Review of Elementary Quantum Mechanics

Review of Elementary Quantum
Mechanics
Unit I & II
Dr Md Kaleem
Department of Applied Sciences
Jahangirabad Institute of Technology (JIT),
Jahangirabad, Barabanki(UP) - 225203
1/28/2017 1DR MD KALEEM/ ASSISTANT PROFESSOR

Objective
• Know the background for and the main features in the historical
development of quantum mechanics.
• Explain, qualitatively and quantitatively, the role of photons in
understanding phenomena such as the photoelectric effect
• Be able to discuss and interpret experiments displaying wavelike
behavior of matter, and how this motivates the need to replace
classical mechanics by a wave equation of motion for matter (the
Schrödinger equation).
• Understand the central concepts and principles of quantum
mechanics: the Schrödinger equation, the wave function and its
physical interpretation.
• Interpret and discuss physical phenomena in light of the uncertainty
relation.
1/28/2017 DR MD KALEEM/ AISSISTANT PROFESSOR

• An idealized body that absorbs all incident EM
radiation incident on it, regardless of frequency or
angle of incidence is called a black body
• Implication: 1. Zero reflection, 2. Zero transmittance
• It is true for : 1. All wavelengths, 2. All incident directions
• Corollary: A black body emits maximum amount of radiation
at a given temperature
Black Body Radiation

Black Body Radiation

Planck’s Radiation Law
The exchange of energy by radiation with
matters do not takes place in continuous
manner but discretely as an integral multiple
of energy E=hv, where h is the Planck’s
constant.
On the basis of his assumption Planck
derived a relation for energy density u(λ) of
resonators emitting radiation of wavelength
lying between λ and λ + d λ as
Max Planck

Photoelectric Effect
The emission, or ejection, of electrons
(Photoelectrons) from the surface of,
generally, a metal in response to incident
light is called photoelectric effect.
This Phenomenon was observed by Lenard
but explained by Albert Einstein in 1905,
by describing light as composed of discrete
quanta, now called photon, rather than
continuous waves.
Albert Einstein

Dual Nature of Radiation
Matter mass
momentum
Two object can’t exist together at same time and place
Wavelength
Frequency
Two wave co-exist together at same time and place
Matter
Wave

de – Broglie Hypothesis
In his 1924, depending on the
source doctoral dissertation, the
French physicist Louis de Broglie
made a bold assertion that just as
light has both wave-like and
particle-like properties, electrons
also have wave-like properties

de – Broglie Relation
λ = h/p
Wave Particle
By rearranging the momentum equation stated in the
above section, we find a relationship between
the wavelength, λ associated with an electron and
its momentum, p, through the Planck constant, h:
λ = h/p

Matter Wave
• A wave associated with the motion of a particle e of atomic
or subatomic size that describes effects such as the diffraction
of beams of particles by crystals.

Wave Packet
• wave packet is a combination of waves with about the same momentum.
• Combining waves into wave packets can provide localization of particles.
• The envelope of the wave packet shows the region where the particle is
likely to be found.
• This region propagates with the classical particle velocity.

Heisenberg's Uncertainty
Principle
The uncertainty principle also called
the Heisenberg Uncertainty
Principle, or Indeterminacy
Principle, articulated (1927) by the
German physicist Werner
Heisenberg, that the position and
the velocity of an object cannot
both be measured exactly, at the
same time.

Heisenberg's Uncertainty
Principle
• The product of the uncertainties in the
momentum and the position of a particle
equals h/(2) or more.

Group Velocity
• The group velocity is dω/dk . It describes how
fast wave packets move. It is the velocity with
which the envelop of the wave packet moves
Group Velocity
Phase Velocity

Phase Velocity
• The velocity of the component waves of a
wave packet is called Phase velocity.
• The phase velocity is ω/k . It describes how
fast the individual wave moves.

Dependency on the medium
• phase velocity vp of waves are typically larger than the
group velocity vg of waves. However, this really
depends on the properties of the medium.
• The media in which vg = vp is called the non-dispersive
medium.
• The media in which vg < vp is called normal dispersive
medium.
• The media in which vg > vp is called anomolous
dispersive media.
• It must be emphasized that dispersion is a property of
the medium in which a wave travels. It is not the
property of the waves themselves.

Wave Function
• It is variable quantity that mathematically describes
the wave characteristics of a particle.
• The value of the wave function of a particle at a given
point of space and time is related to the likelihood of
the particle’s being there at the time.
• By analogy with waves such as those of sound, a wave
function, designated by the Greek letter psi, Ψ, may be
thought of as an expression for the amplitude of the
particle wave (or de Broglie wave), although for such
waves amplitude has no physical significance.

Physical interpretation of wave
function
• The wave function, at a particular time, contains all the
information that anybody at that time can have about the
particle.
• But the wave function itself has no physical interpretation.
• It is not measurable. However, the square of the absolute
value of the wave function has a physical
interpretation. We interpret |ψ(x,t)|2 as a probability
density, a probability per unit length of finding the particle
at a time t at position x.
• The probability of finding the particle at time t in an
interval ∆x about the position x is proportional to
|ψ(x,t)|2∆x.

Schrödinger Equation
• Schrödinger equation is a linear, second order partial
differential equation.
• It is a wave equation in terms of the wave function
which predicts analytically the probability of the
properties the system or events or outcome with
precision.
• The Schrödinger equation is a more general and
fundamental postulate of quantum physics.
• It plays the same role in quantum physics which
Newton's laws and the conservation of energy play in
classical physics i.e., it predicts the time evolution i.e.
future behavior of a quantum dynamical system.

• An important feature of the Schrödinger
equation is that it is linear. Hence it allows for
the superposition of its solutions i.e. wave
functions.
21  ba 

• The Schrodinger equation has two forms’, one in which
time explicitly appears, and so describes how the wave
function of a particle will evolve in time. In general, the
wave function behaves like a, wave, and so the
equation is, often referred to as time dependent
Schrodinger wave equation.
• The other is the equation in which the time
dependence has been removed and hence is known as
the time independent Schrodinger equation and is
found to describe, amongst other things, what the
allowed energies are of the particle.

Schrödinger Time Independent
Equation
  0
8
2
2
2
2





VE
h
m
x
Wave function
Second derivative
wrt x
Position
Energy
Potential Energy
Schrodinger established his equation based on de Broglie’s Hypothesis of
matter wave, Classical plane wave equation & Conservation of Energy.

Schrödinger Time Dependent
Equation
 V
mt
i 

 2
2
2


•The TDSE is consistent with energy conservation.
•The TDSE is linear and singular value, which implies
that solutions can be constructed by superposition of
two or more independent solutions.
•The free-particle solution (U(x) = 0) is consistent with a
single de Broglie wave.

Introduction of LASER
LASER
Light
Amplification
Stimulated
Emission
Radiation

Spontaneous and Stimulated
Emission
• Stimulated Absorption: An atom in a lower
level absorbs a photon of frequency hν and
moves to an upper level.
• Spontaneous Emission: An atom in an upper
level can decay spontaneously to the lower
level and emit a photon of frequency hν if the
transition between E2 and E1 is radiative. This
photon has a random direction and phase.

Spontaneous and Stimulated
Emission

Stimulated Emission
• An incident photon causes an upper level
atom to decay, emitting a “stimulated” photon
whose properties are identical to those of the
incident photon.
• The term “stimulated” underlines the fact that
this kind of radiation only occurs if an incident
photon is present.
• The amplification arises due to the similarities
between the incident and emitted photons.

Stimulated Emission

Metasatable State
It is excited state of an atom,
nucleus, or other system that has a
longer lifetime than the ordinary
excited states and that generally has
a shorter lifetime than the lowest,
often stable, energy state, called the
ground state.
It has a life time between 10-3 to 10-5
sec.

Population Inversion
It is the process in which the
population of upper energy
level is increased in
comparison to lower energy
level. It is used obtain optical
amplification via stimulated
emission needed for laser
action.

Laser Gain
• It is defined as the amount of stimulated emission in
which a can generate as it travels a given distance in
the laser medium.
• It is characterised by the ability of laser medium to
increase the intensity of power of laser.
• It is also the measure of degree of amplification.

Q M in Real Life
At bottom, the entire computer industry is built on
quantum mechanics.
Modern semiconductor-based electronics rely on
the band structure of solid objects. This is fundamentally
a quantum phenomenon, depending on the wave nature
of electrons, and because we understand that wave
nature, we can manipulate the electrical properties of
silicon.
The fibers themselves are pretty classical, but the light
sources used to send messages down the fiber optic
cables are lasers, which are quantum devices.

Review of Elementary Quantum Mechanics

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