The following content is
provided under a Creative Commons license. Your support will help
MIT OpenCourseWare continue to offer high-quality
educational resources for free. To make a donation or to
view additional materials from hundreds of MIT courses,
visit MIT OpenCourseWare at ocw.mit.edu. ROBERT FIELD: Well, we're now
well underway into quantum mechanics. So a lot of the important
stuff goes by very fast. So we represent a quantum
mechanical operator with a little hat, and
it means do something to the thing on its right. And it has to be
a linear operator, and you want to be sure you
know what a linear operator does and what is not a
linear operator. This is an eigenvalue equation. So we have some
function which, when the operator operates
on it, gives back a constant times that function. The constant is the
eigenvalue, and the function is an eigenfunction
of the operator that belongs to this eigenvalue,
and all of quantum mechanics can be expressed in terms
of eigenvalue equations. It's very important, and you
sort of take it for granted. Now, one of the important
things about quantum mechanics is that we have to find
a linear operator that corresponds to the classically
observable quantities. And for x the linear
operator is x, and for the momentum
the linear operator is minus ih bar partial
with respect to x. That should bother you two ways. One is the i, and the other
is the partial derivative. But when you apply this
operator to functions, you discover that out pops
something that has the expected behavior of
momentum, and so this is in fact the
operator that we're going to use for momentum. And then there is
a commutation rule. This commutation rule, xp
minus px, is equal to this. This is really the foundation
of quantum mechanics, and as I've said before,
many people derive everything from a few commutation rules. It's really scary,
but you should be able to work out
this commutation rule by applying xp minus px
to some arbitrary function. And going through the symbolics
should take you about 30 seconds, or maybe it shouldn't. Maybe you can go faster. OK. We have an operator,
and we often want to know what
is the expectation value of the particular
function, which we could symbolize here, but
it's never done that way. So we have some
function, and we want to calculate its expectation
value of operator A, and this is it. And so this is a
normalization integral, and this normalization integral
is usually taken for granted, because we almost always
work with sets of functions which are normalized. And so if you convince
yourself that it is, in fact, they are
normalized, fine, and then this is the thing that
you normally would calculate. Then, we went to
our first problem in quantum mechanics which
is the free particle, and the free particle
has some idiosyncrasies. The wave function
for the free particle has the form e to the ikx
plus e to the minus ikx, and the Hamiltonian
is minus h bar squared over 2m second partial
with respect to x plus v0. So there's no v0 here, and we
have two different exponentials and so is this really going
to be an eigenfunction of the Hamiltonian? This is really p
squared, and so this is going to be an
eigenfunction of p squared too. All right, so let's show
a little picture here. Here is energy, and this
is v0, and let's say this is the 0 of energy. What are the
eigenvalues of this? What does the Hamiltonian
do to this function? Well, in order to do that, you
have to calculate something like where you have to calculate
the second derivative of each of these terms, and the
second derivative of this term brings down a minus k squared. Now, the second
derivative of this term brings down a minus k squared,
and so the energy eigenvalues are going to be given by h bar
squared a squared over 2m plus v0. So these are the
eigenvalues, eigenfunctions, the energies of a free particle,
and they're not quantized. Now this v0 is something
that will often trip you up, because it's hidden here. It's not in here. OK. I'm going to torture
you with something. So why are these
two k's the same? What would happen if the
k for the positive term were different from the
k for the negative term? Simple answer. Yeah? AUDIENCE: There'd
be two eigenvalues? ROBERT FIELD: That's right. It wouldn't be an eigenvalue. It wouldn't be an eigenfunction
of the Hamiltonian. It's a mixture of
two eigenvalues, and so that's simple. But often we might be dealing
with a potential that's not simple like this
but has got complexity. So suppose we had a
potential that did this. The potential is
constant piecewise, and so what do we do? Yes? AUDIENCE: Break down
the function into pieces for each in certain boundaries? ROBERT FIELD: Yes, and
that's exactly right. You do want to
break it up, but one of the things I'm
stressing here is that you want to be
able to draw cartoons. And so we know that if
we choose an energy here, there is a certain momentum, or
a certain kinetic energy here, and a different kinetic
energy here, and so somehow, what you write for the wave
function will reflect that. But now qualitatively,
pictorially, if we have a wave function
in this region which is oscillating like
this, and it'll be oscillating at the same
spatial frequency over here, well what's going to
be happening here? Is it going to be
oscillating faster or slower? AUDIENCE: Faster. Faster. ROBERT FIELD:
Absolutely, and is it going to have amplitude
smaller or larger than here? You're going to answer, yes? Well, let me do a
thought experiment. So I'm going to walk from
one side of the blackboard to the other, and I'm going to
walk at a constant velocity. Then, I'm going to walk
faster and then back to this original velocity. So what's the
probability of seeing me in the middle region
relative to the edge regions? Yes? AUDIENCE: It's less. ROBERT FIELD: That's right. The probability,
local probability, is proportional to
1 over the velocity, and the wave function
is proportional to 1 over the square root
of the velocity. And the velocity is
related to the momentum, and so we have everything. So we know that the wave here
will be oscillating faster and with lower amplitude. This is what I want
you to know, and you'll be able to use that
cartoon to solve problems. If you understand
what's going on here, these pictures
will be equivalent to global understanding,
and these pictures are also part of semi-classical
quantum mechanics. I believe you all know classical
mechanics at least a little, enough to be useful. And what we want to be able to
do in order to draw pictures and to understand stuff is
to insert just enough quantum mechanics into classical
mechanics so that it's correct. Then, all of a sudden, it
starts to make a lot more sense. OK. So the particle in a box, well,
we have this sort of situation, and we have 0 and a. So the length of the box is a,
and the bottom of the box v0 is 0 for this picture. Now, one of the things that
I want you to think about is, OK, I understand. I've solved this problem. I know how to
solve this problem. I know how to get
the eigenvalues, and I know how to get
the eigenfunctions, and I know how to
normalize them. Well, suppose I move
the box to the side. So I move it from
say b to a plus b. So it's the same width, but
it's just in a different place. Well, did anything change? The only thing that changes
is the wave function, because you have to
shift the coordinates. What happens if I raise
the box or lower the box? Will anything change? AUDIENCE: [INAUDIBLE] ROBERT FIELD: You're hot. AUDIENCE: [INAUDIBLE] ROBERT FIELD: I'm sorry? AUDIENCE: [INAUDIBLE] ROBERT FIELD: Yeah. If I move the box so that
v0 is not 0, but v0 is 10. AUDIENCE: Then, the weight
function will oscillate slower. ROBERT FIELD: No. AUDIENCE: [INAUDIBLE] ROBERT FIELD: So if you
move the box up in energy, the wave function is going
to look exactly the same, but the energies are going to
be different by the amount you move the box up or down, and
this is really important. It may seem trivial to some
of you and really obscure to others, but you
really want to be able to take these things apart. Because that will enable
you to understand them in a permanent way, and the
cartoons are really important. So if you have the solution
to the particle in a box, then it doesn't matter
where the box is. You know the solution to
any particle in a box. OK. There is something that I
meant to talk about briefly, but when we write
these solutions-- where did the other blackboard go? All right, well,
I've hidden it-- so when we have solutions
like e to the ikx and e to the minus ikx, so we
have say a here and b here. When we go to normalize
a function like this-- let's put the plus in here-- then we write psi star psi dx. So psi star would
make this go a star and this go to e
to the minus ikx, and this go to b star
e to the plus ikx. So now, we multiply
things together. We get an a, a star which
is the square modulus of a, and we get e to the ikx
and e to the minus ikx. It's 1. This is why we use this form. The integrals for things
involving e to the ikx are either 1 or 0. So if you took e to
the ikx, this term, and multiplied it by this
term, you'd get an a, b star e to the 2 ikx
integrated over a finite region. That's 0. So we really like this
exponential notation, even if you've been brought
up on sines and cosines, and you use the sines and
cosines to impose the boundary conditions. OK, another challenge. So this is v0, and the
only problem is this v0-- well, it looks like this. So this is v1. OK, so we have now a
particle in this straight. It's a hybrid between
the free particle and a particle in a box. So suppose we're at
an energy like this. What's going to happen? Well, everything
that's outside-- everything that's in the
classically-allowed region, we understand. We know how to deal with it, but
in here, well, that's OK too. But inside this
classically-forbidden region, the wave function is going
to behave differently. Now, I'm going to
assert something. It doesn't have nodes. It doesn't oscillate. It's either
exponentially decreasing or exponentially increasing, and
it will never cross 0, never. OK. So now, if we're solving
a problem involving any kind of 1D potential,
number of nodes. So for 2D-bound problems, the
number of nodes starts with 0, and it corresponds to
the lowest energy state. The next state up has 1 node,
and the next state has 2 nodes. So by counting the
nodes, you would know what the energy order is of
these eigenvalues which is also an extremely useful thing. If you're thinking about
it or telling your computer to find the 33rd
eigenvalue of something, because you just run a
calculation that solves for an approximate wave
function, and the 33rd, it needs 32 nodes. And so the computer says,
oh, thank you, master, and here is your wave
function, but you have to find the right thing. OK. Now, here is the picture that
you use to remember everything about a particle in a box. And the wave function
looks like this, and the next wave
function looks like that, and the next wave
function looks like this. And so no nodes, 1
node, 2 nodes, the nodes are symmetrically arranged
in the space available. And the lobes on one side of
the node and the other side have the same amplitude,
different sine, and they're all normalized. And so the maximum value
for each of these guys is 2 over a square root,
where this is 0 to a. So that's a fantastic
simplification, and it also reminds
you of Mr. DeBroglie. He said, you have
to have an integer number of half wavelengths-- well, for the hydrogen--
an integer number of wavelengths around a path. And for here, you
need an integer number of wavelengths for
that round trip which is the same
thing or an integer number of half wavelengths. That's DeBroglie's
idea, and it enables you to say, oh well,
let's see if we can use this concept
of wavelength to approach general problems. OK. Well, if you do something
to the potential by putting a little thing in
it, well, the wave function will oscillate more
slowly in that region, and that causes it to be at
a higher or lower energy? If it's oscillating
more slowly here, it has to make it to an integer
number of half wavelengths, and so that means
it pushes it up. And if you do this,
it'll push it down, and you can do terrible things. You can put a delta
function there, and now you know
everything qualitatively that can happen in a 1D box. OK. One of the things that
bothers people a lot is, OK, so we have
some wave function, it's got lots of nodes, and the
particle starts out over here. How did it get across the node? How does it move
across the node? Well, the answer
is it's not moving. It's here. It's here. It's here. It's everywhere, and this is
just the probability amplitude. There is no motion through
a node, no motion at all. We are going to do
time-dependent quantum mechanics before too long,
and then there will be motion, but that motion is encoded
in a different way. OK. Another thing, suppose you
have a particle in a box, and it's in some state, and
I'm going to draw something like this again. OK, first of all, one,
two, three, four, five, which state is that? I got-- the hands are right,
six, it's the sixth eigenstate. OK. Now, suppose--
nothing is moving. Right? This is a stationary state. How would you
experimentally, in principle, determine that the particle
is in this n equals 6 state? Now, this can be a completely
fanciful experiment, which you would never do,
but you could still describe how you would do it
and what it would tell you. And so, yes. AUDIENCE: Try to find
the n equals 6 to n equals 7 transition by
irradiating it or something? ROBERT FIELD: OK. That's the quantum mechanical-- I agree, spectroscopy
wins always. But if you want to observe
the wave function or something related to the wave function,
like the number of nodes, what would you do? And the reason I'm being
very apologetic about this is because it's a
crazy idea, But this is a one-dimensional system. Right? It's in the blackboard, and
so you could stand out here and shoot particles at it from
the perpendicular direction and collect the number
of times you have a hit. And so you would
discover that you would measure a probability
distribution which had the form-- well, I can't draw
this properly. It's going to have one,
two, three, four, five, six, six regions separated by a
gap, and what it's measuring is psi 6 squared. Well, you can't measure, you
cannot observe a wave function, but you can observe a
probability distribution wave function squared. You can also do a
spectroscopic experiment and find out what is the
nature of the Hamiltonian. And if you know the
nature of the Hamiltonian, you can calculate
the wave function, but you can't observe it. OK. Another thing, this harmonic
oscillator-- this particle in a box has a minimum
energy which is not at the bottom of the box. Well, we have something called
the uncertainty principle. Now, I'm just pulling
this out of my pocket, but I know that x, p is
equal to i h bar not 0, and one can derive some
uncertainty principle by doing a little bit more mathematics. But basically that
uncertainty principle is where sigma x is
expectation value of x squared minus expectation
value of x squared square root. So if we can calculate
this and calculate that, we can calculate
the variance in x, and you can calculate
the variance in p. That's exact, and
that's what you can derive from the computation
rule, but for our purposes, we can be really crude. And so if I'm in this
state, what is delta x? What is the range of
possibilities for x? AUDIENCE: The box link? ROBERT FIELD: Yeah, a. OK, and what is
the possibility-- what is the
uncertainty in p sub x? In an eigenstate, we've got
equal amplitudes going this way and going that way. So we could just say p sub x
positive minus p sub x negative which is 2p. That's the uncertainty. And if we know what
quantum number we're in we know what the expectation
value for p of the momentum is, and what we derive is that
delta x delta P is equal to hn. You can do that, and maybe
I should ask you to really be sure you can do that. In seconds, because
you really know what the possible values of
momentum or momentum squared are for a product of a box. OK. So why is there zero-point
energy, because if you said, I had a level at the
bottom of the box, we would have the momentum 0. The uncertainty and
the momentum is 0, and the product
of the uncertainty of the moment times the
product of the uncertainty in the position has to
be some finite number, and you can't do that here. And so this is a simple
illustration of the uncertainty principle that you have to have
a non-zero zero-point energy. That's true for all
one-dimensional problems. OK. We've got lots of time. One of the beautiful things
about quantum mechanics is that if you
solved one problem, you could solve a whole
bunch of problems, and so to illustrate
that, let's consider the 3D particle in a box. So for the 3D particle
in a box, the Hamiltonian can be written as a little
Hamiltonian for the x degree of freedom y and z. OK. So we have three independent
motions of the particle. They're not coupled. They could be, and we're
interested in letting them be coupled. But that's where we start
asking questions about reality, and that's where we bring
in perturbation theory. But for this, oh,
that's fantastic, because I know the
eigenvalues of this operator and of this operator,
eigenfunctions, and of this operator. So the problem is
basically solved once you solve the 1D box. Now, one proviso, what you can
do this separation completely formally as long
as hx, hy commute, and basically we say the
x, y, and z directions don't interact with each other. The particle is
free inside the box. It's just encountering walls. There are no springs or
anything expressing the number of degrees of freedom. OK. So we have now a wave function,
which is a function of x, y, and z, but we can
always write it as psi x of x, psi
y of y, psi z of z. So it's a product of three
wave functions that we know, and the energy is
going to be expressed as a function of three quantum
numbers, where the box is edge lengths a, b, and c. You didn't see me
looking at my notes. I'm just taking the solution to
1D box, and I'm multiplying it. And so now we have the
particle in a 3D box, and this is where the ideal
gas law comes from, but not in this course. So anyway, this
is a simple thing, and the wave functions
are simple as well, and you can do all
these fantastic things. So there are many problems
like a polyatomic molecule. In a polyatomic molecule,
if you have n atoms, you have 3n minus 6
vibrational modes. You might ask, what
is a vibrational mode? Well, are they're
independent motions of the atoms that satisfy
the harmonic oscillator Hamiltonian, which
we'll come to next time. And so we have 3n minus
6 exactly solved problems all cohabiting in
one Hamiltonian. And then we can say, oh yeah,
we got these oscillators, and if I stretch
a particular bond, it might affect the force
constant for the bending. So we can introduce couplings
between the oscillators, and in fact, that's what we
do with perturbation theory. That's the whole purpose. And with that, we can
describe both the spectrum and how the spectrum encodes
the couplings between the modes. And also, we can describe
what's called intramolecular vibrational
redistribution, which happens when you have
a very high density of vibrational state. Energy moves around, because
all the modes are coupled, and so even if
you've plucked one, the excitation
would go to others. And we can understand that
all using the same formalism that we're about to develop. All right. I'm not using my
notes this time, because I think there's
just so much insight, so I have to keep checking
to see what I've skipped. All right. So what I've been saying is
whenever the Hamiltonian can be expressed as a sum of
individual Hamiltonians, whenever we can write
the Hamiltonian this way, we can write the wave
function as a product of wave functions for
coordinates, xi i1, 2 N. And the energies
will be the sum Ein, i equals little ei n sub i. I equals 1 to N. So this is really easy. If we have simply a
Hamiltonian, which is a sum of individual
particle Hamiltonians, we don't even have
to stop to think. We know the wave functions
and the eigenvalues. OK. Now, suppose the Hamiltonian
is this plus that. So here, we have a Hamiltonian,
and this is this simply the uncoupled Hamiltonian. This is what we'd like nature
to be, but nature isn't so kind, and there are some
coupling terms. And so we know the
eigenfunctions and eigenvalues for this Hamiltonian. We call them the basis functions
and the zero-order energies, and then there is
this thing that couples them and leads
to complications, and that's perturbation theory. We're going to do that. OK. So now, let me just say,
on page nine of your notes, there's the words next
time, and those are going to be replaced by you should. There's a whole bunch of things
that I want you to consider, and I was planning on
talking about them, but they're all pretty trivial. And so there are a
whole bunch of things you should study, because I will
ask you questions about them. And of greatest
importance is the ability to calculate things like that. OK. Now, I'm going to give
you a whole bunch of facts which I may not have derived. But you're going
to live with them, and you can ask me questions. Some of these things are
theorems that we can prove, but the proof of the
theorem is really boring. Understanding what it
is is really wonderful. So all eigenfunctions
that belong to different eigenvalues-- of whatever operator we
want, the Hamiltonian, some other operator-- are orthogonal. That's a fantastic
simplification. So if you have
two eigenfunctions of the Hamiltonian, of the
position operator, anything, those eigenfunctions
are orthogonal. Their integral is 0,
very, very useful. Then, one of the
initial postulates about quantum
mechanics is this idea that the wave functions
are well-behaved. Well, if I were to state
it at the beginning, you wouldn't know what's
well-behaved and ill-behaved, but now I can tell you. One of the things is that the
wave function is continuous, no matter what the
potential does. The derivative is continuous,
except at an infinite barrier. So you come along, and you
hit an infinite barrier, and you've already seen that
with the particle in a box. The wave function is continuous
at the edge of the box, but the derivative
is discontinuous, and it's because it's
an infinite wall. That's a pretty violent thing
to make the first derivative be discontinuous. The secondary derivative
is continuous, except at any sudden
change in the potential. So when you're
solving 1D problems, and you've got a solution that
works in the various regions, and you want to
connect them together, these provide some rules
about the boundary conditions. So now, most real systems
don't have infinite walls or infinitely sharp steps. So for calculation of
physically reasonable things, wave function for the first
derivative, second derivative, they are continuous. But for solving a problem,
we like these steps, because then we know how to
impose boundary conditions, and that gets us a much
easier thing to calculate. OK. Now-- oh, that's
where it went, OK-- semi classical
quantum mechanics. We know that the
energy classically is p squared over 2m. Right? It's 1/2 mv squared, but
that's the same thing as p squared over 2m. In quantum mechanics, when
we talk about Hamiltonians, the variables are x
and p not x and v. So that seems like
a picky thing, but it turns out to
be very important. And so we can say,
well, the momentum can be a function
of x, classically. So I just solved this problem,
and if the potential is not constant, then the momentum,
classical momentum, is not constant. But we know what
it is everywhere, and we also know that
the wavelength is equal to h over p. So we could make a step
into the unknown saying, well, the wavelength for
a non-constant potential is a function of
coordinate, and it's going to be equal
to h over p of x. That's semi-classical
quantum mechanics, everything you would possibly want. Now, for one-dimensional
problems, you can solve in terms of this
coordinate dependent wavelength which is related to
the local momentum. And so it doesn't matter how
complicated the problem is, you know that you can calculate
the spatial modulation frequency, you could
calculate the amplitude, is it big or small,
based on these ideas of the classical
momentum function. So this demonstration
of my walking across the room slow, fast, slow
tells you about probability. So if you use this formula,
you know the node spacings, and you know the amplitudes. Now, what you don't know
is where are the nodes? You know how far
they are apart, but I have to be humble about this. In order to calculate
where the nodes are, I have to do a little bit more
in order to pin them down. But mostly, when you're trying
to understand how something works, you want to know the
amplitude of the envelope, and that's a
probability, and so it's related to 1 over the
square root of the momentum. I'm sorry, the amplitude
is 1 over the square root of the momentum, and
the nodes spacings, those are the things
you want to do, and you want to know
them immediately. And a couple other facts that I
told you earlier, but I think I want to emphasize them-- the energy order,
number of nodes, number of internal nodes. For 1D problems, you
never skip a number of-- so you can't say, there is no
wave function with 13 nodes, even if you don't
like being unlucky. And it's there, and so if you
want the 13th energy level, you want something
with 12 nodes, and that also focuses things. So these are amazingly
wonderful things, because you can
get them from what you know about
classical mechanics, and it's easy to embed them
into a kind of half quantum mechanics. And since, I told
you, this course is for use and insight, not
admiration of philosophy or historical development, and
this is what you want to do. You look at the
problem and sketch how is the wave
function going to behave and perhaps how a
particular thing at some place in
the potential is going to affect the energy
levels or any other observable property. And so the cartoons are really
your guide to getting things right, but you
really have to invest in developing the sense of
how to build these cartoons. OK. I'm finished early again. Does anybody have any questions? OK. We start the harmonic
oscillator next time. OK, so I can say a
couple of things. The wave functions,
the solution to the 1D box and the free particle,
they're really simple. The solution to the
harmonic oscillator involves a complicated
differential equation which the mathematicians
have solved and worked out all the properties. But there is a really
important simplification that enables us to proceed
with even greater velocity in the harmonic
oscillator than we would in a particle in a box. And they are these
things called a and a dagger, creation and
annihilation operators, where when we operate on psi with
this creation operator, it converts it to square root of
v plus 1, psi v plus 1 further. These things, we don't ever
have to do an integral. Once you're in harmonic
oscillator land, everything you need comes from
these wonderful operators. And so even though the
differential equation is a little bit
scary for chemists, these things make
everything trivial. And so we use the
harmonic oscillator, and the particle in
a box to illustrate time-dependent
quantum mechanics. They each have their own special
advantages for simplifications, but it's wonderful, because
we can use something we barely understand for the first time. And actually reach
that level of, yeah, I can understand
macroscopic behaviors too and how they relate to
quantum mechanical behavior of simple systems. OK, so that's where we're going. We're going to have
two or three lectures on harmonic oscillator.
