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Introduction to Quantum Mechanics Unit I

Contents

1. Failures of the classical theory

a. Maxwell’s equations and electromagnetic waves
b. Blackbody radiation
c. Atomic spectra
2. Dual nature of radiation – Compton Effect
3. Matter waves
4. Double slit experiment with particles
5. Heisenberg’s uncertainty principle
6. Concepts of wave functions

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1. Maxwell’s equations in a Dielectric medium

Maxwell summarized the existing ideas of electric and magnetic fields and their inter-related
phenomena into four equations (in 1860) which are known as the Maxwell’s equation. This also
paved the way for describing radiation as an electromagnetic wave.

Integral form Differential form

Gauss’s law for electric charges

1. Gauss’s law states that the integral over the 1. Divergence of the electric field is given by
closed surface of the electric field normal to the charge density divided by
the surface should be a measure of the ( Gauss’ law for electric fields)
charge enclosed by the
surface

Gauss’s law for magnetic charges

2. Since magnetic mono poles do not exist and 2. Divergence of the magnetic field is uniformly
the magnetic lines of force are continuous zero
the integral form of Gauss’s ( Gauss’ law for magnetic fields)
law for magnetic fields This implies the absence of magnetic
monopoles
Faraday’s law of electromagnetic induction

3. A time varying magnetic field induces an 3. The curl of the electric field is equal to the
electric field in a closed loop enclosing a rate of change of the magnetic field which is
surface described by the standard Faraday’s law of
electromagnetic induction.

Ampere - Maxwell circuital law (Ampere’s law modified by Maxwell )

4. An electric current or a changing electric flux 4. The curl of the magnetic field is given by the
through a surface produces a circulating current density through the closed loop and
magnetic field around any path that bounds the displacement current
that surface

This equation is an extension of the

Ampere’s law. The second term represents
the concept of displacement current
associated with time varying electric fields
which is Maxwell’s contribution.
…………………………………………………………………………………………………………………..
[A review of the operations with the operator (nabla operator)

The operator is a partial differential mathematical operator and is given by

where are the unit vectors in the three orthogonal directions.

The operator when operates on a scalar quantity gives rise to a vector. For
example the operator operating on an electric potential gives the electric field at the point

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The dot product of the operator with another vector field gives rise to the divergence of the defined
field or the rate of change of the field in the three orthogonal directions.

The cross product of operator with a vector field gives the curl of the field and results in a vector
which is perpendicular to both and the given vector.

Another important identity with the operator is the curl of the curl of a vector ie.,

where is the scalar Laplacian operator.]

Maxwell’s equations in free space

In the case of free space (which does not have sources of charges and currents) then the
Maxwell’s equations reduce to

…..……………………………………… (1)
…………………………………………. (2)

…………………………………………. (3)

…………………………………………. (4)
Taking the curl of curl of the electric field the equation can be written as

this reduces to

Since this reduces to

Substituting for curl of B the above equation simplifies to .

But is the general form of a wave equation. Maxwell observed

and the equation reduces to concluded that the wave equation should be an electric field in

free space travelling at the speed of light

In a very similar way, starting from the curl of the curl of the magnetic field the analysis yields

. This describes a transverse magnetic field vector travelling at the speed of

light.

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The electric and magnetic waves must therefore be representing light and hence Maxwell
proposed that light could be treated as electromagnetic waves, where the electric and magnetic
vectors are mutually perpendicular and perpendicular to the direction of propagation of the
radiation.

Consider a 1D electric wave Ex associated with radiation propagating in the Z direction which can
be represented as

This implies that the electric field vector has only an x component and the other two components
Ey and Ez are zero.

Hence the associated magnetic component of the EM wave can be evaluated using the Maxwell’s
third equation namely

Evaluating the curl of the electric field

this implies that

Integrating the above equation with respect to time t we get

since is the velocity of the radiations. We note that the magnetic component of the EM
wave has only the Y component and the magnitude of the wave is times the magnitude of the
electric component of the wave. Thus we conclude that the EM waves have a coupled electric and
magnetic field components which are mutually perpendicular to each other and both are
perpendicular to the direction of propagation of radiation.

Energy of EM waves

Classically the energy of waves is equivalent to it’s intensity which is square of the amplitude of
the waves.

The energy associated with an Electric field per unit volume of free space is where E
is the electric field.

The energy content of the electric component of the wave

The energy content of the magnetic component of the wave

Hence the total energy content of the wave is the sum of the two components .
Poynting Vector

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EM waves carry energy in the direction perpendicular to the E and B field variations and is
described by the Poynting vector

Ex is however a time varying component and hence to determine the average energy of the wave
transmitted per unit time through unit area can be found out as

This implies that the average energy content of EM waves to be proportional to the square of the
amplitude of the electric or magnetic vector and is independent of the frequency of the waves.
Thus the classical picture of the EM waves as carriers of energy gives a picture of frequency
independence. For this reason some of the observed phenomena of interaction of light with matter
could not be consistently explained in spite of the fact that all other observed phenomena of
radiation could be explained by the Maxwell’s EM wave theory.

Polarisation states of EM waves

Light is a transverse electromagnetic wave, but natural light is

generally unpolarized, all planes of propagation being equally
probable.

Light in the form of a plane wave in space is said to be linearly polarized. The
addition of a horizontally linearly polarized wave and a vertically polarized wave of the
same amplitude in the same phase result in a linearly polarized at a 45o angle

If an electromagnetic wave is the result of the superposition of two plane waves of equal
amplitude but differing in phase by 90°, then the light is said to be circularly polarized..

If two plane waves of differing amplitude are related in phase by 90°, or if the relative
phase is other than 90° then the light is said to be elliptically polarized.

2. Black body radiation in equilibrium

Classically the interaction of radiation with matter with radiation is manifested in the way materials
absorb radiations, emitted characteristic wavelengths which gives the color of the material.

Gustav Robert Kirchhoff studied the absorption properties of materials and found
materials which absorb all incident rays. If such a material is heated then it would
emit all wavelengths of radiation as it absorbed. Such a material is defined as a
black body.

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A black body is also modeled as a cavity which does not allow

any incident radiation to escape due to multiple reflections
inside the cavity. This cavity when heated radiates emit
radiation of every possible frequency at a rate which increases
with temperature of the body. The amount of radiant energy
does not increase monotonically with time but is limited by the
rate at which the radiation is also absorbed by the cavity. The
amount of energy emitted at a frequency is limited by the
thermodynamic equilibrium of the absorption and emission
processes. It was also observed that the radiation density in an
equilibrium state depends only on the temperature of the walls of the cavity and does not depend on
the material or structure of the wall. The distribution as well as the characteristic maximum
wavelength shifts to the lower wavelength side.

The emission from the blackbody at different temperatures can be modeled as due to emissions from
harmonic oscillators on the surface of the walls of the cavity. Due to physical dimension of the cavity
being large and the number of harmonic oscillators on the surface of the cavity is also large, a large
range of wavelengths can be emitted. The emission from the cavity is limited to those radiations
which can form standing waves in the cavity. Consider a cubical cavity of side a. The wavelengths of
the oscillations that can resonate in the cavity has to satisfy the condition .

Or the frequencies that can sustain in the cavity is given by

The additional no. of frequencies when n changes to dn is

a
In a three dimensional cavity the number of frequency states that can
resonate in the cavity is estimated as where

is the radius of a surface in n space for allowed

values of nx , ny and nz. This surface is an octant of a sphere of radius
r. The volume of this octant is a measure of the number of frequency states in the cavity up to the
allowed nx , ny and nz

The additional no. of frequencies when r changes to dr is

The energy density of radiations can be estimated if the number oscillators and their average energy
can be estimated. In the case of a cavity of volume V we can estimate
the number of oscillators with frequencies between
as .

Thus the density of frequency states (number of states per unit volume)
with frequencies between as

The Boltzmann distribution function describing the probability of a large

number of oscillators with energy E in thermal equilibrium at

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temperature T is given by .

Rayleigh and Jeans estimated the average energy of the oscillators using the Boltzmann distribution
function as

This integral for E varying from 0 to infinity gives the equi-partition of energy .

Thus the energy density (energy per unit volume) of radiations with frequencies between
can be estimated as

This predicts that the intensity of radiations of a particular frequency should increase as and at
very high frequencies the intensities must be infinite at any temperature T. This is the Rayleigh
Jeans law which is in contradiction with the experimental observations and termed as the ultra violet
catastrophe.

A solution to this problem was provided by Max Planck in 1900 when he proposed that the energy of
the harmonic oscillator are restricted to multiples of the fundamental natural frequency times a
constant ) ie., . Thus the radiations are from a collection of harmonic
oscillators of different frequencies and the energy of the radiations from the oscillators has to be
packets of . Hence the average energy has to determined as a summation of the probabilities of

the energy of the individual oscillators to the total probability as

With this concept of energy of the radiations the average energy of the oscillators can be evaluated
as and the energy density of radiations can be evaluated as

The decrease in the intensity at higher frequencies (smaller wavelengths) can be attributed to the fact
that the excitation of the oscillators to the higher energy states is less probable at lower temperatures.
At higher temperatures the thermal energy enables oscillations at higher frequency . This
expression gives excellent co-relation with experimental results which was a milestone. At very low
frequencies this expression reduces to the Rayleigh Jeans expression.

Thus Max Planck had unknowingly laid the foundation for quantization of energy states of a system
though not in the currently understood terminologies.

Atomic Spectra

Atoms of different elements have distinct spectra and therefore atomic spectroscopy allows for the
identification and quantization of a sample's elemental composition.

Robert Bunsen and Gustav Kirchhoff discovered new elements by observing their emission spectra.
Atomic absorption lines are observed in the solar spectrum and referred to as Fraunhofer lines after
their discoverer. The existence of discrete line emission spectra or the absence of discrete lines in an

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absorption spectra puzzled scientists since the

atomic model had not evolved at the time of these
Photo electric effect – an experiment in which radiation
observations. (electromagnetic waves) interact with matter where
emission of electron from the metal when radiation of
Classical physics tried to model the emission from wavelengths lesser than a cutoff wavelength. The
electron emission was instantaneous and the kinetic
atoms as that due to the orbiting electron, since energy of the emitted electrons depended only on the
an accelerated charge should emit wavelength of the incident radiation and not on the
electromagnetic radiation --- light. However intensity.
according to this model the electron should be These results could not be explained on the basis of the
continually losing energy and fall into the atom in classical EM wave theory. The classical theory
an extremely small time interval. suggests that electrons accumulate of energy from the
incident waves on continuous irradiation and when the
energy of the electrons is more than the work function
(The explanation of the line spectrum of atoms in of the material it is emitted from the metal after a
terms of transition between energy states of an delayed time. According to the classical theory the
kinetic energy of the electrons emitted should be
atom evolved after the quantum model of the dependent on the intensity of the radiation and
atom evolved.) independent of the wavelength. All these explanations
were in contradiction to the experimental results.
Dual nature of radiation
Einstein explained the effect considering light to
behave as particles called Photons and the interaction
Radiation is part of the electromagnetic spectrum of the photons with the electrons in the metal can result
and can be described as electromagnetic waves. in transfer of energy to the electron. If the energy
Electromagnetic waves are construed as mutually gained by the electron is greater than the work function
of the metal, then the electron can be emitted and the
perpendicular sinusoidal electric and magnetic kinetic energy of the photo electron would depend on
fields and perpendicular to the direction of the energy of the incident photon. This was a classic
example of radiation displaying a particle nature when
propagation of the waves. The classical concept
the interaction is at atomic / sub atomic particles.
assumed that the energy content of the wave is
proportional to the square of the amplitude of the
waves. The wavelength (and hence the frequency) are not of any consequence with regard to the
energy of the wave. Conventional wave theory of radiation explains the phenomena of reflection,
refraction, interference, diffraction and polarization of light. Interference and diffraction though stands
out as an exclusive wave property.

The Photo electric effect was explained by Einstein as a

particle - particle interaction. This paved the wave for the Relativistic concepts of energy and
momentum of particles
concept of the dual nature of radiations. The particle nature of
Einstein’s concepts of relativistic
radiations could be evident when radiation interacted with
particles (particles moving with
matter at the atomic / sub atomic level. speeds comparable to the speed of
light) are
Compton effect :  A particle it at rest it has a rest
2
mass energy given by E= moc .
Arthur H Compton while studying the scattering of X rays by  A particle moving with a velocity
materials observed that in addition to the emission of an v will have a mass given by
electron, the scattered beam has a different wavelength as
compared to the incident wavelength.
 If p is the momentum of the
The scattering of X rays (the high energy end of the particle then the kinetic energy
of the particle is given by pc
electromagnetic spectrum) if treated classically would not  The total energy of the particle
explain the origin of X rays of higher wavelength. is given by
Compton treated the problem as a particle -particle collision in
which photons of momentum are scattered with an electron at rest. This results in a transfer

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of momentum and energy to the electron which is scattered. The photon loses energy and
momentum which results in a gain in momentum and energy for the electrons. The scattered X-ray
photon has reduced energy which results in an increase in the wavelength.

The analysis of the conservation of energy and momentum conservation before and after the collision
(taking into consideration the relativistic effects for the energy and momentum of the electron) gives
the increase in the wavelength of the scattered photon.

If is the wavelength of the incident X-ray photon, the momentum of the photon is and the
energy of the photon is . Since the electron is at rest the initial momentum of the electron is
zero and the energy of the electron is the rest mass energy moc2.
After the collision the wavelength of the scattered X ray is and the momentum . The energy

of the scattered X ray is .

The momentum of the scattered electron is and the energy is

The energy conservation equation is
From this expression we get …(i)
Since the X ray and the electron are scattered in different directions, the momentum conservation
equation has to be treated as conservation of the momentum of the particles before and after
collisions in the direction of incidence and in a perpendicular direction.
Momentum conservation along the incident direction is
.
Hence ..(ii)
Momentum conservation in a perpendicular direction is

Or …(iii)
Squaring and adding equation (ii) and (iii) we
…(iv)
Comparing equations (i) and (iv) we obtain which can be
simplified to

The change is the wavelength is known as the

Compton Shift.

It is obvious that the Compton shift is

 independent of the incident wavelength of X rays

 independent of the type of the scattering material
 depends only on the angle of scattering of the X rays

The term is termed as the Compton wave length and is a

-12
constant =2.42 x 10 m.

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At θ=0o we notice there is no shift in the wavelength or there is no interaction of the X rays with the
electrons.

At θ=90o the shift

At θ=180o the collision is a head on collision and the shift in the wavelength is maximum .
Thus we conclude that maximum momentum and hence energy transfer happens when the incident
X ray is back scattered.

At other angles the predicted shift were in agreement with the experimental observations.

Thus the Compton Effect was another instance of the particle nature of radiation.

The two experiments of interaction of radiation with matter at sub atomic levels (Photo electric effect
and the Compton effect) led to the conclusion that radiation exhibit a dual nature - show the normal
wave characteristics and a particle at times of interaction of radiation with matter.

Young’s double slit experiment with particles

A double slit experiment with a particles incident on the slits

one at a time reveal some interesting outcomes. Experiments
show that electrons (or photons) as particles are expected to
arrive at some definite location on a screen, unlike a wave.
But if a second electron (or photons) is incident at the slit, the
second electron reaches a different location, often far outside
any experimental uncertainty. If many electrons (or photons)
are incident on the slit but one at a time then the
measurements will display a statistical distribution of
locations that appears like an interference pattern.

The building up of the diffraction pattern of electrons

scattered from a crystal surface. Each electron arrives
at a definite location, which cannot be precisely
predicted. The overall distribution shown at the bottom
can be predicted as the diffraction of waves having the
de Broglie wavelength of the electrons.

This experiment leads to concept of probabilities that

are inherent in quantum mechanical systems. A
quantum particle exhibits probabilistic behavior when there is no effort to detect the particle.

Dual nature of matter

Louis de Broglie (analyzing the results of the dual nature of radiation) put forward the hypothesis that
matter (form of energy) when in motion can display wave characteristics and the wavelength
associated with the moving particle where mv is the momentum of the particle.

Common heavier particles have a wavelength that is beyond the measurement capabilities with the
best of techniques available. For example the wavelength of a carbon atom moving with a velocity of
100m per second could possess a wavelength of the order of 10-10m. This has to be measured with
an experiment characteristic of waves such as diffraction or interference.

The wavelength of the associated waves has to be in the measureable range of an interference or
diffraction experiment to prove the existence of matter waves.

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This concept was experimentally verified by Davisson and Germer who observed unusual scattering
characteristics for electrons scattered by a Ni crystal when the accelerating potential of the electrons
was 54V and angle of scattering 50o.

The de Broglie wavelength of electrons accelerated by 54V can be estimated to be 1.67 x 10-10m. If
the electron wave possess such a wavelength, it should be possible to diffract the waves with a
known crystal.

If the scattering has to be explained as a diffraction phenomena (characteristics of waves) following

Bragg’s law, then where d in the interplanar distance of the Ni crystal, the glancing
angle (angle between the incident ray and the surface of the crystal) and is the wavelength of the
“waves”.

This yields a wavelength which is close to the value obtained using de Broglie’s hypothesis (matter
waves). Since diffraction is characteristic of waves, it was concluded that electrons undergo
diffraction under the set experimental conditions.

Thus it is concluded that matter display dual characteristics at appropriate conditions of interaction.
This concept has been further confirmed by diffraction experiments using heavier particles such as
the slow neutrons from a nuclear reactor.

Hitachi in the 1980s showed the diffraction of electrons when scattered by a thin wire. It was
observed that the electrons scattering patterns are very close to a diffraction pattern produced by a
double slit experiment.

Wave packets

The concept of matter waves requires a wave like (mathematical) representation of the moving
particle where position and momentum of the particle can be estimated with reasonable accuracy.
Sinusoidal representations result in a gross uncertainty in the position while providing a highly
accurate estimation of the momentum.

The superposition of two waves of very close frequency and propagation constant results in a wave
packet, frequency and propagation constant . Let y1 be a sinusoidal wave with angular
frequency ω and propagation constant k and y2 be a wave with frequency and propagation
constant k+Δk.

and + =

The superposition of the two waves gives a resultant

The first part is the original high frequency component and the second term is a low frequency
component. This is the case of an amplitude modulated wave where the amplitude of the high
frequency component is modulated according to the amplitude of the low frequency component.
Since this is the resultant of a group of super imposed waves, it is referred to as a wave packet.

The momentum of the particle could be evaluated with the estimation of the wavelength of the waves
in the wave packet. The position of the particle could be inferred from the region in which the
amplitude (and hence the intensity) of the wave is a maximum.

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This gives a reasonably accurate value of both momentum and position. The momentum is derived
from the wavelength of the high frequency component and the position from the region of maximum
amplitude of the wave packet.

We can define both a phase and group velocity for the wave packet.

The phase velocity of the waves is defined as the velocity of an arbitrary point marked on the wave
(the high frequency component) as the wave propagates and is given be

The velocity of the wave packet (wave group) is defined as the group velocity and given by .

Relation between group velocity and particle velocity

Group velocity of waves =

The angular frequency where E is the energy of the wave and hence
The wave vector where p is the momentum and hence
Therefore the group velocity

Since the group velocity where v is the particle velocity.

Relation between group velocity and phase velocity

The group velocity of the particle is given by

However .

And hence

In a dispersive medium (where the velocity of the waves depends on the wavelength) the group
velocity is given by the above equation.

When the group velocity of the wave packet is equal to the phase velocity the medium in which the
wave propagate is a non dispersive medium. In this case or the phase velocity is a constant
with respect to wavelength.

Evaluate the condition under which the group velocity of a wave packet is

i) Half the phase velocity and ii) twice the phase velocity

The group velocity of a wave packet is given by

Case 1.

This on integration yields or

This implies that the phase velocity is proportional to the square root of the wavelength

Case 2.

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This on integration yields or

This implies that the phase velocity is inversely proportional to the wavelength.

Uncertainty principle

Heisenberg’s analysis of the wave packet revealed the spread

in the estimation of the position and the spread in the
propagation constant of the wave is intrinsically related. A
Fourier transform of the wave functions gives the distribution of
the propagation constant. In summary he product of the The more precisely
standard deviations in the estimates of the position and the
the position is determined, the
propagation constant was shown to be greater than or at the
less precisely the momentum is
most equal to . known in this instant, and vice
This then translates to the standard form of the uncertainty versa.
principle when the propagation constant is transformed to the Heisenberg, uncertainty paper,
momentum through the relation 1927

The position and momentum of a particle cannot be determined simultaneously with unlimited
precision. If one of the parameter is determined with high precision then the other must necessarily
be imprecise, such that the product of the uncertainties is greater than or equal to ie.,

where is the uncertainty in the position and is the uncertainty in the momentum determined
simultaneously.
The uncertainty relation for energy E and time t for a physical system can be written as

Where is the uncertainty in the energy E of a system and is the uncertainty in the time in which
this energy is estimated.

In the case of rotational motion the uncertainty relation between the angular position θ and the
angular momentum L can be written as

where is the uncertainty in the angular position and is the uncertainty in the angular momentum
determined simultaneously.

Heisenberg’s Gamma ray microscope:

Heisenberg proposed the gamma ray microscope (as a thought experiment

to illustrate the uncertainty principle) to locate the position of an electron. To
be able to “observe” the electron, it should be “illuminated” by a radiation
whose wavelength is comparable to the size of the scattering object. Hence
it is evident that one should use rays to observe electrons, which in turn
scatter the radiation onto the objective lens of the microscope. In order that
we “see” the electron, the limit of resolution of the microscope should be
comparable to the position uncertainty .

Thus, we get .

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However, when the high energy rays can impart momentum to the electrons (following the
principles of Compton Effect).

The maximum momentum imparted to the electron can be estimated from the
maximum shift in the scattered photon momentum. If the momentum of the photons
entering the microscope at a half cone angle is then the maximum momentum
gained by the electron in the x direction would be

Thus the minimum momentum of the electron would be uncertain by a factor

Thus the product of the uncertainties conforms to the uncertainty

principle.
This illustrates that in the simultaneous determination of the position and momentum of an electron
results in an inherent uncertainty.

Electron’s existence inside the nucleus

The uncertainty principle can be used to illustrate the impossibilities in physical systems or the
correctness of assumptions. Beta particle emission from radioactive nuclei is one such example.
Experiments show that the beta emission is the emission of an electron with a high energy of about
4MeV by radioactive nuclei. If we assume the electron to be an integral part of the nucleus then we
may be able to estimate the minimum energy of the electron using the uncertainty principle.

If the electron is part of the nuclei, then the position of the electron is uncertain to the extent of the
nuclear diameter. The uncertainty in the position of an the electron

The minimum uncertainty in the momentum of the electron then can be estimated as

Hence the minimum momentum of the electron p has to be at least the uncertainty and hence

The kinetic energy of the electron

This implies that the energy of the electron emitted by the radioactive nuclei should be quite high if
electron had to be integral member of the nuclei. Since the energies of the electron emitted by
radioactive nuclei are very less compared to the estimate, we conclude that the electron cannot be a
permanent part of the nuclei, thus illustrating the power of the uncertainty principle.

Wave functions

A moving particle can be represented by a wave packet. The wave packet can be described by a
function . The function though contains information about the physical state of the
system, has no other physical interpretation. However since the amplitude of the wave gives
information on the probable position of the particle, can be termed as the probability amplitude.

The functions should have the following characteristics if it has to be representing a moving particle.

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(1)  must be finite, continuous and single valued in the regions of interest.

Since is a probability amplitude (as discussed in the definition of a wave packet) it

is necessary that the function is finite and highly localized. The value of the function could
change with the position of the particle but has to remain finite and continuous (as is obvious
from the fact it is representing a wave packet). Since the physical parameters of the system
are single valued the wave function has also to be single valued.

(2) The derivatives of the wave function must be finite, continuous and single valued in the regions of
interest.

Since the wave function is a continuous function, the derivatives of the function with respect to
the variables must exist. For example is the function is a plane wave given by
then the derivative of the function is . It is obvious that the derivative of
the function would inherit the properties of the wave function, and hence it has to be finite
continuous and single valued.

(3) The wave function  must be normalisable. i.e.  

The wave function is a probability amplitude and the intensity of the wave (the point
at which the energy of the wave is likely to be concentrated) is the square of the probability
amplitude. Since the wave function can be a real or imaginary function it is evident that the
square of the wave function . is the complex conjugate of the wave function.
Thus the product is representative of the intensity of the wave or the probability of finding the
particle at any point in the wave packet and is called the probability density. Thus the
summation of all the for the extend of the wave packet should give the total probability
of finding the particle. Thus . Since the wave function is a continuous function it
can be written as

Since the wave function is highly localized this implies that the wave function vanish
at large values of x, ie .

Hence the integral evaluated over all of space . This is called as the
normalization of the wave function. In effect the amplitude of the wave is normalized to ensure
that the total probability of finding the particle is always equal to one.

The wave function satisfying the above conditions are called as well behaved functions and can be
used to represent the physical state of a system in quantum mechanics.

Linear superposition of wave functions

The classical Young’s double slit experiment with light waves demonstrates the diffraction of light.
The experiment conceived photons passing through two closely space slits diffract and forms an
interference pattern at the screen.

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The experiment can be repeated with single photon emitted one at a time allowed to be incident on
the slit with no particular alignment. The photon decides to enter either slit 1 or slit 2. The state of the
photon entering slit 1 is described by the wave function and represents the wavefunction of the
photons passing through slit 2. When slit 2 is closed and slit 1 is opened all photons preferentially go
thru slit 1 and the intensity pattern of the photons on the screen will be given by P 1.Similarly when slit
1 is closed and slit 2 is opened all photons
preferentially go thru slit 2 and the intensity pattern of
the photons on the screen will be given by P2.
However, when both the slits are opened the intensity
pattern do not add up. The resultant is an interference
pattern which can be described as the superposition of
the two photon states given by .

If the photon beam is replaced by an electron bean

then it is interesting to note that the electrons going
through the slit behaves as waves, and form an
interference pattern on the screen.

This clearly demonstrates the wave behavior of electrons. If represents the electrons passing
through slit 1 and represents the electrons passing through slit 2, then the probability densities
.

It is obvious that represents the super imposition of the two states or super position of
the two wave functions. Thus the probability density of the new states should be given by

and is not equal to .

Thus we observe that the superposition of the two states yield a new state.

In general if and are two wave functions of a system then the linear superposition of the two
states is also a possible quantum state

Another interesting observation is that any attempt to observe the events of which or how many
electrons pass through each slit with a detector, the
interference pattern is not observed and the intensity
of is a algebraic addition of the two intensities from
the two slits. Thus the superposed wave function
is non-existent and the wave function is said to have
collapsed.
Schrödinger's cat: a
Thus quantum systems are not subjected to
cat, a flask of poison, and a radioactive source are
observations, since the observations interfere with
placed in a sealed box. If an internal monitor (e.g.
the quantum behavior of the particles being studied.
Geiger counter) detects radioactivity (i.e. a single
This is the classic Schrodinger’s cat paradox.
atom decaying), the flask is shattered, releasing the
Observables, poison, which kills the cat. The Copenhagen
interpretation of quantum mechanics implies that
The physical parameters associated with the particle after a while, the cat is simultaneously alive and
such as energy, momentum, kinetic energy, spin, etc dead. Yet, when one looks in the box, one sees the
are observables of the state of a system. cat either alive or dead not both alive and dead. This
Experimental results can give us values of poses the question of when exactly quantum
observables, and multiple measurements on the superposition ends and reality collapses into one
possibility or the other.
system at the same state should result in the same
value or average values for the observables, if the

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state of the system is not modified by the measurement. Observables have real values and their
accurate measurements would be limited by the principles of uncertainty.

The wave function carries information about the state of the system or the observables can be
extracted from the wave functions with the help of appropriate operators.

Operators

The wave function describing a system in one dimension can be written as . This
function contains information about the observables of the system. The values of the observables can
be inferred using a mathematical operator operating on the wave function.

Differentiating with respect to position yields

Or the operation yields the momentum of the system.

is the momentum operator.

Differentiating the above yields

Rearranging the terms and dividing by 2m we get

Thus the operation } on yields the kinetic energy.

is the kinetic energy operator

Differentiating with respect to time yields

Or the operation yields the total energy of the system

is the total energy operator.

The position operator is equivalent to multiplying the wave function by x itself .

The potential energy operator is not explicitly defined as the potential can be inferred if the total
energy and the kinetic energy of the system is known.

Expectation values:

Quantum mechanics deals with probabilities and hence predicts only the most probable values of the
observables of a physical system which are called the expectation values. These expectation values
could be the average of repeated measurements on the system. The method of evaluating the
expectation values is outlined as below.

Let be an operator that gives the value of the momentum p when it operates on the wave function

The operation where p is the value of the observable extracted from . If the
many values of p are extracted from the wave function has to be averaged for the extend of the wave
packet. The same can be obtained by integrating the expression which gives

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where is the most probable value of the momentum.

Thus the expectation value of the momentum is written as .

If the integral is over all of space then the integrals could be evaluated between limits of . In this
case it is observed that the denominator would be the total probability and can be written as 1.
However, it is customary to write the expectation values in the standard form.

In the general case for an operator when and hence the expectation value of the parameter can
be written as .

Conclusions of Unit I

 Brief review of Maxwell’s equations and the concept of EM waves propagating in

free space lead to the conclusion that the energy of EM waves is proportional to the
square of the amplitude of the waves – which does not explain some interactions of
radiation with matter
 The blackbody radiations as due to a cavity of harmonic oscillators to explain the
energy density of radiations by a blackbody was understood. The explanations by
Max Planck on the nature of emission as packets of energy hv – the fundamental
idea of quantization.
 The Dual nature of radiation explains experiments of interaction of radiation with
matter at sub atomic levels. Discussions of the Compton Effect (relativistic
interactions) to support the dual nature of radiations.
 Dual nature of matter – Louis de Broglie’s hypothesis
 Matter waves – wave packets to represent moving particle – superposition of
waves. Ideas of phase and group velocities of wave packets
 Heisenberg’s Uncertainty Principle – evolution of the uncertainty principle from the
discussion of the wave packet. Implications on physics at small scales.
 Well behaved wave functions, Observables, operators and expectation values in
quantum physics.

Some typical solved Numericals

1. Estimate the Compton shift for X rays scattered at 95o with respect to the incident direction. If the
momentum of the scattered X rays is 4.5x10 -24 kg m s-1 estimate the wave length of the incident X Rays.

The Compton shift is given by .

At 95o the shift is

Wave length of the scattered X ray is

Wavelength of the incident X ray is

2. In a Compton scattering of electrons with X rays discuss the condition under which the energy gained by
the electron is maximum. If the wave length of the incident X ray is 0.1nm calculate the maximum
energy gained by the electron.

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In Compton effect the maximum shift in the wavelength of the scattered X ray is when the angle of
scattering is 180o. At this angle there is a head on collision with the electron and the electron moves
along the incident direction. The momentum and hence the energy transfer is maximum under this
condition.
The maximum momentum transfer is the maximum momentum loss of the X ray
= = 3.07 x10-25 Kg m s-1.
The maximum energy transfer is the maximum energy loss of the X ray
= = 575.16eV
3. Estimate the energy of a non relativistic electron if it to confined in a region of width 10 -14 m and
calculate the de Broglie wavelength of the electron with this energy. Comment on the results obtained.
The position of the electron is uncertain to the extent of 10-14 m.

The minimum uncertainty in the momentum =p

The kinetic energy of the electron

.

The de Broglie wavelength > the width of the region, which implies the electron cannot be confined in
this interval.
4. Find the spread in the wavelength of a photon whose life time in the excited state is uncertain to 10-10s
if the wavelength of emission is 541nm.

The uncertainty relation for energy E and time t for a physical system can be written as
where is the uncertainty in the energy E of a system and is the uncertainty in
the time in which this energy is estimated.

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Numericals

1. Find the de Broglie wavelength of electrons moving with a speed of 107 m/s (Ans 7.28 x 10-11 m)
2. Compare the momenta and energy of an electron and photon whose de Broglie wavelenth is
650nm (Ans Ratio of momenta =1; ratio of energy of electron to energy of photon = )
3. Calculate the de Broglie wavelength of electrons and protons if their kinetic energies are
i) 1% and ii) 5% of their rest mass energies. (Ans Rest mass energy of electron = 8.19x 10-14 J; rest
-10
mass energy of protons = 1.503 x 10 J. The de Broglie wavelength Electron 1%
Electron 5% Proton 1% Proton 5%
)
4. An electron and a photon have a wavelength of 2.0 A. Calculate their momenta and total
energies.
5. What is the wavelength of an hydrogen atom moving with a mean velocity corresponding to
the average kinetic energy of hydrogen atoms under thermal equilibrium at 293K?

6. The frequency of Surface tension waves in shallow water is given by ν= (2πT/ρλ 3)½, where T
is the surface tension, ρ is the density of the medium and λ the wavelength of the waves. Find
the group velocity of the waves.
7. The relation between the wavelength λ and frequency ν of electromagnetic waves in a wave
guide is given by λ=c/√(ν2 - νo2). Find the group velocity of the waves.
8. The speed of an electron is measured to be 1 km/s with an accuracy of 0.005%. Estimate the
uncertainty in the position of the particle.
9. The spectral line of Hg green is 546.1 nm has a width of 10-5 nm. Evaluate the minimum time
spent by the electrons in the upper state before de excitation to the lower state .
(Ans: )

10. The uncertainty in the location of a particle is equal to it's de Broglie wavelength. Show that
the corresponding uncertainty in its velocity is approx one tenth of it's velocity. (Ans:
Hence )
11. Determine the maximum wavelength shift in the Compton scattering of photons from protons.
(Ans = 2.64×10−5 Å)
12. Show that for a free particle the uncertainty relation can also be written as
where Δx is the uncertainty in location of the wave and Δλ the simultaneous uncertainty
in wavelength.

https://www.youtube.com/watch?v=yy6TV9Dntlw

https://www.youtube.com/watch?v=EA-wcFtUBE4

https://www.youtube.com/watch?v=vkVnnN0MjIE

https://www.youtube.com/watch?v=cugu4iW4W54

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