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PROFESSOR: Today, we begin
with our study of scattering.

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And so scattering
is the next chapter

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in our study of approximations.

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What is the purpose
of scattering?

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Physicists want to study the
interactions between particles,

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the forces.

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We know, for example,
in a hydrogen atom

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there is an electromagnetic
force within the proton

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and the electron.

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We understand it rather well.

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But there are other
forces in nature

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that we don't understand very
well, are harder to understand.

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Those at higher energies, the
interactions between particles

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can be rather complicated.

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Also, it may happen that you
have a situation where there's

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a potential affecting particles,
maybe a trap or something that

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holds particles.

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And there's a potential.

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And you really cannot measure
directly this potential.

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There's no simple
way of doing so.

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And then what you do is
you send particles in.

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And you see how
they get affected.

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And by studying the
reflection or the scattering

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of the particles off
of this potential,

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you'll learn a lot about
what this potential can be.

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So this subject
today is scattering.

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And in general, you have a
beam of particles, a target,

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and a number of detectors.

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That's what happens
in a big accelerator.

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And in general, you
have such situations.

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OK?

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So a beam, some sort of target,
and particles get scattered.

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And you have detectors.

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And in general, the detectors
are all over in all directions

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because particles
may back scatter,

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may go in different ways.

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So collisions, in general,
are rather intricate.

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Particles can change
identity, can do

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all kinds of different things.

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For example, you can have a
proton colliding with a proton.

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This could be the case when
the beam is a beam of protons.

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And here, maybe there's
a target in which

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you study what happens when
a proton encounters a proton.

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Or it may be that in
some cases there's

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another beam that is
coming also with protons.

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And they collide.

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But when protons
collide with protons,

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funny things can happen.

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For example, you can
get the two protons

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plus a pion, a neutral pion,
0 denoting zero charge.

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That's a hadron, a strongly
interacting particle,

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like the proton is.

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It's made of quarks.

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And this can happen.

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There's no such thing
as a conservation

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of the number of particles.

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That is one of the reasons you
need quantum field theory to do

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this kind of computation.

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And this reaction is
allowed because it,

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at least the first
thing you can check

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is that it conserves charges.

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Proton and proton go
to proton and proton,

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so a charge is conserved.

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And pi 0 is neutral.

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So that, for
example, can happen.

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Or proton plus proton can
go to proton plus neutron.

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Now, charge is not conserved.

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But this time, a positively
charged pion called pi plus.

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The particles can
completely change identity.

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You can have an electron
and a positron colliding.

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So that's electron.

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This is the positron, the
anti-particle of the electron,

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equal mass, oppositely charged.

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And then they can go into
a mu plus plus a mu minus.

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These are charged laptons.

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Again, just like the
electrical laptons, as opposed

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to these particles here.

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They're called hydrons.

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These particles are, as far
as we know, are elementary.

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These particles are
made out of quarks.

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And in this case, you have a
complete change of identity.

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The original particles
have disappeared.

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And new particles
have been created.

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And you can look forward
to study these processes

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as you continue your studies
of quantum mechanics.

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But these are like reactions.

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Now, in our name, in our
nomenclature for scattering,

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we will call the process
a scattering process

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where the identity of the
particles is unchanged.

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So we will have
scattering, strictly

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speaking, when there is
no change of identity

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in the initial and final states.

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No change of identity of
the particles in the process

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of going from initial
to final state.

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So a scattering process looks
like a plus b goes to a plus b.

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We don't change identity.

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Those processes where you change
identity are harder to discuss.

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And we won't discuss them here.

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Even here, there is still a
lot of things that can happen.

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It's very intricate.

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So we will demand even more.

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We will call or look
for elastic scattering.

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And that means that if these
particles or objects have

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internal states, those internal
states are not changed.

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So the internal states of
the particles do not change.

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Of course, that can
happen essentially

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when the particles are complex.

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They're bound states
of other particles.

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So you may have a bound
state of one form.

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And then when you
have bound states,

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you know you have all kinds of
energy levels internal states.

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And then when we say we
have an elastic process,

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it means that those internal
states are not changed.

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A classic example of
an inelastic scattering

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experiment, the most
famous historically

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of those experiments, is an
experiment by Frank and Hertz

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in 1914.

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They were having a
chamber with mercury gas.

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And they shot electrons
from one side to the other.

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And the electrons and the
mercury atoms collided.

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And they found that the
electrons were slowed down

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by some quantized
amount of energy.

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And that was the first evidence
that atoms had energy levels.

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Bohr's theory of atoms
had been proposed one year

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before that, 1913.

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And then, the process in which
these electrons hit the atoms

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and lost energy corresponding
to producing transitions

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in the atoms.

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And that would be an
inelastic scattering

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because the atom changed
its internal state.

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This is a collision between
an electron and an atom,

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but the atom has
changed internal states.

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So we will want
to consider cases

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when we don't change
the internal state.

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And that will be when we
have elastic scattering.

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So a few more things that
we're going to assume as we

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do elastic scattering.

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We will work without spin.

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All we will do in the
next lectures, one or two

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more lectures in this, will
be particles without spin.

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As we're doing in
our course, we also

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work in the nonrelativistic
approximation.

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So these are the first
things that we will assume.

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One, no spin.

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This doesn't complicate
matters as far

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as the scattering is concerned.

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It just complicates the
algebra and the calculations

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you have to do because you
have more degrees of freedom.

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We will be working
nonrelativistically.

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Moreover, we will assume
there are interactions.

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There are interactions
between the particles

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that produce the scattering.

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And those interactions
are simple enough

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that they just depend on
the difference of position.

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So the interaction
potentials are

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of the form v of R1 minus R2.

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So processes in which will
have two incoming particles

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and they interact, and
they scatter elastically.

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And the potential depends
on just the difference

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of the two positions.

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Eventually, we will
even assume it's only

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on the magnitude of
these differences

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for particular cases.

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But if you have a
potential like this,

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like when we were
studying with hydrogen

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atom for the first
time, a potential

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that just depends on the
differences of positions

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can be treated in the
center of mass frame.

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It's a nice frame where
you can work on it.

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And then you can think of
it as a single particle

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scattering of a potential.

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So you translate the
Schrodinger equation

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into a center of mass
degrees of freedom

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that are generally simple.

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And we don't worry about and
a relative degree of freedom.

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So this can be done here.

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So this makes a process
equivalent to scattering

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of a particle of a
potential V of r.

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And that particle
has the reduced mass,

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just like happened
with a hydrogen atom.

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The real mass of the
equation we solve

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for showing the Hamiltonian
for a hydrogen atom,

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the mass that shows up
there is the reduced mass,

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which is approximately
equal to the electron mass.

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But in general, in this
scattering process,

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it may be between two identical
particles in which case

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a reduced mass would be half
the mass of the particle.

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So again, we will
work scattering

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with energy eigenstates.

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You may have studied
already in 804

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a little bit of problems
of scattering off

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of rectangular barriers,
tunneling, all these things.

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And we work with energy
eigenstates in those cases.

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And we will work with energy
eigenstates as well here.

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Now, in some versions of 804,
you discuss a lot wave packets.

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I used to discuss
a lot wave packets.

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We would build wave
packets and send them in.

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And use a transmission on
reflection coefficients

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to figure out what
the wave packets do.

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And in general, a wave packet
is a somewhat more physical way

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of thinking of the processes.

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Because the energy states,
eigenstates, are unnormalizable

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and don't have a
direct interpretation

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in terms of particles.

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Rigorously speaking,
you should always

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do things with wave packets.

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But the fact is that we all
work with energy eigenstates.

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And most of the times, what
happens with wave packets

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can be more or less gleaned
from what is happening

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with energy eigenstates.

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So we will not new
wave packets here.

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We will not bother to
construct wave packets.

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They would not teach us
too much at this moment.

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So we will work with
energy eigenstates.