Lecture-XXII

Introduction to Quantum Mechanics

Why Quantum Mechanics?

Universe consists of material particles and radiation.Dynamics of material particles is described by classical mechanics.Radiation is described by electromagnetism.

However, both fail at the atomic level. (Several examples in quantum theory)

Need of a new mechanics, a theory to explain physics at the atomic level.

Electromagnetic radiation is quantized as photons. (Photoelectric effect,Compton scattering, etc)

Need of a new mechanics, a theory to explain physics at the atomic level.

Quantum Mechanics

Do particles behave like waves?

Double Slit Experiment: with light waves

( )12 * 2

1 1 1 1 1 1 1Slit 1 open: ,i t kx

A e I Aωφ φ φ φ− −

= = = =

( )

( ) ( )

( )

( ) ( )

2

1 2

2 * 2

2 2 2 2 2 2 2

1 2 1 2

22 2 2

1 2 1 2 1 2

1 2 1 2

Slit 2 open: ,

Both the slits open:

2 cos ,

2

i t kx

i t kx i t kx

A e I A

A e A e

I A A A A

k x x x x

ω

ω ω

φ φ φ φ

φ φ φ

φ φ φ δ

πδ

λ

− −

− − − −

= = = =

= + = +

= = + = + +

= − = −

Double Slit Experiment: with bullets

Imagine an experimental setup in which a machine gun is spraying bullets at a screen in

which there are two narrow openings, or slits. Suppose, in the first instance, that this

experiment is carried out with only one slit opened, slit 1 say.

Slit1 is open: P1 (x) is the probability that a bullet landing at x.

Slit2 is open: P2 (x) is the probability that a bullet landing at x.

Both the slits open: It simply shows that the two probabilities are added up.

Double Slit Experiment: with electrons

The wave nature of electrons is also confirmed by Davisson and Germer bypassing electron through a thin gold foil.

Matter Wave

• Wave-particle duality- matter can behave both

like particles as well as waves.

Louis de Broglie

• If a particle of mass m moves with a velocity v

then it behaves like a matter wave having a

wavelength λ given by,

h h

mv pλ = =

Why are bullets different from electrons?

Bullets are far more massive than the electrons. One can observe them as long as one likes but it would not make any difference to them.

There are interference wiggles in the case ofThere are interference wiggles in the case ofbullets also. The are so crowded that it isphysically impossible to resolve them, onesees an average behaviour.

Wave Particle

De Broglie wave length 0

Electron diffraction with spies

Electrons are no more waves, they are particles now.

The Heisenberg’s uncertainty principleSets limit on what we can observe.

Consider our attempt to view the electrons in the double slit system by shining light of

wavelength λ on them with photon momentum pphoton = h/λ . If we manage to see an

electron it will be because one of these photons has struck it. Clearly the electron

momentum will be affected by this interaction with the photon. Let the change in the

electron's momentum is ∆pelectron and certainly it is proportional to pphoton. Let us assume

the constant of proportionality to be 1 in the present context. Thus, ∆pelectron = h/λ.

Now the precision with which we can determine a distance is limited by the size of the

wavelength λ of the light which we use to measure the distance. In fact this uncertainty

No precise coordinates, no trajectory.

electron ~x λ∆

wavelength λ of the light which we use to measure the distance. In fact this uncertainty

in position ∆x is directly proportional to the wavelength of the light; again, at our present

level of accuracy, we can set the proportionality constant to 1. Thus,

Caution: Uncertainty principle is not an explanation

to the null result of double slit experiment as we try

to to observe it. Rather it is entirely due to the

unavoidable disturbance that we make when we

observe the experiment.

The Uncertainty Principle implies a built-in, unavoidable limit to the accuracy with which

we can make measurements.

. ~h

x p hλλ

∆ ∆ =Therefore:

2x p∆ ∆ ≥ �