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Wave Particle Duality _ The Uncertainity Principle

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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 : •A minute positively charged nucleus. •An equal and opposite negative charge is distributed around the nucleus in the form of electrons.
Nucleus
10 -8 cm

10 -12 cm

Electrons

10 -8 cm

Can the electrons be stationary?

_ +2 a _

2a

Can the atom contain stationary positive and negative charges ?

NO !

Can the electrons be stationary?

The attractive electrostatic force 2 2 due to the nucleus 2e / a
_ +2 a _

2a

is much greater than

The repulsive electrostatic force due 2 2 e / 4a to the other electron

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
There are four lines in the 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.

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)

1 1   RH ( 2  2 ) 2 n

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. Distribution of energy in The maximum shifts to the spectrum of a smaller wavelengths. blackbody radiation at
different temperatures.

Planck’s theory
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.
1918

Max Planck (1858 – 1947)

Planck’s theory
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. 1918

Max Planck (1858 – 1947)

Planck’s Theory
Oscillators can radiate energy only in discrete amounts like 0, 0 , 20, 30….. n 0. 0  h  is a QUANTUM of energy. h is a universal constant. (Planck’s constant)

Planck’s Theory
Drastic departure from classical ideas. Average energy of the oscillator :

 ε  (e

h
h k BT

 1)

h=6.62618 X 10-34 Js

Classical result : <> = kB T h 0

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 nh 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 > 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  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 doesn’t this work?

Increasing wavelength λ

Decreasing frequency ν

Theory of Photoelectric effect.
Any given source could 1921 absorb or emit radiant energy only in units or quanta all exactly equal to h ν. Light itself consisted of quanta of energy h ν which move through space with the velocity of light. Albert Einstein: 1879 - 1955 This quantum of electromagnetic radiation is called a photon.

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 ν.

The photoelectric equation

1 2 mv  h  W 2

Workfunction of the metal W= hν0 So 1 mv 2  h(   0 )
2

Photoelectric Effect
1 mv 2  h(   0 ) 2 Electrons can be emitted from the metal only when ν > ν0. The energy of the electrons varies linearly with the frequency (ν - ν0). 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
Rutherford’s model of the atom was intrinsically unstable. Bohr applied the quantum ideas of Planck and Einstein to Rutherford’s nuclear atom. His model for the atom is a hybrid of classical and quantum ideas

1922

Niels Bohr (1885-1962)

Postulates of Bohr’s Atomic Model
I. 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 Bohr’s Atomic Model
II. 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 ђ (h/2π).

| L | mvr  n

Postulates of Bohr’s Atomic Model
III. 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 Bohr’s 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. Einstein’s frequency relation hν = Ei - Ef

A logical progression
• The frequency of the emitted radiation is

1 1   RH ( 2  2 ) 2 n

• The energies vary as 1/n2 • Radii of the orbits vary as n2 • Angular momentum varies as nh

Radius & Energy for Bohr’s Orbits
With these postulates we can write
mvn Ze 2  2 rn 4 0 rn Centripeta l force  Electrosta tic attraction
2

mvrn  n Quantizati on of Angular momentum

Radius & Energy for Bohr’s Orbits
Radius of the nth orbit:

nh rn  2 2 4 me Z
Energy of the nth stationary state:
1 Ze Z e m 2 E n  mvn   2 2 2 2 4 0 r 8 0 h n
2 2 4

2

2

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,

Z e m 1 1  v mn  2 2 3  2 n  8 0 h  m
2 4

Successes of the Bohr Model
Bohr’s 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 Bohr’s Theory
•

The spectra of the neutral hydrogen atom and the singly ionized helium atom also have fine lines. This could not be explained within Bohr’s theory which has only a single quantum number n.

Extension of Bohr’s 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 • Bohr’s 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
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

1929

λ = 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 Light

Particle : Photon Momentum p = h/ Energy E=h = hc/=pc
Particle p=mv

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 human being weighing 70 kg, and moving with a speed of 25 m/s ~ 3.79 x 10-37 meters. 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 (I)One at the incident wavelength (II)One at a longer wavelength (  dependent)

Compton Effect
Elastic Collision between a Photon and an Electron

h /c h /c = h/
Incident Photon 

Scattered Photon



Recoil 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 Bohr’s Atomic Model
If electrons behave as waves the concept of Bohr’s orbit must change.

Such a wave could exist is if a whole number of its wavelengths fit exactly around the circle.

Revisiting Bohr’s 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

mvr=nh/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. Bohr’s theory of atomic structure gave a physical interpretation for atomic spectra. Matter was proposed to have wave-like properties.


				
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