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Wave - Particle Duality

- Subhalakshmi Lamba

Till the nineteenth century

- Established theories in Physics
- Analytical Mechanics
- Thermodynamics
- Maxwellian Electrodynamics

Why a new theory ?

By the end of the nineteenth century and the

early years of the twentieth century a number of

experiments had been carried out which could not

be explained by the classical theories. So A

new way of thinking was required !

Experimental problems confronting classical

physics

- Discovery of Electrons
- Alpha Scattering
- Atomic Spectroscopy
- Blackbody Radiation
- Photoelectric Effect
- Compton Effect
- Electron Diffraction

Cathode Rays

Cathode rays are deflected off their paths by

magnetic and electric fields.

Discovery of electrons

- He proposed that
- Cathode rays are actually streams of tiny
- negatively charged particles (much smaller than

atoms). - Their charge to mass
- ratio 1.7 1011 C/kg.

1906

J. J. Thomson (1856 1940)

Electrons are a fundamental constituent of

matter.

Alpha Scattering Experiment

- Established that
- Almost all the mass
- of an atom was concentrated in a

positively charged - nucleus.
- Most of the atom was empty space.

Ernest Rutherford

Nuclear Model of the Atom

An atom is made up of

Nucleus

- A minute positively charged nucleus.
- An equal and opposite
- negative charge is distributed around the nucleus

in the form of electrons.

Electrons

Can the electrons be stationary?

- Can the atom contain
- stationary positive
- and negative charges ?

NO !

Can the electrons be stationary?

is much greater than

There is a problem..

The electron would fall into the nucleus.

An electric charge cannot be in equilibrium, at

rest under the action of electric forces alone.

Next ?

- Consider that the electron revolves around the

nucleus and the - attractive electrostatic force provides the

necessary centripetal force.

There is a problem

An accelerating electron, however, would

radiate energy.

Eventually it would fall into the nucleus.

Atomic Spectroscopy

- Balmer series of Hydrogen
- Line spectrum is obtained when light from a gas

through which an electric discharge is passed is

dispersed by a prism or a grating - spectrometer.

There are four lines in the Balmer series of

Hydrogen.

What is surprising is that

- Instead of a continuous band of colors only a few

colors appear. - The wavelengths of the lines are characteristic

of the element that is emitting the light. - Each element has its own particular line spectra.

Johannes Rydberg (1854-1919)

Blackbody Radiation

- At very low and very high temperatures the

emissive power is very small. - At intermediate temperatures there is a maximum.
- The height of the maximum increases with

temperature. - The maximum shifts to smaller wavelengths.

Distribution of energy in the spectrum of a

blackbody radiation at different temperatures.

Plancks theory

1918

Emission and absorption of radiation caused by

oscillators present in the walls of the black

body. The walls of the blackbody contain

oscillators of all frequencies.

Max Planck (1858 1947)

Plancks theory

1918

- He sought
- To find the average energy of a harmonic

oscillator at a given temperature. - To modify the statistical distribution of energy

between the oscillators.

Max Planck (1858 1947)

Plancks Theory

- Oscillators can radiate energy only in discrete

amounts like 0, ?0 , 2?0, 3?0.. n ?0.

?0 h n is a QUANTUM of energy. h is a

universal constant. (Plancks constant)

Plancks Theory

- Drastic departure from classical ideas.
- Average energy of the oscillator

h6.62618 X 10-34 Js

How much is a quantum of energy ?

What is the magnitude of energy E associated

with a quantum ? E hc/? (?c/ ? ) For a

quantum of visible light of wavelength 5000 Å

the energy is E 4 10-19 J

The birth of QUANTUM PHYSICS

The year was 1900

- Energy of an oscillator can vary only in discrete

jumps. - Emission discontinuous transition
- between states n h n
- In general any physical system capable of

emitting electromagnetic radiation has a discrete

set of allowed energy values - or energy levels.

Photoelectric effect

Light with a frequency gt threshold frequency

Electrons emitted

Metal surface

- Electron current varies with the intensity of

the light. - The emission of electrons is immediate (3 10 -9

secs). - The maximum kinetic energy of the emitted

electrons is a linear function of n and is

independent of the intensity .

More about the threshold frequency

- For any given metallic surface,
- if the frequency of the incident light
- is less than the threshold frequency,
- then, no matter
- how long the light is incident or
- how great its intensity,
- electrons are not emitted.

Classical Roadblock

- In metals the outermost electrons in the atoms

are not tightly bound to the nucleus and can be

removed. - Just sufficient energy is required.

Light is, after all, an electromagnetic wave

Increase energy by increasing the amplitude

Why doesnt this work?

Increasing wavelength ?

Decreasing frequency ?

Theory of Photoelectric effect.

- Any given source could
- absorb or emit
- radiant energy only in units
- or quanta all
- exactly equal to h ?.
- Light itself consisted of
- quanta of energy h ?
- which move through space
- with the velocity of light.
- This quantum of electromagnetic radiation is

called a photon.

1921

Albert Einstein 1879 - 1955

So Photoelectric Effect is

A collision between a photon and a bound

electron in which a photon is completely

absorbed and the energy of the electron increases

by h ?. The photoelectric equation

Workfunction of the metal W h?0 So

Photoelectric Effect

- Electrons can be emitted from the metal only

when ? gt ?0. - The energy of the electrons varies linearly
- with the frequency (? - ?0).
- The energy of the electrons is independent of

the intensity of the radiation. - The number of electrons ejected is proportional

to the intensity of the radiation.

Structure of the atom

Rutherfords model of the atom was intrinsically

unstable. Bohr applied the quantum ideas of

Planck and Einstein to Rutherfords nuclear atom.

His model for the atom is a hybrid of

classical and quantum ideas

1922

Niels Bohr (1885-1962)

Postulates of Bohrs Atomic Model

- Orbit of the electron around the nucleus.
- The electrostatic attraction between the nucleus

and the electron, similar to the gravitational

attraction in its spatial properties could lead

to stable circular or elliptical orbits for the

electron. - Electron in an atom moves in a circular orbit

about the nucleus with the centripetal force

being supplied by the Coulomb attraction between

the nucleus and the electron

Postulates of Bohrs Atomic Model

- Allowed Orbits.
- The allowed orbits are separated from the

forbidden ones by a quantum condition, which is

imposed on the angular momentum and not on the

energy. - Only those orbits are allowed for which the

angular momentum of the electron L is an

integral multiple of ? (h/2p).

Postulates of Bohrs Atomic Model

- Stationary States.
- The problem of the stability for the circular

orbit of the electron was solved by postulating

that in an allowed orbit, the electron must have

a constant energy. - An electron in an allowed orbit does not emit

any radiation. These constant energy states are

called stationary states.

Postulates of Bohrs Atomic Model

- IV. Emission/Absorption of Energy.
- The mechanism of emission /absorption of energy

from/by an electron in an atom was by transitions

between these constant energy states. - Energy is emitted (or absorbed) from an atom

only when the electron jumps from one allowed

orbit to another. - Einsteins frequency relation
- h? Ei - Ef

A logical progression

- The frequency of the emitted radiation is
- The energies vary as 1/n2
- Radii of the orbits vary as n2
- Angular momentum varies as nh

Radius Energy for Bohrs Orbits

- With these postulates we can write

Radius Energy for Bohrs Orbits

Radius of the nth orbit

Energy of the nth stationary state

Explanation of Atomic Spectra

- The frequency of the emitted radiation when the

electron jumps from a state n to am state m can

be found from the Einstein frequency relation by

substituting for the energies in the two states.

So,

Successes of the Bohr Model

- Bohrs theory could explain
- The spectra of one electron atoms.
- (hydrogen and single ionized Helium)
- And
- Gave a physical interpretation for the spectral

lines in terms of the stationary states of the

atom.

Extension of Bohrs Theory

- The spectra of the neutral hydrogen atom and

the singly ionized helium atom also have

fine lines. - This could not be explained within Bohrs

theory which has only a single quantum number

n.

Extension of Bohrs Theory

- It was explained by Sommerfield by
- 1. Postulating elliptic as well as

circular orbits thus adding a new quantum

number. - 2. Accounting for the relativistic

variation of electronic mass .

Serious Discrepancies

- Were found between theory and experiment when
- Bohrs theory was applied to
- two-electron atoms and
- In trying to account for the splitting of

spectral lines in a magnetic field. Another

quantum number was required. - The atomic model itself was held to be at fault

and QUANTUM MECHANICS developed.

Let us examine the following

- Our understanding of the physical world
- is that it is made up of two basic distinct

entities. - Waves
- Like sound waves,ripples on the surface of

water, electromagnetic waves. - Material objects
- Like a particle,a ball, a car, the planets.

Are they very different ?

- Material Objects
- Can be located at a definite position at a given

time. - Can be at rest or moving or accelerating under

an external force. - When they collide, they either scatter or

shatter. They definitely cannot pass through each

other.

It would seem so.

- Waves
- They are spread out in space and time.
- They are defined by their velocity, wavelength,

frequency or amplitude. - They can pass through one another. In the process

the waves are either enhanced or reduced.

Wave Nature of Material Objects

- Light, believed to be an electromagnetic wave

shows both - Wave like behavior interference

diffraction and - Particle like behavior photoelectric effect.
- Should not material particles then show wave

like behaviour ?

De Broglie Hypothesis

1929

- Particles of matter should exhibit both particle

and wave nature. - A material particle of energy E and a momentum p

may exhibit the characteristics of a wave of

wavelength - ? h/p

Louis de Broglie (1892-1987)

Louis de Broglie, Nobel Prize Speech

Determination of the stable motion of electrons

in the atom introduces integers, and up to this

point the only phenomena involving integers in

physics were those of interference and of normal

modes of vibration. This fact suggested to me the

idea that electrons too could not be considered

simply as particles, but that frequency (wave

properties) must be assigned to them also.

Unification of Two Concepts

- This required the unification of two concepts
- Wavelength which has a clear cut meaning only

for waves. - Momentum which has a natural interpretation

only for a moving particle.

De Broglie Hypothesis

To complete the analogy we write

Particle Photon Momentum p h/? Energy

E h ? hc/?pc

Light

Wave matter wave Wavelength ?h/p Momentum

p h/?

Is Light a Particle or a Wave ?

On a macroscopic scale (a large number of

photons) light can still be thought of as a

wave. In the interaction of light with matter on

the subatomic scale we must look at the

particle description of light.

Some Typical De Broglie wavelengths

A ball weighing 100 g, and moving with a speed

of 25 m/s 2.65 x 10-34 meters.

An electron accelerated through a potential

difference of 50 V 1.73 x 10-10 meters.

Compton Effect

Monochromatic X-rays were scattered by a

graphite block and the wavelength of the

scattered radiation was measured.

1929

Arthur Compton

Compton Effect

- incident X-rays scattered X-rays
- electron
- At each scattering angle peaks
- were observed at two wavelengths
- One at the incident wavelength
- One at a longer wavelength
- ( ? dependent)

Compton Effect

Elastic Collision between a Photon and an

Electron

Applying the principles of conservation of

energy and momentum he could derive the

expression for the wavelength shift.

Waves could behave like particles!

Diffraction of Electrons from a Crystal

Accelerated electrons impinging on a Ni crystal

create a diffraction pattern.

1937

Davisson and Germer

Diffraction of Electrons from a Crystal

Electrons are associated with a wave of

wavelength ? 1 A . If the planes of the Ni

crystal are considered to be a diffraction

grating then we can look upon the process as the

Diffraction of Electron Waves.

Particles could behave like Waves!

Low Energy Electron Diffraction

LEED has developed as the principle technique

for examining surface structures.

Uses a beam of electrons (typically in the range

20 - 200 eV) incident normally on a crystal

sample. The diffraction pattern provides accurate

information about the atomic positions and the

unit cell.

Low Energy Electron Diffraction

Revisiting Bohrs Atomic Model

If electrons behave as waves the concept of

Bohrs orbit must change. Such a wave could

exist is if a whole number of its wavelengths fit

exactly around the circle.

Revisiting Bohrs Atomic Model

So orbits can have only certain sizes,

depending on the wavelength of the electron

--which is decided by their momentum.

Circumference of the circle (2 ? r) is an

integral multiple of the wavelength (? h/p )

of the electron. So 2 ? r n h/p n h / m v

m v r n h / 2 ? which is the condition

for quantization of angular momentum.

Summary

- The idea of a quantum of energy for a harmonic

oscillator is needed to explain blackbody

radiation - The concept of a quantum of light- the photon was

able to explain photoelectric effect. - Bohrs theory of atomic structure gave a

physical interpretation for atomic spectra. - Matter was proposed to have wave-like properties.

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