PARTICLE PROPERTIES OF WAVES

Quantum Mechanics: (i) Particle properties of waves: Black body radiation, photoelectric effect, x-ray Diffraction and Compton effect, pair production, photon and gravity.

References:A. Beiser, Concept of Modern Physics (or Perspective of Modern

Physics), Tata-McGraw Hill, 2005

Course Outline...

ELECTROMAGNETIC SPECTRUM

Particle aspect of radiation

Classical physics: a particle : energy E and momentum p Wave : an amplitude and wave vector k

4.3 x 1014 7.5 x 1014

Principle of superposition

A characteristic properties of all waves

Particle: e.g e-: charge, mass : laws of particle mechanics

General Consensus:

Waves: em waves: diffraction, interference, polarization etc

Young’s experiment: Light consists of waves

Black Body RadiationClassical approach

Only the quantum theory of light can explain its origin

An ideal “blackbody”, is a material object that absorbs all of the radiation falling on it, and hence its appears as black under reflection when illuminated from outside.

Black Body

When an object is heated , it radiates electromagnetic energy as a result of the thermal agitation of the electrons in its surface

Blackbody radiation: radiation leaving the hole of a heated hollow cavity or the radiation emitted by a blackbody when hot.

Classical: continuous energy distribution

EXPERIMENTAL

Radiation inside a cavity: series of standing waves.

Density of standing waves in cavity:

Classical average energy per standing wave:

Rayleigh’s assumption: a standing wave can exchange any amount(continum) of energy with matter,

Rayleigh & Jeans

2

28)(c

ddG

kT

Rayleigh-Jeans formula: dckTdGdu 22

8)()(

Classical physics failure introduction of Q.M.

As increases toward the ultraviolet end of the spectrum, the energy density should increase as 2. In the limit of infinitely high , u()d should also go to . In really, of course, the energy density falls to 0 as . This discrepancy called ultraviolet catastrophe.

Consequences:Ultraviolet catastrophe

Planck’s formulaThe energy exchange between radiation and matter must be discrete and energy of radiation E = nh

1/

3

2

8)( kThed

chdu

1/ kTheh

Average energy per standing wave

Planck’s modifications

h = 6.626 x 10-34 J.sMax Planck (1918 Nobel prize)

SUMMARYAt low At high

dckT

dhkT

chdu

23

32

8

8)(

Rayleigh & Jeans

h >> kT

kThe /

0)( du

h << kT

hkTkThe kTh

/1/1

111/

# of modes per unit frequency per unit volume

Probability of occupying modes

Average energy per mode

CLASSICAL Equal for all modes kT

QUANTUM

Quantized modes: require h energy to excite upper modes, less probable

3

28c

3

28c

1kTh

e

h

Particle properties of waves:

[1] Photoelectric effect[2] Compton effect[3] Pair production

EXPERIMENTAL CONFIRMATION

Photoelectric effectProvides a direct confirmation for the energy quantization of light.

The phenomenon of ejection of electron from the surface of a metal when light of a suitable frequency strikes on It, is called photoelectric effect. The emitted electrons are called photo-electrons.

The time between the incidence and emission of a photoelectron is very small, 10–9 s.

V0

Heinrich Hertz in 1887later on, it was explained by Albert Einstein in 1905

It was during this study of electromagnetic waves that he saw that if you send ultraviolet light onto metals, you'll get sparks coming off

Experimental

Classical: raise intensity more electrons independent of frequency

Observed: No electron below threshold frequency

The following experimental laws were discovered:

The kinetic energy of the ejected electrons depends on the frequency but not on the intensity of the beam; the kinetic energy of the ejected electron increases linearly with the incident frequency.

If the frequency of the incident radiation is smaller than the metal’s threshold frequency(a frequency that depends on the properties of metal), no electron can be emitted regardless of the radiation’s intensity.

No matter how low the intensity of the incident radiation, electrons will be ejected instantly the moment the frequency of the radiation exceeds the threshold frequency.

At a fixed frequency, the number of ejected electrons increases with the intensity of the light but doesn’t depend on its frequency.

Classical physics fails: for dependence of the effect on the threshold frequency

K = h - W = h - h0

0 = threshold frequency

The stopping potential : at which all of the electrons will be turned back before reaching the collector

eW

ehc

eW

ehVS

Work function: W = h0

Larger W needs more energy needed for an electron to leave

Photoelectric effect:

metal

E = h0

KE = 0

KEmax = h - h0

E = h

Einstein interpretation(1905)

Light comes in packets of energy (photons)E = h

An electron absorbs a single photon to leave the material

0

K

When two UV beams of wavelengths 1 = 280nm and 2 = 490nm fall on a lead surface, they produce photo electrons with maximum kinetic energies 8.7eV and 6.67eV, respectively. (a) calculate the value of planck constant. (b) Calculate the work function and the cutoff frequency of lead.

ANS.(a)

K1= hc/ 1 – W and K2= hc/ 2 – W

Js

cKKh

34

12

2121

1062.6

Use 1eV = 1.6 x 10-19 J

(b) W = hc/ 1 –K1 = - 4.14eVThe cutoff frequency of the metal is

0 = W/h = 1015 Hz

Exercise(estimation of the Planck constant)

EM radiation with 0.01 to 10 nm x-rays

Photons transfer energy to electron: photoelectric effect

Can reverse be true?

X-rays

V

16

min1024.1

V.mX-ray production:

X-ray diffraction

d

2

d sin

2d sin = Path difference

Incident wave

Bragg’s Law:2d sin = n

Crystal structure determination

Compton effectCompton scattering is another one of those really important events that happened at the beginning of the 20th century that indicated that photons were real. They really, like really does behave like a particle.

Large wave length small wave length collision

c

h

cosc

h

sinc

h

cosp

sinpp

chphE

/

Scattered photon

0

20

pcmE

chphE

/

Incident photon

Target electron

pp

cpcmE

22420

Scattered electron

-

-

Further confirmation of photon model

1927 Nobel

Scattering of x-rays from electrons in a carbon target and found scattered x-rays with a longer wavelength than those incident upon the target.

In the original photon direction:Initial momentum = final momentum

coscos0 pc

hc

h

In the perpendicular direction: sinsin0 pc

h

Collisions: conserve energy and conserve momentumElastic process

We've got to know what the energy of the photon is and what its momentum is. Photon energy: h Photon momentum: h/= E/c

sinsincoscos

hpc

hhpc

2222 )(cos))((2)( hhhhcp

Form the total energy expression we have:

)(2)())((2)( 20

2222 hhcmhhhhcp

)cos1)()((2)(2 20 hhhhcm

Compton wavelength:

cmh

C0

gives the scale of the wave length change of the incident photon

cos/

sintan

Angle of recoil electron:

Compton effect or shift:

In order to conserve energy and momentum, I've got to have a shift in the wavelength. that means, the outgoing photon has a different wavelength than the incoming one. this is impossible classically. You cannot have a wave do that. But in quantum, you can.

this shift in wavelength is associated with h/mc. This is dimensionless

h/mc is really really small compared to the wavelength of light, then the shift in wavelength is relatively really small.

If h/mc is very very large compared to the wavelength of the radiation that you're sending in to hit the electron, then this shift in wavelength is going to be huge. So the Compton scattering is determined by how big the wavelength of the light that you're sending in is, in comparison to the Compton wavelength.

Small effect if >> c

Large effect if << c

)cos1(0

cm

h

Compton wavelength is really really small. 2.4 pm. So a pm is a trillionth of a meter. So this actually is about 100 times smaller than the atom. So you got to send in light with a really tiny wavelength, 100 times smaller than an atom in order to get an appreciable Compton scattering effect. But we've done it and this is the way it comes out.

Experimental Demonstration

A.H. Compton, Phys. Rev. 22 409 (1923)

Ex-1High energy photos (-rays) are scattered from electrons initially at rest. Assume the photons are backscattered and their energies are much larger than the electron’s rest mass energy, E >> mec2. (a) calculate the wavelength shift, (b) show that the energy of the scattered photons is half the rest mass energy of the

electron, regardless of the energy of the incident photons, (c) calculate the electrons recoil kinetic energy if the energy of the incident photons

is 150 MeV.ANS:

(a) Here = , wave length shift or Compton shift:

mcmc

cmh

cmh

ee

122

0

108.442)cos1(

(b) Energy of scattered photon E :

2/2)/()/(2 2

2

2

2

Ecmcm

hccmcm

cmhhchcE

e

e

e

e

e

MeVcmEcmcm

EcmcmE eeeee 25.0

2)(

221

2

2222122

(c) Kinetic energy of recoil electron: Ke = E - E 150 MeV-0.25 MeV = 149.75 MeV

If E >> mec2 we can approximate by

Pair production

+

-

Nucleus

Photon Electron

Positron

p

p

cosp

cosp

ch /

Electromagnetic energy is converted into matterEnergy and linear momentum could not both be conserved if pair production were to occur in empty space, so it does not occur there

Pair production requires a photon energy of at least 1.02 MeV.

e- or e+ m0c2 = 0.51 MeV (rest mass energy),

additional photon energy becomes K.E of e- and e+.

Pair annihilation:

e- + e+ +

The pair production : direct consequences of the Einstein mass-energy relation; E = mc2.

Ex-1Calculate the minimum energy of a photon so that it converts into an electron-positron pair. Find the photon‘s frequency and wavelength

The Emin of a photon to produce e- and e+ pair is equal to sum of rest mass energy of both. So Emin = 2mec2 = 1.02 MeV.

The photon’s frequency and wavelength can be obtained at once from Emin = h = 2mec2 and =c/:

= 2.47 x 1020 Hz and = 1.2 x 10-12 m

Ans:

Photon and gravity

H

hgHchhE 2

hE

KE = mgH

KE = 0 Gravitational behaviour of light: follows from the observation that, although a photon has no rest mass, it nevertheless interacts with electrons as though it has the inertial mass

Photon “mass” 2ch

vpm

Photon energy after falling through height H

21cgHhh

Although they lack rest mass, photons behave as though they have gravitational mass

Assignment-IVPH101: Physics-I Due Date: 28/10/2013 S. N. Dash

1. A 45 kW broadcasting antenna emits radio waves at a frequency of 4 Mhz. (a) How many photons

are emitted per second? (b) Is the quantum nature of EM radiation important in analyzing the radiation emitted from this antenna? [2+2]

2. When a light of a given 1 is incident on a metallic surface, the stopping potential for the photoelectrons is 3.2 V. If a second light source whose wavelength is double that of the first is used, the stopping potential drops to 0.8 V. From this data, calculate (a) the of the first radiation (b) the work function and cutoff frequency of the metal. [4+2]

3. Consider a photon that scatters from an electron at rest. If the Compton’s wavelength shift is observed to be triple the wavelength of the incident photon and if the photon scatters at 60o , calculate (a) the wavelength of the incident photon, (b) the energy of the recoiling electron, (c) the angle at which the electron scatters. [2+2+2]

4. What voltage must be applied to an x-ray tube for it to emit x-rays with a minimum wave length of 30 pm (10-12 m) ? [2]

5. The distance between adjacent atomic planes in calcite is 0.3 nm. Find the smallest angle of Bragg scattering for 0.03 nm x-rays ? [2]

6. A positron collides head on with an electron and both are annihilated. Each particle had a kinetic energy of 1 MeV. Find the wavelength of the resulting photons. [2]