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Quantum Mechanics

by

Dr. Amit Kumar Chawla

Introduction

1. Stability of an atom
2. Spectral series of Hydrogen atom
3. Black body radiation

There are a few phenomenon which the classical mechanics

failed to explain.

Max Planck in 1900 at a meeting of German Physical

Society read his paper “On the theory of the Energy

distribution law of the Normal Spectrum”. This was the start

of the revolution of Physics i.e. the start of Quantum

Mechanics.

Quantum Mechanics

1. Photo electric effect
2. Black body radiation
3. Compton effect
4. Emission of line spectra

It is able to explain

The most outstanding development in modern science was

the conception of Quantum Mechanics in 1925. This new

approach was highly successful in explaining about the

behavior of atoms, molecules and nuclei.

Photo Electric Effect

The emission of electrons from a metal plate when illuminated

by light or any other radiation of any wavelength or frequency

(suitable) is called photoelectric effect. The emitted electrons

are called ‘photo electrons’.

V

Evacuated

Quartz

tube Metal

plate

Collecting

plate

Light

^^^^^^^^

A

+

_

Photo Electric Effect

4. Increase in frequency of light increases the energy of the

electrons. At frequencies below a certain critical frequency

(characteristics of each particular metal), no electron is

emitted.

Voltage

Photo Current

v 1

v

2

v

3

Einstein’s Photo Electric Explanation

The energy of a incident photon is utilized in two ways

1. A part of energy is used to free the electron from the atom

known as photoelectric workfunction (W

o

).

2. Other part is used in providing kinetic energy to the emitted

electron.











 2

2

1

mv

2

h W mv

o

This is called Einstein’s photoelectric equation.

It is in form of. The graph with on y-axis

and on x-axis will be a straight line with slope

o o

eV  h   h 

o

V

If is the stopping potential, then

max o

KE  h   

e

h

e

h

V

o

o

 

y  mx  c

h e

o

V



Photons

Einstein postulated the existence of a particle called a photon,

to explain detailed results of photoelectric experiment.





hc

E h

p

 

Photon has zero rest mass, travels at speed of light

Explains “instantaneous” emission of electrons in photoelectric

effect, frequency dependence.

From Theory of Relativity, total energy of the recoiled electron

with v ~ c is

2 2

E mc K m c

o

Similarly, momentum of recoiled electron is

2 2

K mc m c

o

2

2 2

2

m c

v c

m c

K

o

o

2 2

2

v c

K m c

o

2 2

1 v c

m v

mv

o

Now from Energy Conversation

cos

cos

2 2

v c

m v

c

h

c

h

o

2 2

2

v c

h h m c

o

  (i)

From Momentum Conversation

(ii) along x-axis

 



sin

sin

2 2

v c

m v

c

h

o

  (iii)

along y-axis

and

On squaring, we get

Subtracting (vi) from (vii)

(vii)

2 2

2 2

2

2

2 2

2 2

v c

m c

hm

c

h

m c

c

h

c

h

o

o o

( 1 cos ) 2 ( ') 0

2

2

o

hm

c

h

( 1 cos )

2

2

c

h

hm

o

( 1 cos )

2

c

h

m

o

But

is the Compton Shift.

c

( 1 cos )

h

m c

o

and





c

So,

( 1 cos )

 h

m c

o

 '     ( 1  cos  )

m c

h

o

It neither depends on the incident wavelength nor on the

scattering material. It only on the scattering angle i.e.

is called the Compton wavelength of the electron

and its value is 0.0243 Å.

m c

h

o

   ( 1  cos )  0.0243 (1- cosθ) Å

m c

h

o

max

So Compton effect can be observed only for radiation having

wavelength of few Å.

Compton effect can’t observed in Visible Light

is maximum when (1- cosθ) is maximum i.e. 2.

0.05 Å

For 1Å ~ 1%

For 5000Å ~ 0.001% (undetectable)

Pair Production

When a photon (electromagnetic energy) of sufficient

energy passes near the field of nucleus, it materializes into

an electron and positron. This phenomenon is known as pair

production.

In this process charge, energy and momentum remains

conserved prior and after the production of pair.

Photon

Nucleus (+ve)



e



e