PHYS-2402

Sep. 22, 2016

Lecture 8

Announcement

HW3 (due 9/27)

Chapter 3

13, 17, 23, 25, 28, 31, 37, 38, 41, 44

Answer Keys for HW1 & 2

Course webpage

http://highenergy.phys.ttu.edu/~slee/2402/

Textbook

Chapter. 3Wave & Particles I

Outline:

• Blackbody Radiation (Plank; 1900; 1918*)• The Photoelectric Effect (Einstein; 1905; 1921*)• The Production of X-Rays (Rontgen;1901; 1901*)• The Compton Effect (Compton; 1927; 1927*)• Pair Production (Anderson; 1932; 1936*)• Is It a Wave or a Particle? à Duality?

EM-“Waves” behaving like “Particles”

Is It a Wave or a Particles?

Duality

Ch.3 E&M-Waves behaving like Particles

Ch.4 Particles behaving like Waves

Wave D>!

Particle D<<!

Is It a Wave or a Particles?

Duality

Ch.3 EM-Waves behaving like Particles

Ch.4 Particles behaving like Waves

Wave D>!

Particle D<<!

Double-slit Diffraction

Experiment

light

INDIVIDUAL

PHOTON

HITS

Although diffraction of light is

a wave phenomenon,

there is no smooth distribution of light

in the diffraction pattern,

but

the pattern is rather formed of many

individual hits of particles – the photons

A single photon

DOES NOT

get “disintegrated” in the

Diffraction process

to make a smooth

diffraction pattern

Coming back to that soon…

Electromagnetic

waves (light)

Particles

(photons)

Massive

ParticlesWaves

?

Chapter. 4

Wave & Particles II

Outline:

• A Double-Slit Experiment (watch “video”)• Properties of Matter Waves• The Free-Particle Schrödinger Equation• Uncertainty Principle

• The Bohr Model of the Atom

• Mathematical Basis of the Uncertainty Principle –

The Fourier Transform

“Matter” behaving as “Waves”

Electron passing through a wide slit arrive at a screen

one by one and produce a bright region essentially

the same width as the slit

Very strange

With a single narrow slit,

Electrons arrive one by one all over the screen.

They don’t avoid any points …

Very strange

Like a

Wave

…

But when 2 slits

are open,

there are certain

points at

which no

electron ever

arrives.

Each electron

wave is

destructively

interfering

with itself.

Electrons producing a double-slit interference pattern - one particle at a time

Actual double-slit pattern produced by electrons

Bragg’s Law• Bragg's law states that when X-rays hit an atom (in a x-tal), they

make the electronic cloud move as does any EM-wave. The movement of these charges re-radiates waves with the same frequency; this phenomenon is known as the Rayleigh scattering.

• These re-emitted wave fields interfere with each other either constructively or destructively, producing a diffraction pattern on a detector or film. The resulting wave interference pattern is the basis of diffraction analysis. [..excellent probe for small length scale]

Diffraction of a beam from multiple atomic planes

Bragg’s Law• The interference is constructive when the phase shift is a multiple of

2π; this condition can be expressed by Bragg's law,

• n is an integer determined by the order given, λ is the wavelength of the X-rays, d is the spacing between the planes in the atomic lattice, and θ is the angle between the incident ray and the scattering planes.

• Bragg's Law is the result of experiments into the diffraction of X-raysoff crystal surfaces at certain angles. Bragg's law confirmed the existence of real particles at the atomic scale, as well as providing a powerful new tool for studying crystals in the form of X-ray diffraction.

Bragg’s Law

Electron wave diffracted by AlMn quasicrystal De Broglie HypothesisThe E&M waves can be described using the language of quantum particles (photons). Q:: Can particles behave as waves?

De Broglie (1923) suggested that a plane mono-energetic wave is associated with a freely moving particle:

( ) ( )0i t kxx e ω −Ψ =Ψ

This wave travels with the phase velocity

This is a solution of the wave equation in 1-dimension:

2 22

2 2vt xψ ψ∂ ∂=

∂ ∂

vkω

=

for photon, E = pc and E = hfhf = pc => h = pc/f = pλ so, λ = h/p

We’ll apply the same logic which helped us to establish the relationship between p and λ for photons:

2 hk pπ

λ = =

- depends on the momentum rather then energy (e.g., for an object at rest, λ = infinity)

Compare with Compton wavelength of the particle

De Broglie wavelength

p - the object’s m/m

Chmc

λ =

4.2 Properties of Matter Waves• What properties characterize a wave?• λ, f, v, also A that varies with position/time (i.e. (x,t))• Wave function: we use different symbol for wave function

of different kinds of wave

(1) For transverse wave on string; y(x,t)-- string’s transverse “displacement”

(2) For E&M plane wave; E(x,t) &B(x,t)-- describe how the oscillating E- & B-field vary with (x,t)

(3) For matter wave;ψ(x,t) = probability amplitude|ψ(x,t)|2 è tell us probability of finding the particle(see next slide for some details)

Properties of

Matter Waves

Amplitude =

function of (x,t)),( tx!

Probability

Amplitude

Amplitude =

function of (x,t)

),( tx!

Probability

Amplitude

The probability density

to find

a particle

at coordinate x, at time t

2|),(| tx!

2|),(| txdxdtP != !!

The probability

to find

a particle

in an interval """"x, """"t

Integrate over """"x, """"t

x

t 2|),(|)( txdtxP != !

4.2 Properties of Matter Waves• Wave length

– De Broglie’s hypothesis: ç=è– λ = Wavelength of matter wave– Experimentally confirmed; e.g. x-tal diffraction– de Broglie got Nobel prize in 1929

2 hk pπ

λ = =C

hmc

λ =

• Frequency (of matter waves)– f = E/h– now, it’s open more convenient to express λ = h/p and f = E/h in

terms of the following quantities;k = 2π/λ (wave number), w = 2π/T (angular frequency)

– Another convenient definition: h = h/2π = 1.055 x 10-34 Js– express the fundamental wave-particle relationship as

!p = "!k E = !ω

Properties of

Matter Waves

Wavelengthp

h=!

De Broglie (1929)

Frequencyh

Ef =

Properties of

Matter Waves

Speed !fv =

BUT

v is NOT the velocity

of the massive particle,

it’s the velocity of the

Matter wave

It does give the wave speed

v = (E/h)(h/p) = E/p

c.f. for photon, E = pc

Electromagnetic

waves (light)

Particles

(photons)

Massive

ParticlesWaves

?

The Free-Particle

Schrodinger Wave

Equation

Analogy:

Waves on a string

[Q] How do we determine the ψ(x,t) of a matter wave?

wave speed

Waves on a string

Solution = function

“Plane Wave”

wave speed

E&M Wave E&M Wave