Light is . . .•Initially thought to be waves

•They do things waves do, like diffraction and interference•Wavelength – frequency relationship

•Planck, Einstein, Compton showed us they behave like particles (photons)•Energy comes in chunks•Wave-particle duality: somehow, they behave like both

•Photons also carry momentum•Momentum comes in chunks

c f

E hf

p E c hf c h p h

Electrons are . . .•They act like particles

•Energy, momentum, etc., come in chunks•They also behave quantum mechanically•Is it possible they have wave properties as well?

The de Broglie Hypothesis

•Two equations that relate the particle-like and wave-like properties of light

E hf

p h 1924 – Louis de Broglie postulated that theserelationships apply to electrons as well•Implied that it applies to other particles as well•de Broglie could simply explain the Bohr quantization condition

•Compare the wavelength of an electron in hydrogen to the circumference of its path

eL n m vr pr hr

2 r

cancel 2n r C

Integer number of wavelengths fit around the orbit

Measuring wave properties of electronsp h •What energy electrons do we want?

212E mv

2

2

p

m

2 2

2 22

h c

mc

18 2

2

1.504 10 eV m

2 215 8

6 2

4.136 10 eV s 3.00 10 m/s

2 0.511 10 eV

2nm

1.504 eV

For atomic separations, want distances around 0.3 nm energies of 10 or so eVHow can we measure these wave properties?•Scatter off crystals, just like we did for X-rays!•Complication: electrons change speed inside crystal

•Work function increases kinetic energy in the crystal•Momentum increases in the crystal•Wavelength changes

The Davisson-Germer ExperimentSame experiment as scattering X-rays, except•Reflection probability from each layer greater

•Interference effects are weaker•Momentum/wavelength is shifted inside the material•Equation for good scattering identical

d

2 cosd m

e-

Quantum effects are weird•Electron must scatter off of all layers

The Results:•1928: Electrons have both wave and particle properties•1900: Photons have both wave and particle properties•1930: Atoms have both wave and particle properties•1930: Molecules have both wave and particle properties•Neutrons have both wave and particle properties•Protons have both wave and particle properties•Everything has both wave and particle properties

Dr. Carlson has a mass of 82 kg and leaves this room at a velocity of about 1.3 m/s. What is his

wavelength?h

p

h

mv

346.626 10 J s

82 kg 1.3 m/s

366.22 10 m

Waves: How come we don’t notice?•Whenever waves encounter a barrier, they get diffracted, their direction changes•If the barrier is much larger then the waves, the waves change direction very little•If the barrier is much smaller then the waves, then the effect is enormous, and the wave diffracts a lot

Light waves through a big hole Sound waves through a small hole

When wave-lengths are short, wave effects are

hard to notice

l

•Simple waves look like cosines or sines:•k is called the wave number

•Units of inverse meters is called the angular frequency

•Units of inverse seconds•Wavelength is how far you have to go in space before it repeats

•Related to wave number k•Period T is how long you have to wait in time before it repeats

•Related to angular frequency •Frequency f is how many times per second it repeats

•The reciprocal of period

•cos and sin have periodicity 2•If you increase kx by 2, wave will look the same•If you increase t by 2, wave will look the same

2 k

2 T

Simple Waves

2 f

, cos

, sin

x t A kx t

x t A kx t

Math Interlude: Partial Derivatives•Ordinary derivatives are the local “slope” of a function of one variable f(x)

0

limh

d f x f x h f x

dx h

•Partial derivatives are the local “slope” of a function of two or more variables f(x,y) in one particular direction

0

, , ,limh

f x y f x h y f x y

x h

•Partial derivatives are calculated the same way as ordinary derivatives, except other variables are treated as constant

2 2 2Ax B Ax Bd de e Ax B

dx dx

Calculate the partial derivative below:

2 2Ax Ayex

2

2 Ax BAxe

2 2 2 2Ax Aye Ax Ayx

2 2

2 Ax AyAxe

cos kx tx

sin kx t kx t

x

sink kx t

Dispersion Relations•Waves come about from the solution of differential equations

•For example, for light•These equations lead to relationships between the angular frequency and the wave number k

•Called a dispersion relation

2 22

2 20z zE E

ct x

k

What is the dispersion relationship for light in vacuum?

Need to find a solution to wave equation, let’s try: coszE A kx t

2

22

sin

cos

z

z

EkA kx t

x

Ek A kx t

x

2

22

sin

cos

z

z

EA kx t

t

EA kx t

t

2 2 2cos cos 0A kx t c k A kx t

2 2 2c k ck

pvT

•The wave moves a distance of one wavelength in one period T•From this, we can calculate the phase velocity denoted vp

•It is how fast the peaks and valleys move

2k

1

2f

T

Phase velocity

f 2

2k

k

pv fk

What is the phase velocity for light

in vacuum? pv

k

ck

k c Not constant for

most waves!

•Real waves are almost always combinations of multiple wavelengths•Average these two expressions to get a new wave:

Adding two waves

1 1 1

2 2 2

cos

cos

k x t

k x t

1 11 1 2 22 2, cos cosx t k x t k x t

•This wave has two kinds of oscillations:•The oscillations at small scales•The “lumps” at large scales

Analyzing the sum of two waves: 1 1

1 1 2 22 2, cos cosx t k x t k x t Need to derive some obscure trig identities:•Average these:•Substitute:

cos cos cos sin sin

cos cos cos sin sin

1 12 2cos cos cos cos

12

12

A B

A B

1 1 1 12 2 2 2cos cos cos cosA B A B A B

Rewrite wave function:

c s, ocos kx t k xx tt

11 22

11 22

k k k

11 22

11 22

k k k

Small scale oscillations

Large scale oscillations

The “uncertainty” of two waves

Our wave is made of two values of k:•k is the average value of these twok is the amount by which the two values of k actually vary from k

•The value of k is uncertain by an amount k

k1 k2

kk

k k

Plotted at t = 0

•Each “lump” is spread out in space also•Define x as the distance from the center of a lump to the edge•The distance is where the cosine vanishes

x

cos 0k x 12k x

1k x First hint of uncertainty principle

Group Velocity c s, ocos kx t k xx tt

The velocity of little oscillations governed by the first factor•Leads to the same formula as before for phase velocity: pv

k

Small scale oscillations

Large scale oscillations

The velocity of big oscillations governed by the second factor •Leads to a formula for group velocity:

gvk

These need not be the same!

More Waves One wave

Two waves

Three waves

Five waves

Infinity waves

•Two waves allow you to create localized “lumps”•Three waves allow you to start separating these lumps•More waves lets you get them farther and farther apart•Infinity waves allows you to make the other lumps disappear to infinity – you have one lump, or a wave packet•A single lump – a wave packet – looks and acts a lot like a particle

Wave Packets•We can combine many waves to separate a “lump” from its neighbors•With an infinite number of waves, we can make a wave packet

•Contains continuum of wave numbers k•Resulting wave travels and mostly stays together,like a particle•Note both k-values and x-values have a spread k and x.

Phase and Group velocityCompare to two wave formulas:•Phase velocity formula is exactly the same, except we simply rename the average values of k and as simply k and •Group velocity now involves very closely spaced values of k (and ), and therefore we rewrite the differences as . . .

pvk

pv

k

gvk

g

dv

dk

What is the phase and group velocity

for this wave?

Sample ProblemWhat is the phase and group velocity for this wave?

Start, t = 0 sFinish, t = 30 s

Moved 60 m

Moved 30 m 30 m1.0 m/s

30 spv

60 m2.0 m/s

30 sgv

Phase and Group velocityHow to calculate them:•You need the dispersion relation: the relationship between and k, with only constants in the formula•Example: light in vacuum has

pvk

g

dv

dk

ck g

p

ckv c

d dv ck c

dk

k k

dk

What’s wrong with the following proof?

Theorem: Group velocityalways equal phase velocity

doesn’t

pkv

g

dv

dk

p

dv k

dk pv pdv

kdk

pv

If the dispersion relation is = Ak2, with A a constant, what are the

phase and group velocity?

2

p

Akv Ak

k k

2 2g

d dv Ak Ak

dk dk

The Classical Uncertainty Principle•The wave number of a wave packet is not exactly one value

•It can be approximated by giving the central value•And the uncertainty, the “standard deviation” from that value

•The position of a wave packet is not exactly one value•It can be approximated by giving the central value•And the uncertainty, the “standard deviation” from that value

k

k k

x

x x

These quantities are related:•Typically, x k ~ 1

Precise Relation: (proof hard)

12x k

Uncertainty in the Time DomainStand and watch a wave go by at one place•You will see the wave over a period of time t•You will see the wave with a combination of angular frequencies •The same uncertainty relationship applies in this domain

12t

Estimating Uncertainty: Carlson’s RuleA particle/wave is trapped in a box of size L•What is the uncertainty in its position x?

?

•Best guess: The particle is in the center, x = L/2•But there is an error x on this amount

•It is no greater than L/2•It is certainly bigger than 0•Carlson’s rule: use x = L/4•This rule can be applied in the time domain as well

L

Guess of position

L/2L/2

L/4

Exact numbers for x:•Particle in a box: 0.181L•Uniform distribution: 0.289L

Sample Problem:A student is supposed to measure the frequency of an object vibrating at f = 147.0 Hz, but he’s late for his next class, so he only spends 0.100 s gathering data. How much error is he likely to have due to his hasty data sampling?

•Since the data was taken during 0.100 s, the date fits into a time box of length 0.100 s•By Carlson’s rule, we have t = 0.0250 s•By the uncertainty principle (time domain), we have:

12t 11

20.0 s2 t

•Since f = /2, this causes an estimated error of

120.0 s

2f

3.17 Hz

•Of course, the error could be much larger than this

Wave Equations You Need:

2k 1

2f

T

p

g

vkd

vdk

12

12

x k

t

•These equations always apply

, cos

, sin

x t A kx t

x t A kx t

•Two equations describing a generic wave

83 10 m/sp gv v c

•Light waves only

Math Interlude: Complex Numbers•A complex number z is a number of the form z = x + iy, where x and y are real numbers and i = (-1).

•x is called the real part of z and y is called the imaginary part of z.•The complex conjugate of z, denoted z* is the same number except the sign of the imaginary part is changed

z x iy

Re

Im

x z

y z

Note: no i

What’s the imaginary part of 4 + 7i?

*z x iy •Adding, subtracting, and multiplying complex numbers is pretty easy:

3 4 2 6i i 26 18 8 24i i i 6 10 24 30 10i i •To divide complex numbers, multiply numerator and denominator by the complex conjugate of the denominator

2 6

3 4

i

i

2 6 3 4

3 4 3 4

i i

i i

2

2

6 8 18 24

9 16

i i i

i

18 26

25

i

A Useful Identity

42 3 41 1 12! 3! 4!0 0 0 0 0f x f xf x f x f x f

Taylor series expansion

Apply to sin, cos, and ex functions

3 5 7 91 1 1 13! 5! 7! 9!

2 4 6 81 1 1 12! 4! 6! 8!

2 3 4 51 1 1 12! 3! 4! 5!

sin

cos 1

1xe x x x x x

In last expression, let x i

2 3 4 5 6 71 1 1 1 1 12! 3! 4! 5! 6! 7!1ie i i i i i i i

2 3 4 5 6 71 1 1 1 1 12! 3! 4! 5! 6! 7!1 i i i i 2 4 6 3 5 71 1 1 1 1 1

2! 4! 6! 3! 5! 7!1 i

cos sini cos sinie i

Complex WavesTypical waves look like:1. We’d like to think about them both at once2. We’d like to make partial derivatives as simple as possible

, cos

, sin

x t A kx t

x t A kx t

A mathematical trick lets us achieve both goals simultaneously:• Real part is cosine• Imaginary part is sine

, i kx tx t Ae

This makes the derivatives easier in differential equations:

, ,

, ,

i kx t i kx t

i kx t i kx t

x t A e ikAe ik x tx x

x t A e i Ae i x tt t

ikx

it

2 22

2 20z zE E

ct x

What is the dispersion relationship for light in vacuum?

2 22 0z zi E c ik E 2 2 2c k

Magnitudes of complex numbersThe magnitude of a complex number z = x + iy denoted |z|, is given by:•This formula is rarely used•The square of the magnitude can be written

2 2z x y

2 2 2z x y 2 2 2x i y x iy x iy 2*zz z

This is the easiest way to calculate it

Sample ProblemWhat’s the magnitude squared

of the following expression?

22

exp4

a ixaa

a it a

2 22 2 2

* exp exp4 4

a ix a ixa aa a

a it a it a a

2 2 22

2 2 2

2 2 8exp

4

a ix a ix aa

a i t a

2 2

2 2

2exp

4

a i x

aa t

2

2 2exp

2

a x

aa t