BARTON ZWIEBACH: De Broglie,

as we discussed last time, we spoke about waves. Matter waves. Because people

thought, anyway light is waves so the surprising

thing that would be that matters are waves. So a free particle with momentum

p can be associated to a wave– to a plane wave, in fact– plane wave– with

wavelength lambda equals Planck’s constant over p. So this wave is what eventually

becomes a famous wave function. So de Broglie was

writing the example or trying to write the example

of what eventually would become wave functions, and the

equations for this wave would become the

Schrodinger equation. So really, this is a pillar

of quantum mechanics. You’re getting there when

you talk about this wave. So Schrodinger’s equation is a

wave equation for these matter waves and this plane

wave eventually will become the wave

function and there is a Schrodinger equation for it. So it’s a wave of what? he was asking– de Broglie had

little idea what that wave was. When you have waves, like

electromagnetic waves, you have polarization, you

have directional properties, the electric field

points in some direction, the wave is polarized. Is there a same property

for the wave function? The answer is yes. We’ll have to wait a

little in 8.04 to see it, but it has to do with spin. When the particles

have spin, there are directional

properties of the wave and typically, you use

several wave functions that correspond to directional

components of this wave. So photons are spin

1 particle electrons or spin 1/2 particles,

so there will be directional properties to it. But to begin with,

let’s consider cases where this directional

properties don’t matter so much, and for the

case of electrons, if the electrons

have small velocities or they are inside

small magnetic fields where some of these properties

of the spin is important, we can ignore that and work

with a wave function that will be a complex number. So it will be a wave function–

we’ll denote it by the letter psi, capitol psi– that depends on

position and time, and that’s the wave function. And to begin with, simplicity

will be one of them, and it’s a complex number. And it’s just one wave function. And the obvious questions

about this wave function are, is it measurable and

what it’s meaning is? So is it measurable? And what is its meaning? But to understand some

of that– in fact, to get to realize that these

waves are no ordinary waves, we’re going to think

a little about what it means to have a wave

whose wavelength is inversely proportional to the

momentum of a particle. That’s certainly a

strange statement and probably these are

strange waves as you will see. And by understanding that

these are strange waves, we are ready to admit later on

that the interpretation could be somewhat surprising as well. And the nature of this number

is, again, a little strange as you will see. So all of that will come by just

looking a little more in detail at this formula of

de Broglie and asking a very simple question– you have this particle

moving with some momentum and I say, OK, it has

this much wavelength. How about the person? If one of you is moving

relative to me, like you usually do with Einstein,

these observers that are boosted, but let’s just

do non-relativistic physics, what is called the Galilean

transformation, in which there will be another observer

moving with constant velocity with respect to you and

you and that other observer compare the results on the

momentum and the wavelength and see if you find a reasonable

agreement or things make sense. So we’re going to try to

think of p is h over lambda. And 2 pi’s are very

useful sometimes. So you put an h over

2 pi here and a 2 pi over lambda and you rewrite

this in terms of quantities that are a little more common– one is h-bar and the other

is called the wave number k. So these are these two

constants and this one is called the wave number. The 2 pi’s are all

over the place. If you have a wave

with some frequency nu, there’s also a frequency

omega, which is 2 pi nu. So we’re going to

look at this wave, and it has some momentum

and some wave number, therefore it has

some wavelength, and let’s see– if

we compare things between two different

frames, what do we find? So we’ll put the

frame S and a frame S prime moving with some velocity

plus v in the x direction. So the setup is

relatively common. We’ll have one frame

here that’s the S frame, and it’s the x-axis

of the S frame. And the S prime frame coincided

with the S frame at time equals 0– now it’s moving,

so it’s now over here, it’s S prime. It has moved a distance of vt– it’s moving with

velocity v and there’s t. And S prime has– and x is x prime. On this, we’re going

to write a few things. We’re going to say we

have a particle of mass m. It has velocity v

underbar, otherwise I’m going to get all

my velocities confused. So this velocity v is the

velocity of the frame, v underbar is the

velocity of the particle, and v underbar prime,

because the velocity depends on the frame of reference. Similarly, it will

have a momentum– and all the things we’re

doing are nonrelativistic, so momentum p or p prime. Here is the particle. And that’s the position x

prime with the particle. And that’s a position

x of the particle. So that’s our system. This particle is moving with

some velocity over here, and we’re going to compare

these observations. So it’s simple to

write equations to relate the coordinates. So x prime, for

example, is the value of the corner at x of the

particle minus the separation. So x minus vt. And I should say it

here, we’re assuming that t prime is

equal to t, which is good nonrelativistically. It’s fairly accurate. But that’s the exact

Galilean answer– when you talk about

Galilean transformations and Galilean physics,

it’s very useful. Even in condensed

matter physics, people write these days lots of

papers about Galilean physics, so when you have particles

moving with low velocities, it’s accurate enough, so

might as well consider it. And these are the two ways

you transform coordinates, coordinates and time. So from this, we can take a

time derivative talking about the particle– so we have

dx prime and dt prime or t, it’s your choice– I guess I should

put dt prime here, dx/dt minus v, which means that

the velocity v prime underbar is equal to v underbar

minus little v. And that’s what you expect. The difference of

velocities is given by the subtraction

of the velocity that the frame is moving. So if this particular has some

high velocity with respect to the lab frame with

respect to this frame, it will have a smaller velocity. So this sine seems right. And therefore,

multiplying by m, you get that p prime is

equal to p minus mv. So if you have

that, we would have that lambda prime, the de

Broglie wavelength measured by either running person,

is equal to h over p prime is equal to

h over p minus mv, and it’s quite different, quite

substantially different from h over p, which is equal to the

de Broglie wavelength seen in the lab. So these two de

Broglie wavelengths will differ very substantially. If this would be a

familiar type of wave– like a sound wave

that propagates in the medium, any kind of wave

that propagates in a medium, like a water wave or

any wave of that type– this would simply not happen. In the case of those waves,

you get a Doppler shift– omega is changed– but

the wavelength really doesn’t change. The wavelength is

almost like something you look at when

you take a picture and whether you take

a picture of the wave as you run or you take

a picture of the wave as you are sitting still, you’ll

measure the same wavelength. Let me convince you of that. It’s an opportunity to

just do a little more formal transformations,

because these are going to be Galilean

transformations, simple transformations. So our first observation is

that the de Broglie wavelength don’t agree, which

pretty much, I think, intuitively is saying that

if you could just sort of see those waves and measure

the distance between peaks, they should agree, but they

don’t, so there’s something very strange happening here.

This guy is awesome he goes into the history behind the subject matter and has great in-depth explanations. My little pet peeve is that De Broglie is pronounced "De Broy" like boy except with an r in there.

can we use E=hf for an electron moving with velocity v ? If yes ,why ? If no,why too ?

Thanks for the help :))

brillant ! the best lecture I have seen in my life !!

I am keeping watching. Great

Professor seems as astonished as deBroglie's advisor who almost threw him out of the PhD program when deBroglie proposed matter was a wave for his one-page dissertation. When the advisor couldn't disprove deBroglie, he asked Einstein who told him to give deBroglie his PhD. deBroglie's dissertation was expanded by committee to six-pages and to this very day, physics fails to comprehend deBroglie's brilliance. Because you see, there are no particles, there are only waves. And just as a photon is a sinusoidal wave of finite temporal dimension, so too are what we incorrectly refer to as "particles."