WEBVTT
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PROFESSOR: So here is
of something funny.
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You might say, OK,
what is simpler?
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A theory that is linear or
a theory that is not linear?
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And the answer, of course, a
linear theory is much simpler.
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General-- Maxwell's
equations are linear.
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Einstein's theory of
relativity is very nonlinear,
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very complicated.
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How about classical mechanics?
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Is classical mechanics
linear or nonlinear?
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What do we think?
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Can't hear anyone.
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Linear, OK.
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You may think it's linear
because it's supposed
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to be simple, but it's not.
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It's actually is very nonlinear.
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Newton could solve
the two body problem
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but he couldn't solve
the three body problem.
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Already with three bodies, you
cannot superpose solutions that
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you get with two bodies.
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It's extraordinarily
complicated,
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classical mechanics.
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Let me show you.
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If you have motion in
one dimension, in 1D,
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you have the equation of
motion, motion in one dimension,
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and there are potential V of
x, that this time independent--
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a particle moving
in one dimension x
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with under the influence
of a potential, V of x.
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The second-- the dynamical
variable is x of t.
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The dynamical variable.
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And the equation of motion is--
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so let me explain this.
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This is force equal
mass times acceleration.
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This is mass, this
is acceleration,
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the second derivative
of the position,
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and V force is minus the
derivative of the potential
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evaluated at the position.
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You know, derivatives
of potentials--
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if you think of a potential,
the derivative of the potential
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is here positive, and you
know if you have a mass
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here, it tends to
go to the left,
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so the force is on the
left, so it's minus.
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So V prime is the derivative of
V with respect to its argument.
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And the problem is that while
this, taking derivatives,
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is a linear operation.
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If you take two derivatives
of a sum of things,
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you take two
derivatives of the first
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plus two derivatives
of the second.
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But yes, its-- this
side is linear,
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but this side may not be linear.
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Because a potential
can be arbitrary.
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And that the
reverse-- so suppose
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the potential is cubic in x.
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V of x goes like x cubed.
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Then the derivative of
V goes like x squared,
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and x squared is not
a linear function.
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So this, Newton's equation,
is not a linear equation.
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And therefore, it's
complicated to solve.
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Very complicated to solve.
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So finally, we can get to
our case, quantum mechanics.
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So in quantum mechanics,
what do we have?
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Quantum mechanics is linear.
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First, you need an equation,
and whose equation is it?
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Schrodinger's equation, 1925.
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He writes an equation for
the dynamical variable,
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and the dynamical
variable is something
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called the wave function.
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This wave function
can depend on t--
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depends on time-- and it may
depend on other things as well.
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And he describes the dynamics
of the quantum system
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as it evolved in time.
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There is the wave function,
and you have an equation
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for this wave function.
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And what is the equation
for this wave function?
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It's a universal
equation-- i hbar
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partial derivative with
respect to time of psi
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is equal to H hat of psi, where
H hat is called the Hamiltonian
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and it's a linear operator.
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That's why I had to
explain a little bit what
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the linear operator is.
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This is the general structure
of the Schrodinger equation--
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time derivative and
the linear operator.
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So if you wish to write the
Schrodinger equation as an L
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psi equals 0, then L psi would
be defined i hbar del/del
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t of psi minus H hat psi.
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Then this is the
Schrodinger equation.
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This equation here is
Schrodinger's equation.
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And as you can see,
it's a linear equation.
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You can check it, check
that L is a linear operator.
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Therefore, it is naturally
linear, you can see,
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because you do it differently,
because the derivative
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with respect to time
is a linear operation.
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If you have the ddt of a
number of times a function,
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the number goes out, you
differentiate the function.
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ddt of the sum of two functions,
you differentiate the first,
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you differentiate.
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So this is linear and
H we said is linear,
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so L is going to be linear
and the Schrodinger equation
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is going to be a
linear equation,
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and therefore, you're going
to have the great advantage
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that any time you find
solutions, you can scale them,
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you can add them, you can put
them together, combine them
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in superpositions, and
find new solutions.
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So in that sense,
it's remarkable
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that quantum
mechanics is simpler
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than classical mechanics.
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And in fact, you will see
throughout this semester
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how the mathematics and
the things that we do
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are simpler in
quantum mechanics,
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or more elegant, more
beautiful, more coherent,
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it's simpler and very nice.
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OK, i is the square root of
minus 1, is the imaginary unit,
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and that's what we're
going to talk next
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on the necessity
of complex numbers.
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hbar, yes, it's a number.
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It shows up in quantum
mechanics early on.
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It it's called Planck's
constant and it
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began when Planck tried to
fit the black value spectrum
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and he found the need to
put a constant in there,
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and then later, Einstein figured
out that it was very relevant,
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so yes, it's a number.
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For any physical
system that you have,
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you will have a wave
function and you
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will have a Hamiltonian,
and the Hamiltonian
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is for you to invent
or for you to discover.
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So if you have a particle
moving on a line,
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the wave function will
depend on time and on x.
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If you have a particle
moving in three dimensions,
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it will depend on x vector.
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It may depend on
other things as well
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or it maybe, like, one particle
has several wave functions
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and that happens when you
have a particle with spin.
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So in general, always
time, sometimes position,
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there may be cases where it
doesn't depend on position.
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You think of an electron at some
point in space and it's fixed--
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you lock it there and
you want understand
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the physics of that
electron locked into place,
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and then position
is not relevant.
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So what it does with
its spin is relevant
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and then you may need more
than one wave function-- what
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is one describing
the spin up and one
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describing the spin down?
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So it was funny that
Schrodinger wrote this equation
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and when asked, so what
is the wave function?
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He said, I don't know.
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No physical interpretation
for the wave function
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was obvious for the people that
invented quantum mechanics.
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It took a few months
until Max Born said
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it has to do with
probabilities, and that's
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what we're going to get next.
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So our next point is the
necessity of complex numbers
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in quantum mechanics.