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PROFESSOR: Very good.
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So it's time to start.
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So today, I want to talk about
general features of quantum
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mechanics.
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Quantum mechanics is something
that takes some time to learn,
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and we're going to be doing some
of that learning this semester.
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But I want to give you a
perspective of where we're
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going, what are the
basic features, how
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quantum mechanics looks,
what's surprising about it,
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and introduce some
ideas that will
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be relevant throughout
this semester and some
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that will be relevant for
later courses as well.
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So it's an overview
of quantum mechanics.
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So quantum mechanics,
at this moment,
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is almost 100 years old.
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Officially-- and we will hear--
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this year, in 2016, we're
celebrating the centenary
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of general relativity.
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And when will the centenary
of quantum mechanics be?
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I'm pretty sure it
will be in 2025.
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Because in 1925,
Schrodinger and Heisenberg
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pretty much wrote down the
equations of quantum mechanics.
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But quantum mechanics
really begins earlier.
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The routes that led to quantum
mechanics began in the late
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years of the 19th century
with work of Planck,
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and then at the
beginning of the century,
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with work of Einstein and
others,m as we will see today
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and in the next few lectures.
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So the thoughts, the
puzzles, the ideas
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that led to quantum
mechanics begin before 1925,
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and in 1925, it
suddenly happened.
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So what is quantum mechanics?
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Quantum mechanics is really
a framework to do physics,
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as we will understand.
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So quantum physics has
replaced classical physics
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as the correct description
of fundamental theory.
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So classical physics may
be a good approximation,
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but we know that at some
point, it's not quite right.
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It's not only not
perfectly accurate.
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It's conceptually very
different from the way things
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really work.
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So quantum physics has
replaced classical physics.
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And quantum physics
is the principles
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of quantum mechanics applied to
different physical phenomena.
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So you have, for example,
quantum electrodynamics,
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which is quantum mechanics
applied to electromagnetism.
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You have quantum
chromodynamics, which
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is quantum mechanics applied
to the strong interaction.
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You have quantum optics when
you apply quantum mechanics
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to photons.
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You have quantum
gravity when you
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try to apply quantum
mechanics to gravitation.
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Why the laughs?
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And that's what gives rise
to string theory, which
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is presumably a quantum
theory of gravity,
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and in fact, the quantum
theory of all interactions
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if it is correct.
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Because it not only
describes gravity,
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but it describes
all other forces.
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So quantum mechanics
is the framework,
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and we apply it to many things.
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So what are we going
to cover today?
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What are we going to review?
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Essentially five topics--
one, the linearity
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of quantum mechanics, two, the
necessity of complex numbers,
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three, the laws of
determinism, four,
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the unusual features
of superposition,
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and finally, what
is entanglement.
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So that's what we
aim to discuss today.
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So we'll begin with
number one, linearity.
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And that's a very
fundamental aspect
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of quantum mechanics,
something that we have
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to pay a lot of attention to.
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So whenever you
have a theory, you
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have some dynamical variables.
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These are the variables
you want to find
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their values because they are
connected with observation.
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If you have dynamical
variables, you
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can compare the values
of those variables,
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or some values deduced
from those variables,
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to the results of an experiment.
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So you have the equations
of motion, so linearity.
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We're talking linearity.
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You have some equations
of motion, EOM.
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And you have
dynamical variables.
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If you have a theory,
you have some equations,
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and you have to solve for
those dynamical variables.
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And the most famous example
of a theory that is linear
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is Maxwell's theory
of electromagnetism.
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Maxwell's theory
of electromagnetism
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is a linear theory.
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What does that mean?
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Well, first, practically,
what it means
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is that if you have a solution--
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for example, a plane wave
propagating in this direction--
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and you have another solution--
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a plane wave propagating
towards you--
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then you can form
a third solution,
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which is two plane waves
propagating simultaneously.
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And you don't have
to change anything.
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You can just put them together,
and you get a new solution.
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The two waves propagate
without touching each other,
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without affecting each other.
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And together, they
form a new solution.
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This is extraordinarily
useful in practice
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because the air
around us is filled
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with electromagnetic waves.
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All your cell phones send
electromagnetic waves
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up the sky to satellites
and radio stations
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and transmitting stations, and
the millions of phone calls
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go simultaneously without
affecting each other.
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A transatlantic cable can
conduct millions of phone calls
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at the same time, and as much
data and video and internet.
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It's all superposition.
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All these millions
of conversations
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go simultaneously
through the cable
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without interfering
with each other.
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Mathematically, we have
the following situation.
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In Maxwell's theory, you
have an electric field,
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a magnetic field, a charge
density, and a current density.
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That's charge per unit
area per unit of time.
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That's the current density.
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And this set of data
correspond to a solution
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if they satisfy
Maxwell's equations,
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which is a set of equations
for the electromagnetic field,
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charged densities,
and current density.
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So suppose this is a solution,
that you verify that it
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solves Maxwell's equation.
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Then linearity
implies the following.
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You multiply this by alpha,
alpha e, alpha b, alpha rho,
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and alpha j.
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And think of this as
the new electric field,
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the new magnetic field,
the new charged density,
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and the new current
is also a solution.
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If this is a solution,
linearity implies
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that you can
multiply those values
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by a number, a constant number,
a alpha being a real number.
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And this is still a solution.
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It also implies more.
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Linearity means
another thing as well.
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It means that if you have two
solutions, e1, b1, rho 1, j1,
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and e2, b2, rho 2, j2--
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if these are two
solutions, then linearity
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implies that the sum e1 plus e2,
b1 plus b2, rho 1 plus rho 2,
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and j1 plus j2 is
also a solution.
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So that's the meaning, the
technical meaning of linearity.
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We have two solutions.
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We can add them.
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We have a single solution.
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You can scale it by a number.
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Now, I have not shown
you the equations
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and what makes them linear.
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But I can explain this
a little more as to
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what does it mean to
have a linear equation.
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Precisely what do we mean
by a linear equation?
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So a linear equation.
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And we write it schematically.
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We try to avoid details.
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We try to get
across the concept.
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A linear equation, we write
this l u equal 0 where
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u is your unknown and l is what
is called the linear operator,
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something that acts on u.
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And that thing, the equation,
is of the form l and u equal 0.
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Now, you might say,
OK, that already
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looks to me a little strange,
because you have just one
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unknown, and here we
have several unknowns.
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So this is not very general.
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And you could have
several equations.
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Well, that won't change much.
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We can have several
linear operators
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if you have several equations,
like l1 or something,
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l2 on something, all
these ones equal to 0
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as you have several equations.
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So you can have several
u's or several unknowns,
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and you could say something
like you have l on u, v,
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w equals 0 where you
have several unknowns.
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But it's easier to just
think of this first.
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And once you understand this,
you can think about the case
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where you have many equations.
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So what is a linear equation?
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It's something in which this l--
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the unknown can
be anything, but l
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must have important properties,
as being a linear operator
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will mean that l on a times
u, where a is a number,
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should be equal to alu and l
on u1 plus u2 on two unknowns
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is equal to lu 1 lu 2.
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This is what we mean by
the operator being linear.
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So if an operator
is linear, you also
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have l on alpha u1 plus beta u2.
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You apply first the second
property, l on the first plus
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l on the second.
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So this is l of alpha
u1 plus l of beta u2.
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And then using the
first property,
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this is alpha l of
u1 plus beta l of u2.
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And then you realize that
if u1 and u2 are solutions--
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which means lu 1
equal lu 2 equals 0
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if they solve the equation--
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then alpha u1 plus
beta u2 is a solution.
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Because if lu1 is 0 and lu2 is
0, l of alpha u1 plus beta u2
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is 0, and it is a solution.
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So this is how we write
a linear equation.
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Now, an example
probably would help.
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If I have the
differential equation
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du dt plus 1 over
tau u equals 0,
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I can write it as an
equation of the form lu
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equals 0 by taking
l on u to be defined
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to be vu vt plus 1 over tau u.
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Now, it's pretty much--
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I haven't done much here.
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I've just said, look, let's
define l [? active ?] [? on ?]
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u to be this.
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And then certainly, this
equation is just lu equals 0.
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The question would be maybe
if somebody would tell you
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how do you write l alone--
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well, l alone, probably
we should write it
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as d dt without anything
here plus 1 over tau.
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Now, that's a way
you would write
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it to try to understand
yourself what's going on.
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And you say, well, then when
l acts as the variable u,
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the first term takes
the derivative,
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and the second term, which is
a number, just multiplies it.
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So you could write
l as this thing.
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And now it is
straightforward to check
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that this is a linear operator.
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l is linear.
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And for that, you have to
check the two properties there.
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So for example, l on
au would be ddt of au
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plus 1 over tau au,
which is a times du d tau
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plus 1 over tau u, which is alu.
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And you can check.
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I asked you to check
the other property
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l on u1 plus u2 is
equal to lu 1 plus lu 2.
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Please do it.