Download Need for Development of Quantum Mechanics

QUANTUM
MECHANICS
By: Harleen Kaur
Lecturer in Physics
P.G.G.C.G.
Sector-11, Chandigarh.
Physics Paper C
BSc II
CHAPTER 1
ORIGIN OF QUANTUM
MECHANICS
NEED FOR DEVELOPMENT OF
QUANTUM MECHANICS
Origin of Quantum mechanics had its roots at
the failure of Newton’s laws of motion and
Special theory of relativity to explain the
behavior of microscopic world of atom.
 Some key phenomena which are responsible
for development of quantum mechanics are:Black body radiation, Specific heat of solids,
Photoelectric effect, Compton effect, Hydrogen atom
spectra etc.

PHOTOELECTRIC EFFECT The phenomenon of ejection of electrons from the surface of the metals when
light of certain threshold frequency strikes it.
THE EXPERIMENT:
By varying the voltage on a negatively charged
grid between the ejecting surface and the
collector plate, Lenard was able to:

Determine that the particles had a negative
charge.

Determine the kinetic energy of the ejected
particles.
He found that by turning up the voltage he reached a point where the
current in his circuit ceased to flow because no more electrons were
making it to the other side of the gap. This was noted and dubbed the
stopping voltage.
Larger frequency, means smaller wavelength, and larger Energy=h.
LENARD’S FINDINGS:

Thus he theorized that stopping voltage must be
equal to the maximum kinetic energy of the ejected
particles, or:
KEmax = eVstopping
Perplexing Observations:
 The
intensity of light had no effect on
energy
 There
was a threshold frequency for
ejection
Classical physics failed to explain this,
Lenard won the Nobel Prize in Physics in 1905.
Larger light intensity means larger number of photons at a given frequency
(Energy)
EINSTEIN INTERPRETATION
A new theory of light:



Electromagnetic waves carry discrete
energy packets
The energy per packet depends on
wavelength, explaining Lenard’s threshold
frequency.
More intense light corresponds to more
photons, not higher energy photons.
EINSTEIN’S RELATIONS:
Einstein predicted that a graph of the maximum
kinetic energy versus frequency would be a
straight line, given by the linear relation:
KE = h - Φ
where Φ is known as work function and it is characteristics of given
metal.
…Therefore light energy comes in multiples of h.
GRAPH OF KEMAX VS. FREQUENCY
PHOTOELECTRIC EFFECT 

The photoelectric effect provides evidence for the
particle nature of light.
It also provides evidence for quantization.

If light shines on the surface of a metal, there is a
point at which electrons are ejected from the metal.

The electrons will only be ejected once the threshold
frequency is reached .

Below the threshold frequency, no electrons are
ejected.

Above the threshold frequency, the number of
electrons ejected depend on the intensity of the light.
DE-BROGLIE
HYPOTHESIS OR WAVE-
PARTICLE DUALITY
Wave-Particle Duality set the stage for the 20th century
Quantum Mechanics.
 Just as light possesed both particle and wave properties
then matter particles must also possess wave properties.
 The wave associated with matter particles are known as
Matter waves or de-Broglie waves.
 De-Broglie wavelength is given by:
where
h = Planck’s constant
p=mv is the momentum of the
particle.
DE-BROGLIE WAVELENGTH ASSOCIATED
WITH AN ELECTRON

de-Broglie wavelength for an electron accelerated
through potential difference V is given by relation
A0

As value of Planck’s constant is very small
(h=6.62x10-34 Js), the order of the wavelength
associated with ordinary objects is negligible . Thus,
Wave-Particle duality is not appreciable for objects
that we interact in daily life(macroscopic objects).
WAVE PACKET
Wave packet is formed by the superposition of waves
having slightly different frequency and amplitude.
They interfere constructively in such a way that
amplitude of resultant wave motion is very large in
small region and negligibly small in rest of space.
 A wave packet is localized – a good representation of
particle!

PHASE VELOCITY AND GROUP VELOCITY

Phase velocity – Velocity of an individual wave comprising the wave packet. It
describes the rate at which the phase of the wave propagates in space.
It is always greater than velocity of light and can’t be accepted.
 Group velocity – Velocity of a wave packet. It is equal to the velocity of particle. It
describes the rate at which the envelope of the wave packet propagates.
where =2 is the angular velocity.
k=2/ is the propagation vector.
HEISENBERG UNCERTAINTY PRINCIPLE
It is the direct consequence of Wave-Particle duality. It
states that:
It is impossible to measure both position and momentum of a
particle simultaneously with accuracy.
The product of the uncertainties is approximately of the
order of h/2.
x.px  h/2
where x is the uncertainty in measurement of position and
px is the uncertainty in measurement of momentum.
HEISENBERG-BOHR THOUGHT
EXPERIMENT:
• It shows that a measurement
itself introduces the
uncertainty.
• When we “look” at an object
we see it through the
photons that are detected by
the microscope.
– These are the photons that are
scattered in the angle < 2θ
– Momentum of electron is changed
HEISENBERG-BOHR thought experiment:
p max
ph  2 p ph sin 
pelectron  p ph  2 p ph sin 
Trying to determine the electron
position, we introduce the
uncertainty of the momentum
p ph 
h

pelectron  2 p ph sin  
2h

sin 
HEISENBERG-BOHR thought experiment:
Image is not the point, but the diffraction
pattern
The uncertainty of the position is
approximately the width of the
central maximum
x   sin 
px 
2h

sin  

sin 
 2h
TIME - ENERGY UNCERTAINTY RELATION
Heisenberg’s Uncertainty principle is applicable to all
“conjugate variables”.
 The Time-Energy Uncertainty principle states:
Et  h/2
where E is the uncertainty in measurement of energy and
t is the uncertainty in measurement of time.

It explains the natural broadening of spectral lines. Since
every excited state has a finite life span, t will be finite so
that no spectral line is infinitely sharp (E0).
 Energy conservation law can be violated but for
the short time interval!

As value of Planck’s constant is very small, Uncertainty
principle is relevant only for microscopic objects.

APPLICATIONS OF UNCERTAINTY
PRINCIPLE

Non-existence of electrons in the nucleus.

Existence of Protons, neutrons and alpha particles in
the nucleus.

Zero point energy- The minimum energy of the
system at 0K. It gives finite value of zero point energy.

Binding energy of an electron in atom.
PROBLEMS
a)
b)
c)
d)
An electron and a proton have the same de-Broglie
wavelength. Prove that the energy of the electron is
greater.
Calculate de-Broglie wavelength of a proton moving
with 1/10th velocity of light.
The electron in hydrogen atom may be thought of as
confined to a radius of 5x10-11m.Calculate the minimum
uncertainty in the momentum of the electron. Also
calculate the minimum kinetic energy of the electron.
A radar pulse lasts for 0.25s.What is the order of
uncertainty in the energy of photons?