PH 301

Dr. Cecilia VogelLecture

Review

Outline

Wave-particle duality wavefunction probability

Photon photoelectric effect Compton scattering

When do We See Which?Wave-particle duality: Light can show wave or particle

properties, depending on the experiment. While propagating, light acts as a wave while interacting, light acts as a

particle.

When do We See Which? Two-slit experiment

Light will propagate through both slits and waves through slits interfere with

each other, but when it strikes the screen,

it interacts with the screen one photon at a time.

When do We See Which? Interference

seen if waves are coherent Diffraction

seen if obstacle/opening about size of wavelength

Why is the sky blue? The sky is blue, because more

blue light is scattered by the air to our eye (than red, yellow, etc).

Blue light is more likely to scatter than red, because red is more likely to diffract instead. Less diffraction occurs for shorter wavelengths.

Blue light has shorter wavelength, so it diffracts less and scatters more.

Why are the clouds white? The water droplets are much

larger than the wavelength of all visible light

(not just blue/violet)

almost no visible light is diffracted by clouds

every color of visible light is scattered by clouds

all colors scattered, so scattered light is white

Matter Matter particles, like electrons,

have particle properties (of course) individual, indivisible particles energy & momentum

(paintball)

Duality of Matter Matter particles also have wave

properties! They diffract! They interfere! Diffract from a

crystal, interference pattern depends on crystal structure

...from a powder, pattern depends on molecular

structure

Duality equations Light/photons Matter, e.g.

electrons

/hp

hfE

ph

hEf

/

/

c

E

hc

p

E

mv

mc 2

p

E

c

E

hc

p

E

Only for lightCue: ‘c’

mv

mc 2

p

EOnly for matterCue: ‘m’

Sameeqns

ExampleWhat is the wavelength of an

electron which has 95 eV of kinetic energy?Note: K<<moc2, so we can use classical equations.

Note: DO NOT USE E=hc/.

m1.26X then

kgm/s5.27X10 then

5784790m/s0.019/eV10511.0

)95(2/2 so

10-

24-

262

21

h/p

mv p

ccX

eVmKvmvK

Wave Function

For light, the wavefunction is E(x,t) electric field (and B(x,t) = magnetic

field). For matter the wave function is

(x,t) like nothing we’ve encountered before. Not an EM wave. The matter itself is not oscillating.

Wavefunction Interpreted For light beam, where the wave

function (E-field) is large, the light is bright there are lots of photons

For beam of matter particles, where the wave function is large there are lots of particles.

The bright spots in interference pattern are where lots of photons or matter

particles strike.

Probability Interpretation If you have one particle, rather than

a beam, the wavefunction only gives probability P(x,t) = |(x,t)|2. there is no way to predict precisely

where it will be. Where the wave function is large

the particle is likely to be. The bright spots in interference

pattern are where a photon or matter particle is

likely to strike.

Probability Interpretation P(x,t) = |(x,t)|2. If we repeat an experiment many,

many times, the probability tells in what fraction of the experiments, we will find the particle at position x at time t.

Do we have to do the experiment many, many times for the probability to have meaning? NO! With one particle, you can still determine

probabilities

Averages and Uncertainty P(x,t) = |(x,t)|2. If you have many possibilities with

known probabilities Average <x> = xave=x= probability

weighted sum of possibilities

<x> = Uncertainty x=rms dev = root mean

square deviation x = Also x =

dxx 2||

2)( xx

22 xx

Imaginary Exponentials What is the meaning of iye

You can do algebra and calculus on it just like real exponentials; just remember i2 = -1.

It is a complex number, with real and imaginary parts. Can be rewritten as: For example

yiye iy sincos

sincos iei

1ie but iei 2/

Complex Algebra

To add or subtract complex numbers, add or subtract real parts (a), add or subtract imaginary parts (b).

To multiply, use distributive law. To get the absolute square |z|2,

multiply z by its complex conjugate, z*. To get the complex conjugate of z,

change the sign of all the i’s.

ibaz

2 2 2z a b

a and b real

Complex Algebra In general, with

2 2z c

idz ce c and d real

Complex Example

Find the absolute square, ||2, which is the probability density.

Need the complex conjugate, *.

)( tkxiAe

22A

The probability density is constant, it is the same everywhere, all the

time. this particle is as likely to be a million

light years away, as here. Not localized.

Complex Example Given ||2 = ¼

show that

works as well as ½.

3 1

4 4A i

2 22 3 1 3 1 1

4 4 16 16 4A

PAL Probability Given the wavefunctionwhere x is in nm and ranges from 0 to 3 nm.1) Find the probability density as a function of x.

2) Find <x> = the average value of x.

3)Find < x2 > = the average value of x2.

4)Find x = the uncertainty in x.

311.5nm( ) sin( )i xx e x