a second-order
partial differential equation. This is not a math course. If you have a second-order
differential equation, there will be two linearly
independent solutions to it. And that's important
to remember. Now, there are
three steps that we use in approaching
a problem like this. Does anybody want to tell me
what those three steps are? Yes? AUDIENCE: [INAUDIBLE] it
has separate functions that take only one variable. ROBERT FIELD: OK,
that's most of it. You want to solve
the general equation, and one way to solve
the general equation is to try to separate variables. Always you want to
separate variables. Even if it's not
quite legal, you want to find a way to do that
because that breaks the problem down in a very useful way. So the first step is
the general solution, and it involves trying
something like this where we say u of x
and t is going to be treated in the separable form. If it doesn't work,
you're going to get 0. You're going to
get the solution. The only solution with separated
variables is nothing happening. And so that's unfortunate
if you do work and you get nothing for it,
but life is complicated. So then after you do the
general solution, what's next? Yes? AUDIENCE: You need to
set boundary conditions? ROBERT FIELD: Yes. Now if it's a second-order
differential equation, you're going to need
two boundary conditions. And when you impose two
boundary conditions, the second one gives some
sort of quantization. It makes the solutions discrete
or that there is a discrete set rather than a continuous set. The general solution is more or
less continuous, or continuous possibility. So we have now the
boundary conditions, and that gives us something
that we can start to visualize. So what are the important
things that, if you're going to be drawing pictures
rather than actually plotting some complicated mathematical
function, what are the first questions you ask? Yes? AUDIENCE: Is it
symmetric or asymmetric? ROBERT FIELD: Yes, if
the problem has symmetry, the solutions will
have symmetry. But there's more. I want another, my favorite. AUDIENCE: Where's it start? ROBERT FIELD: I'm sorry? AUDIENCE: Where does it start? So t equals 0 or x equals 0. ROBERT FIELD: The initial
condition is really the next thing I'm
going to ask you about, and that's called the pluck. You're right on target. But now if you're going
to draw a picture, the best thing you want
to do to draw a picture is have it not move. So you want to look at
the thing in position, and what are the things
about the position function that you can immediately
figure out and use in drawing a cartoon? AUDIENCE: Nodes. ROBERT FIELD: Yep. So how many nodes
and where are they? Are they equally spaced? And that's the most important
thing in drawing a picture, the number of zero crossings
that a wave function has, and how are they distributed? Their spacing of nodes
is half the wavelength, and the wavelength is
related to momentum. And so I'm jumping
into quantum mechanics, but it's still valid for
understanding the wave equation. So we want the number of nodes
for each specific solution satisfying the boundary
conditions and the spacing and the loops between nodes. Are they all identical? Is there some
systematic variation in the magnitude of each
of the loops between nodes? Because if you have just a
qualitative sense for how this works, you can draw
the wave function, and you can begin to make
conclusions about it. But it all starts with nodes. Nodes are really important. And quantum mechanics, the
wave functions have nodes. You can't do better
than focus on the nodes. And then I like
to call it pluck, but it's a superposition
of eigenstates. And those superpositions
for this problem can move. And so-- you're
missing a great-- anyway. And so you want to know,
what is the kind of motion that this thing can have? And so one of the things-- this is really the three
steps you go through in order to make a picture
in which you hang up your insights. And so if there's one
state or two states, one state is just going
to be standing waves. Two states is all the complexity
you're ever going to need. And if two states have
different frequencies, there will be motion,
and the motion can be side-to-side motion
or sort of breathing motion where amplitude moves in
from the turning-point region to the middle and
back out again. And so you can be
able to classify what you can understand and
to imagine doing experiments based on this simplified
version of the flux. In my opinion, the
most important thing you can do as a professional
quantum machinist and in preparation
for exams is to be able to draw these cartoons
quickly, really quickly. That means you have to
think about them in advance. And so this recipe
is how you're going to understand quantum
mechanical problems too. And this differential
equation is actually a little more complicated
than the first few that we're going to encounter
because the first few problems we're going to face
are not time dependent. There may still be a separation
of variables situation and imposing boundary
conditions and so on, but there is no motion. But eventually we'll get motion
because our real world has motion, and quantum mechanics
has to reproduce everything that our real world does. So, we're going to
begin quantum mechanics, and first of all I
will describe some of the rules we have
to obey in building a quantum-mechanical picture. And then I'll approach
two of the easiest problems, the free
particle and the particle in an infinite box. So we have the one-dimensional
Schrodinger equation, and the one-dimensional
Schrodinger equation looks like the wave equation. And why? Because waves interfere
with each other. We can have constructive and
destructive interference. Almost everything that is
wonderful about quantum mechanics is the solutions
to this Schrodinger equation also exhibit constructive
and destructive interference. And that's essential to our
understanding of how quantum mechanics describes the world. The next thing I want to
do is talk a little bit about postulates. Now I'm going to be introducing
the quantum-mechanic postulates as we need them
as opposed to just a dry lecture of these
strange and wonderful things before we're ready for them. But a postulate is something
that can't be proven right. It can be proven wrong. And we build a system of logic
based on these postulates. Now one of the great
experiences in my life was one time when I visited the
Exploratorium in San Francisco where there are rather crude,
or at least when I visited almost 50 years ago,
there were rather crude interactive
experiments where people can turn knobs and push
buttons and make things happen. And the most wonderful
thing was really young kids trying to break these exhibits. And what they did is
by trying to break them they discovered
patterns, some of them, and that's what
we're going to do. We're going to try to break or
think about breaking postulates and then see what we learn. So, let's begin. We have operators in
quantum mechanics. And we denote them either with
a hat or as a boldface object. We start using this
kind of notation when we do matrix mechanics,
which we will do, but this is just a general
symbol for an operator. And an operator
operates on a function and gives a different function. It operates to the
right, or at least we like to think about it as
operating to the right. If we let it
operate to the left, we have to figure out
what the rules are, and I'm not ready to
tell you about that. So this operator
has to be linear. And so if we have an
operator operating on a function, A f plus
bg, It has to do this. Now you'd think, well,
that's pretty simple. Anything should do that. So taking the
derivative does that. Doing an integral does that, but
taking the square root doesn't. So taking the square
root-- an operator says take the square root, well,
that's not a linear operator. Now the only operators
in quantum mechanics are linear operators. We have eigenvalue equations. So we have an operator
operating in some function. It gives a number and
the function back again. And this is called
the eigenvalues, and this is the eigenfunction. AUDIENCE: Dr. Field? ROBERT FIELD: Yes? AUDIENCE: In the
[INAUDIBLE] that should be b times
A hat [INAUDIBLE].. ROBERT FIELD: What did I do? Well, it should just be-- sorry about that. I'm going to make
mistakes like this. The TAs are going
to catch me on it, and you're going to do it too. All right, so now here we
have an operator operating on some function. And this function is
special because when the operator operates
on it, it returns the function times a
number, the eigenvalue and the eigenfunction. We like these. Almost all quantum mechanics
is expressed in terms of eigenvalue equations. Operators in quantum mechanics-- so for every physical quantity
in non-quantum-mechanical life there corresponds an operator
in quantum mechanics. So for the coordinate,
the operator is just the coordinate. For the momentum,
the operator is minus ih bar partial with respect to
x or derivative with respect to x. Now this is not too
surprising, but this is really puzzling because why
is there an imaginary number? This is the square root of
minus 1, which we call i. Why is that there and why
is the operator a derivative rather than just
some simple thing? Another operator is
the kinetic energy, and the kinetic energy
is p squared over 2m. And so that comes out to
be minus h bar squared over 2m second partial
with respect to x. Well, it's nice that I
don't have to memorize this because I can just square
this and this pops out, but you have to be
aware of how to operate with complex and
imaginary numbers. And there are so many
exercises on the problem set, so you should be
up to date on that. And now the potential
is just the potential. And now the most
important operator, at least when we start
out, is the Hamiltonian, which is the operator that
corresponds to energy, and that is kinetic energy
plus potential energy. And this is called
the Hamiltonian, and we're going to be
focusing a lot on that. So these are the operators
you're going to care about. The next thing we talk
about is commutation rules or commutators. And one really
important commutator is the commutator
of the coordinate with the conjugate
momentum, conjugate meaning in the same direction. And that is defined
as xp minus px. And the obvious thing is that
this commutator would be 0. Why does it matter which
order you write things? But it does matter. And, in fact, one approach
to quantum mechanics is to start not
with the postulates that you normally deal with
but a set of commutation rules, and everything can be derived
from the commutation rules. It's a much more
abstract approach, but it's a very
powerful approach. So this commutator is not zero. And how do you find out
what a commutator is? Well, you do xp minus px,
operate on some function, and you find out. And you could do that. I could do that. But the commutator is going
to be equal to ih bar. Now there is a little
bit of trickiness because the commutator xp is
ih bar and px is minus ih bar. And so I don't
recommend memorizing it. I recommend being able
to do this operation at the speed of light
so you know whether it's plus ih bar or minus
ih bar because you get into a whole lot of
trouble if you get it wrong. So this is really where
it all begins, and this is why you can't
make simultaneously precise measurements of
position and momentum, and lots of other good things. And then we have wave functions. So wave function for when the
time independent Hamiltonian is a function of one variable, and
it contains everything we could possibly know about the system. But this strange
and wonderful thing, which leads to all sorts
of philosophical debates, is that this guy, which contains
everything that we can know, can never be directly measured. You can only
measure what happens when you act on something
with a given wave function. You cannot observe
the wave function. And for a subject area where the
central thing is unobservable is rather spooky. And a lot of people
don't like that approach because it says we've got this
thing that we're relying on, but we can't observe it. We can only observe what
we do when we act on it. And usually the
action is destructive. It's destructive of the
state of the system. It causes the
state of the system to give you a set
of possible answers, and not the same
thing each time. So it's really weird. So we have wave functions. And we can use the
wave function to find the probability of the system
at x with a range of x, x to x plus dx. And that's psi
star x psi of x dx. So you notice we have two wave
functions, the product of two, and this star means takes
a complex conjugate. So if you have a complex number
z is equal to x plus iy-- real part, imaginary part-- and if we take z star,
that's x minus iy. So these wave functions
are complex functions of a real variable. And so we do things like
take the complex conjugate, and you have to become
familiar with that. Now we have what we call
the expectation value or the average value,
and we denote this as A. So for the state
function psi, we want the average value of the
operator A. Now in most life, that symbol is not included
just because people assume you know what you're doing. And this is psi star A hat
psi dx over psi star psi dx. And this is integral from
minus infinity to infinity. So this down here is a
normalization integral. Now we normally deal
with state functions which are normalized to
1, meaning the particle is somewhere. But if the particle
can go anywhere, then normalization to 1 means
it's approximately nowhere. And so we have to think
a little bit about what do we mean by
normalization, but this is how we define the average
value or the expectation value of the quantity
A for the state psi. So this is just a
little bit of a warning that, yeah, you would
think this is all you need, but you also need to at
least think about this. That's great. I'm at the top of the board
and we're now at the beginning. So the Schrodinger
equation is the last thing, and that's the Hamiltonian
operating on the function and gives an energy
times that function. And if it's an
eigenvalue, then we have this eigenvalue equation. We have these symbols here. So that's the energy associated
with the psi n function. So now we're ready to
start playing games with this strange new world. And so let's start out
with the free particle. Now because the free particle
has a complicated feature about how do we
normalize it, it really shouldn't be the first
thing we talk about. But it seems like the
simplest problem, so we will. So what's the Hamiltonian? The Hamiltonian is the kinetic
energy, minus h bar squared, or 2m second derivative
with respect to x plus the potential energy, V0. Free particle, the
potential is constant. We normally think of it
as the potential is zero, but there is no absolute
scale of a zero of energy, so we just need to specify this. And so we want to write
the Schrodinger equation, and we want to arrange it in
a form that is easy to solve. There is two steps
to the rearrangement, and I'll just write
the final thing. So the second
derivative of psi is equal to minus 2m over h bar
squared times E minus V0 psi. So this is the differential
equation that we have to solve. So there was a little bit
of rearrangement here, but you can do that. So the second derivative
of some function is equal to some constant
times that function. We've seen that problem before. It makes a lot of difference
whether that constant is positive or
negative, and it better, because if we have
a potential V0 and we have an
energy up here, well, that's perfectly reasonable. The particle can be
there, classically. But suppose the
energy is down here. If the zero of energy is
here, you can't go below it. That's a classically
forbidden situation. And so for the classically
allowed situation, the quantity, this
constant, is negative. For the classically
forbidden situation, this constant is positive. You've already seen
the big difference in the way a second derivative,
this kind of equation, works when the constant
is positive or negative. When this constant is
negative, you get oscillation. When this constant is positive,
you'll get exponential. Now we're interested
in a free particle, so free particle
can be anywhere. And we insist that
the solution to our quantum-mechanical
problem, the wave function is what we say well behaved. So well behaved has many
meanings, but one of them is it never goes to infinity. Another is that when
you go to infinity, the wave function
should go to zero. But there's also things about
continuity and continuity of first derivatives
and continuity of second derivatives. We'll get into those,
but you know immediately that if this
constant is positive, you get an exponential
behavior, and you get the e to the ikx and e to the-- not ik-- e to the kx
and e to the minus kx. And one of those blows up
at either positive infinity or negative infinity. So it's telling you
that in agreement with what you expect
for the classical world, an energy below the constant
potential is illegal. It's illegal when this
situation persists to infinity. But we'll discover
that it is legal if the range of coordinate
for which the energy is less than the potential is finite. And that's called
tunneling, and tunneling is a quantum-mechanical
phenomenon. We will encounter that. So we know from our experience
with this kind of differential equation that the solutions
will have the form sine kx and cosine kx. But we choose to use
instead e to the ikx and e to the minus ikx
because this cosine kx is 1/2 e to the ikx
plus e to the minus ikx. And so we can use
these functions because they're more
convenient, more memorable. All the integrals and
derivatives are trivial. And so we do that. So the differential equation-- and we saw before that we
already have what k is. So minus k squared is minus
2m over h bar squared-- minus 2m over h bar
squared E minus V0. We take the derivative
of this function. This is the function, and
this is the eigenvalue. We take the second
derivative with respect to x. We get an ik from
this term and then another ik, which
makes minus k squared. And we get a minus ik
and another minus ik, and that gives a
minus k squared. And so, in fact, this is
an eigenvalue equation. We have the form where this
equation is an eigenfunction. With this, we have everything. So the energies for the free
particle, h bar k over h bar k squared over 2m plus V0, so
this is an eigenfunction, and this is the eigenvalue
associated with that function. We're done. That was an easy problem. I skipped some steps
because it's an easy problem and I want you to go
over it and make sure that you understand
the logic and can come to the same solution. Let's take a little side issue. Suppose we have psi
of x is e to the ikx. Well, we're going to find that
this is an eigenfunction of p, and the eigenvalue or the
expectation value of p is h bar k. And if we had minus e to the
minus ikx, then what we'd get is minus h bar k. So we have this relationship
between p, expectation value, and h bar k. So this corresponds
to the particle going in the positive x direction,
and this corresponds to the particle going in
the negative x direction. Everything is
perfectly reasonable. We have solutions to
the Schrodinger equation for the free particle. The solutions to
the free particle are also solutions to
the eigenvalue equation for momentum. And the two possible
eigenvalues for a given k are plus h bar k minus h bar k. Now that's fine. So everything works out. We're getting
things, although we have the definition of the
momentum having a minus 1, an i factor, and a
derivative factor. Everything works. Everything is as
you would expect. And the general solution
to the Schrodinger equation can have two different values,
the superposition of these two. Right now, this wave function
is the localized overall space. Now if we want to
normalize it, we'd like to calculate integral
minus infinity to infinity of psi star x psi of x dx. This is why we
like this notation because suppose we have a
function like this, psi star-- well, actually like this. Psi star is equal to a star e
to the minus ikx plus b star e to the ikx, and psi
is a e to the plus ikx. This would be e
to the minus ikx. And so when we write this
integral, what we get is integral of
psi star psi dx is equal to a squared integral
minus infinity to infinity of a squared plus b squared dx. So we have two constants
which are real numbers because they're square modulus. They're additive. And we're integrating this
constant from minus infinity to infinity. We'll get infinity. We can't make this equal to 1. So we have to put
this in our head and say, well,
there's a problem when you have a wave function
that extends over all space. It can't be normalized
to 1, but it can be normalized so that for
a given distance in real space, it's got a probability
of 1 in that distance. So we have a different
form of normalization. But when we actually
calculate expectation values, we can still use this naive idea
of the normalization interval and we get the
right answer, even though because both the
numerator and denominator go to infinity and
those infinities cancel and everything works out. This is why we don't
do this first usually because there's
all of these things that you have to
convince yourself are OK. And they are and you should. But now let's go to the
famous particle in a box. It's so famous that we
always use this notation. This is particle
in an infinite box, and that means the particle
is in a box like this where the walls go to infinity. And so we normally
locate this box at a place where this
is the x coordinate and this is the
potential energy, and the width of the box is a. And we normally put the
left edge of the box at zero because that problem
is a little easier to solve than the
more logical thing where you say, OK, this
box is centered about zero. And that should bother
you because anytime you're interested in asking
about the symmetry of things you'll want to choose a
coordinate system which reflects that. Don't worry. I am going to ask
you about symmetry, and it's a simple thing to take
the solution for this problem and move it to the
left by a over 2. So we have basically a
problem where the potential is equal to 0 for 0 is less
than or equal to x less than or equal to a, and it's equal to
infinity when x is less than 0 or greater than a. So inside the box it looks
like a free particle, but it can't be a free
particle because there's got to be nodes at the walls. We know that outside the
box, the wave function has to be 0 everywhere because
it's classically forbidden, strongly forbidden. We know that the wave
function psi is continuous. So if it's at 0 outside, it's
going to be 0 at the wall. And so the wave functions
have boundary conditions where, at the wall, the
wave function goes to 0. So now we go and we
solve this problem. And so the Schrodinger equation
for the particle in the box where V of x is 0. Well, we don't need it. We just have the
kinetic-energy term, h bar squared over 2m second
derivative with respect to x. Psi is equal to e psi. Again, we rearrange it. And so we put the
derivative outside, and we have minus 2me
over h bar squared psi. And this is a number. And so we just call it
minus k squared psi. We know what that k is as long
as we know what the energy is, and k squared is equal to
2me over h bar squared. Now we have this thing which
is equal to a negative number times a wave function,
and we already know we have
exponential behavior. But in this case, we
use sines and cosines because it's more convenient. So psi of x is going to be
written as A sine kx plus B cosine kx. This is the general solution
for this differential equation where we have a negative
constant times the function. So the boundary
condition, psi of 0-- well, psi of 0, this is
0, but this part is 1. So that means that
psi of 0 has to be 0, so B has to be equal to 0. And here, the other
boundary condition, that also has to be equal to 0,
and that has to be A sine ka. And ka has to be equal
to n times pi in order to satisfy this equation. Sine is 0 at 0, pi, 2
pi, 3 pi, et cetera. So k is equal to n pi over a. And so we have the solutions. Psi of x is equal to
A sine np a over x. And so we could put
a little n here. And this is starting
to make you feel really good because for
all positive integers there is a solution. There's an infinite
number of solutions. And their scaling with
quantum number is trivial. And it's really great
when you solve an equation and you are given an
infinite number of solutions. Well, there's one thing
more you have to do. You have to find out what the
normalization constant is, so you do the
normalization integral. And when you do
that, you discover that this is equal to 2 over a
square root sine n pi over a. So these are all the
solutions for the particle in an infinite box, all of them. And the energies you can write
as n squared times h squared over 8 m a squared or
n squared times E1. There's another thing. n equals 0. If n is equal to 0, the wave
function corresponding to n equals 0 is 0 everywhere. The particle isn't in the box. So n equals 0 is not a solution. So the solutions we have-- n equals 1 2, et
cetera up to infinity, and the energies are
integer square multiples of a common factor. This is wonderful because
basically we have a problem. Maybe it's not that
interesting now because why do we have infinite
boxes and stuff like that? But if you ask, what
about the ideal gas law? We have particles that don't
interact with each other inside a container which
has infinite walls. And I can tell you
that in 5.62, there's a three-line derivation
of the ideal gas law based on solutions
to the particle in a box. Also, we often have
situations where you have molecules where
there's conjugation so that the molecule looks
like a not quite flat bottom box with walls. And this equation enables
you to learn something about the electronic
energy levels for linear conjugated molecules. And this leads to a lot
of qualitative insight into problems in photochemistry. Now the most important
thing, in my opinion, is being able to draw
cartoons, and these cartoons for the solutions the particle
in a box look like this. So what you frequently
do is draw the potential, and then you draw the energy
levels and wave functions. I have to cheat a little bit. So number of nodes-- number of nodes, internal nodes. We don't count
zeros at the walls. The number of
nodes is n minus 1. The maximum of the wave
function is always 2 over a. So here 2 over a, here
2 over a, here minus 2 over a i square
root of 2 over a. This slope is identical
to this slope. This slope is identical
to this slope which is identical to that slope. So there's a tremendous
amount that you can get by understanding
how the wave function looks and drawing these cartoons. And so now if instead we were
looking at problems where, instead of a particle
in a box like this, we have a little dimple
in the bottom of the box or we have something at the
bottom or the bottom of the box is slanted. You should be able to
intuit what these things do to the energy
levels, at least have the beginning of an intuition. So, we have an infinite number
of oscillating solutions. That means that we
could solve the problem for any kind of a box as long
as it has vertical walls, and that's called
a Fourier series. So for a finite range, we
can describe the solution to any quantum-mechanical
infinite box with a terrible
bottom, in principle, by a superposition of
our basis functions. That's what we call them. Now, there are several
methods for doing the solution of a problem
like this efficiently. And you're going to see
perturbation theory. And at the end of
the course, you're going to see something that
will really knock your socks off which is called the discrete
variable representation. And that enables you
to say, yeah, well, potential does terrible things. I solved the problem by not
doing any calculation at all because it's already been done. So these things are
fantastic that we have an infinite number of
solutions to a simple problem. We're always looking for a way
to describe a simple problem or maybe not so simple problem
with an infinite number of solutions where the energies
for the solutions and the wave functions behave in a simple
n-scaled, quantum-number-scaled way. And this provides us with
a way of looking at what these things do in real life. You do an experiment
on an eigenfunction of a box like this, and it will
have certain characteristics. And it tells you, oh,
if I measured the energy levels of a pathological box,
the quantum-number dependence of the energy levels
has a certain form, and each of the constants
in that special form sample a particular feature
of the pathological potential. And that's what we do
as spectroscopists. We find an efficient way to
fit the observables in order to characterize what's going on
inside something we can't see. And that's the game
in quantum mechanics. We can't see the
wave function ever. We know there are eigenstates. We can observe energy levels
and transition probabilities, and between those two things,
we can determine quantitatively all of the internal structure
of objects that we can't see. And this is what I do
as a spectroscopist. And I'm really a little
bit crazy about it because most people
instead of saying let's try to understand
based on something simple, they will just solve the
Schrodinger equations numerically and get a
bunch of small results and no intuition, no
cartoons, and no ability to do dynamics except
another picture where you have to work things
out in a complicated way. But I'm giving you the
standard problems from which you can solve almost anything. And this should sound
like fun, I hope. OK, so have a great weekend.
