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BARTON ZWIEBACH: De Broglie,
as we discussed last time, we

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spoke about waves.

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Matter waves.

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Because people
thought, anyway light

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is waves so the surprising
thing that would

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be that matters are waves.

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So a free particle with momentum
p can be associated to a wave--

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to a plane wave, in fact--

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plane wave-- with
wavelength lambda

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equals Planck's constant over p.

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So this wave is what eventually
becomes a famous wave function.

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So de Broglie was
writing the example

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or trying to write the example
of what eventually would become

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wave functions, and the
equations for this wave

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would become the
Schrodinger equation.

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So really, this is a pillar
of quantum mechanics.

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You're getting there when
you talk about this wave.

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So Schrodinger's equation is a
wave equation for these matter

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waves and this plane
wave eventually

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will become the wave
function and there is

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a Schrodinger equation for it.

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So it's a wave of what?

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he was asking-- de Broglie had
little idea what that wave was.

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When you have waves, like
electromagnetic waves,

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you have polarization, you
have directional properties,

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the electric field
points in some direction,

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the wave is polarized.

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Is there a same property
for the wave function?

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The answer is yes.

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We'll have to wait a
little in 8.04 to see it,

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but it has to do with spin.

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When the particles
have spin, there

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are directional
properties of the wave

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and typically, you use
several wave functions

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that correspond to directional
components of this wave.

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So photons are spin
1 particle electrons

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or spin 1/2 particles,
so there will be

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directional properties to it.

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But to begin with,
let's consider cases

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where this directional
properties don't matter

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so much, and for the
case of electrons,

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if the electrons
have small velocities

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or they are inside
small magnetic fields

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where some of these properties
of the spin is important,

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we can ignore that and work
with a wave function that

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will be a complex number.

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So it will be a wave function--
we'll denote it by the letter

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psi, capitol psi--

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that depends on
position and time,

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and that's the wave function.

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And to begin with, simplicity
will be one of them,

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and it's a complex number.

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And it's just one wave function.

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And the obvious questions
about this wave function are,

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is it measurable and
what it's meaning is?

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So is it measurable?

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And what is its meaning?

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But to understand some
of that-- in fact,

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to get to realize that these
waves are no ordinary waves,

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we're going to think
a little about what

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it means to have a wave
whose wavelength is inversely

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proportional to the
momentum of a particle.

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That's certainly a
strange statement

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and probably these are
strange waves as you will see.

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And by understanding that
these are strange waves,

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we are ready to admit later on
that the interpretation could

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be somewhat surprising as well.

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And the nature of this number
is, again, a little strange

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as you will see.

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So all of that will come by just
looking a little more in detail

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at this formula of
de Broglie and asking

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a very simple question--

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you have this particle
moving with some momentum

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and I say, OK, it has
this much wavelength.

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How about the person?

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If one of you is moving
relative to me, like you usually

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do with Einstein,
these observers

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that are boosted, but let's just
do non-relativistic physics,

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what is called the Galilean
transformation, in which there

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will be another observer
moving with constant velocity

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with respect to you and
you and that other observer

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compare the results on the
momentum and the wavelength

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and see if you find a reasonable
agreement or things make sense.

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So we're going to try to
think of p is h over lambda.

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And 2 pi's are very
useful sometimes.

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So you put an h over
2 pi here and a 2 pi

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over lambda and you rewrite
this in terms of quantities

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that are a little more common--

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one is h-bar and the other
is called the wave number k.

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So these are these two
constants and this one

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is called the wave number.

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The 2 pi's are all
over the place.

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If you have a wave
with some frequency nu,

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there's also a frequency
omega, which is 2 pi nu.

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So we're going to
look at this wave,

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and it has some momentum
and some wave number,

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therefore it has
some wavelength,

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and let's see-- if
we compare things

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between two different
frames, what do we find?

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So we'll put the
frame S and a frame S

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prime moving with some velocity
plus v in the x direction.

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So the setup is
relatively common.

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We'll have one frame
here that's the S frame,

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and it's the x-axis
of the S frame.

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And the S prime frame coincided
with the S frame at time

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equals 0-- now it's moving,
so it's now over here,

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it's S prime.

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It has moved a distance of vt--

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it's moving with
velocity v and there's t.

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And S prime has--

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and x is x prime.

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On this, we're going
to write a few things.

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We're going to say we
have a particle of mass m.

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It has velocity v
underbar, otherwise

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I'm going to get all
my velocities confused.

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So this velocity v is the
velocity of the frame,

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v underbar is the
velocity of the particle,

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and v underbar prime,
because the velocity depends

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on the frame of reference.

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Similarly, it will
have a momentum--

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and all the things we're
doing are nonrelativistic,

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so momentum p or p prime.

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Here is the particle.

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And that's the position x
prime with the particle.

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And that's a position
x of the particle.

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So that's our system.

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This particle is moving with
some velocity over here,

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and we're going to compare
these observations.

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So it's simple to
write equations

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to relate the coordinates.

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So x prime, for
example, is the value

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of the corner at x of the
particle minus the separation.

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So x minus vt.

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And I should say it
here, we're assuming

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that t prime is
equal to t, which

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is good nonrelativistically.

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It's fairly accurate.

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But that's the exact
Galilean answer--

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when you talk about
Galilean transformations

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and Galilean physics,
it's very useful.

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Even in condensed
matter physics,

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people write these days lots of
papers about Galilean physics,

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so when you have particles
moving with low velocities,

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it's accurate enough, so
might as well consider it.

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And these are the two ways
you transform coordinates,

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coordinates and time.

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So from this, we can take a
time derivative talking about

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the particle-- so we have
dx prime and dt prime or t,

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it's your choice--

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I guess I should
put dt prime here,

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dx/dt minus v, which means that
the velocity v prime underbar

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is equal to v underbar
minus little v.

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And that's what you expect.

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The difference of
velocities is given

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by the subtraction
of the velocity

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that the frame is moving.

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So if this particular has some
high velocity with respect

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to the lab frame with
respect to this frame,

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it will have a smaller velocity.

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So this sine seems right.

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And therefore,
multiplying by m, you

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get that p prime is
equal to p minus mv.

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So if you have
that, we would have

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that lambda prime, the de
Broglie wavelength measured

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by either running person,
is equal to h over

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p prime is equal to
h over p minus mv,

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and it's quite different, quite
substantially different from h

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over p, which is equal to the
de Broglie wavelength seen

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in the lab.

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So these two de
Broglie wavelengths

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will differ very substantially.

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If this would be a
familiar type of wave--

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like a sound wave
that propagates

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in the medium, any kind of wave
that propagates in a medium,

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like a water wave or
any wave of that type--

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this would simply not happen.

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In the case of those waves,
you get a Doppler shift--

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omega is changed-- but
the wavelength really

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doesn't change.

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The wavelength is
almost like something

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you look at when
you take a picture

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and whether you take
a picture of the wave

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as you run or you take
a picture of the wave

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as you are sitting still, you'll
measure the same wavelength.

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Let me convince you of that.

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It's an opportunity to
just do a little more

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formal transformations,
because these

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are going to be Galilean
transformations,

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simple transformations.

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So our first observation is
that the de Broglie wavelength

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don't agree, which
pretty much, I think,

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intuitively is saying that
if you could just sort of see

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those waves and measure
the distance between peaks,

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they should agree, but they
don't, so there's something

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very strange happening here.