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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.
Quantum Mechanics
Electromagnetic radiation is quantized as photons.
Compton scattering, etc)
Do particles behave like waves?
(Photoelectric effect,
Double Slit Experiment: with light waves
Slit 1 open: φ1 = A1e
− i (ωt − kx1 )
Slit 2 open: φ2 = A2 e
2
, I1 = φ1 = φ1φ1* = A12
− i (ωt − kx2 )
2
, I 2 = φ2 = φ2φ2* = A22
Both the slits open:
φ = φ1 + φ2 = A1e −i(ωt −kx ) + A2 e−i(ωt − kx )
1
2
2
2
I = φ = (φ1 + φ2 ) = A12 + A22 + 2 A1 A2 cos δ ,
δ = k ( x1 − x2 ) =
2π
λ
( x1 − x2 )
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 by
passing 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,
λ=
h
h
=
mv p
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 of
bullets also. The are so crowded that it is
physically impossible to resolve them, one
sees an average behaviour.
De Broglie wave length
Wave
0
Particle
Electron diffraction with spies
Electrons are no more waves, they are particles now.
The Heisenberg’s uncertainty principle
Sets 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
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, ∆xelectron ~ λ
Therefore: ∆ x . ∆ p ~ λ
h
λ
∆x∆p ≥ 2
= h
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.
No precise coordinates, no trajectory.