To experimentally verify the Bohr’s theory of quantized energy levels in an tom and the critical
potentials an experiment was designed . The experiment is known as the Franck-Hertz experiment.
Experimental set up:
Gas of the element to be studied and mercury vapor inside the gas tube T
Filament F produces electrons
Electronsare accelerated by applying a potential between F and G (grid)
Potential difference is varied from 0 to 60 volts bya potentiometer .
P is plate where the electronsare collected
Plate P is kept at a small negative potential sothat it can collect electrons.
Electrons with kinetic energies greater than the potential between G and P can reach the plate P.
The milliammeter measures the current flowing in the circuit.
The readings are taken with accelerating potential on the x axis and plate current on yaxis
The collector current first increases almost linearly andthen drops at 4.9 V and then again increases and
again drops at 9.8 V .The pattern continues as the accelerating potential is increased.
The electron gun produces a beam of electrons that move slowly from F towards the plate. In course of
motion they collide with mercury atoms . Only those electrons will reach the plate which do not suffer
any energy loss due to collision. If the accelerating potential is increased the energy of the electrons
increase and they reach plate producing current. The increasing current from zero shown in the graph is
due to increase in accelerating potential. As the accelerating potential reaches 4.9 volts the lectrons
gain 4.9 V of energy. The electrons with 4.9 V of energy when collide with the mercury atoms they lose
all their energy by inelastic collision with mercury atom. Thus the electron loses all its energy and cannot
reach plate P. This causes a sharp fall in current at aceelrating voltage equal to 4.9 V. The mercury atom
absorbs the energy from the electron to rise to an higher energy state . This suggests that the critical
potential or the energyrequired to raise a mercury atom to high energy state is 4.9eV. This confirms the
critical potential .
There are a large number of electrons that are emitted from the filament. It is not necessary that every
electron is undergoing collision with mercury atoms and they can reach the plate as the potential is
increased this causes an increase in the current after 4.8 V. At 9.8Volts the electrons collide inelastically
with two mercury atoms and lose their energy in exciting two mercury atoms thus the current again fall
at 9.8 volts .
At each dip (point where current falls ex 4.9 V, 9.8v) Hg atoms are being excited to higher excited states.
This leads to the conclusion that the energy levels of Hg atoms are quantized .The HG atoms will go to
excited state only when an electron of exact energy will undergo inelastic collision with the Hg atom.
The graph pattern will continue as more andmore number of Hg atoms will absorb energy to go to
higher energystate. Thus Franck Hertz experiment proves the quantization of energy levelsof Hg atoms.
Sommerfield correction (model)
(1) Fine structure of Hydrogen spectra : Fine structure means that the lines in the hydrogen spectra
that appeared as single lines were actually made of more than one lines, For example H alpha
line was observed to be made of two lines separated by 0.13 angstorm.
To explain the fine structure of the H alpha line Sommerfield introduced some corrections in the
existing Bohr’s theory.
General principle of quantization
1 general principle of quantization as explained by Sommerfield states that
For any physical system for which the coordinates are periodic functions of time then for each
coordinate there exists a quantization condition
Q is one coordinate and pq is the momentum asscociated with q, nq denoted the quantum number for
the coordinated q the integration sign denotes that the integral is taken over one period of the
coordinate.
We consider an example of harmonic oscillator where a mass is attached with a spring and constitutes
shm the total energy is
The mass is along the x axis so we consider the x component of momentum and the x ccordinate the
above equation can be expressed as
The equation is that of an ellipse. At a particular time the state of anysystem of the harmonic oscillator
is defined by the momentum and position so x and px are the coordinate that define the system at any
instant of time a space where the system is defined by coordinates x and px is called phase space since
the equation is of an ellipse so we can write
Area of ellipse is piab.
The second diagram shows the allowed energies of the harmonic oscillator or any physical system.
Thus sommerfield concluded from the general general quantization rule that the orbits are elliptical in
shape this was Sommerfield’s first correction that is the concept of elliptical orbits . thus for the area we
have two equations
For a simple harmonic oscillator we have the condition
Nu is the natural frequency of oscillation
Using this relation we have
And from Sommerfield’s general condition of quantization
This planck’s law
By similar theory Bohr’s postulate of quantization of angular momentum can be explained on the
basis of sommerfield’s general condition of quantization . L is angular momentum . The equation says
that the angular momentum along an orbit will be quantized
The allowed orbits arethose orbits for which the circumference is exactly equal to the De-Broglie
wavelength of the moving electron
Sommerfield’s concept of elliptical orbits
Sommerfield using his concept of general quantization rule explained the presence of elliptical orbits
for the electro in an hydrogen atom. Sommerfield evaluated the energy radius and angular
momentum of the electron on elliptical orbits . He used the quantization condition of Bohr to find the
quantum numbers asscociated with angular momentum and energy , As the motion in the circular
path is described by two coordinates r and theta so according to general quantization rule the
momenta corresponding to r and theta will be quantized thus we have two equations of quantization
for two coordinates r and theta
So now there are two quantum numbers nr and ntheta so the geometrical interpretation of this
condition is that the electron orbit will be such that there are two radii. Thus the path will be an
ellipse. The first equation leads to the condition
The second condition is the relation between L angular momentum and the axes a and b (semi major
and semi minor axes of the ellipse) in terms of the semi major and semi minor axes the quantization
of angular momentum was expressed by sommerfield as
thus we now have two
quantum numbers to define the state of an electron in a given orbit . Using the same formalism as
that in the Bohr’s theory Sommerfield calculated the energy radius and other quantities using the
general quantization rule and using the two quantum numbers nr and ntheta
Sommerfield’s elliptical orbit theory defines the principal quantum number of Bohr’s theory as the sum
of nr and ntheta
So sommerfield suggested from his calculations that the orbit in Bohr’s theory that was specified by a
quantum number n is actually made up of more than one orbit which means the quantity n of Bohr’s
theory is a sum of nr and n theta
For example in Bohr’s theory suppose an electron makes a transition from n=3 to n=2 then one line will
appear corresponding to this transition . In sommerfield’s correction of general quantization rule the
levels n=3 and n=2 are composed of many levels or orbits
n=3, according to Sommerfield’s correction this level can result from the following combinations
1st possibility n=3: nr=0 and ntheta =3
2nd possibility nr=1 ntheta =2
3rd possibility nr=2 and ntheta =1
The above three possibilities denote the possible orbits for a given value of n of which one is necessarily
circular and the others are elliptical similarly the orbit n=2 has the following possibilities
N=2: nr=0 ntheta =2
N=2: nr=1 ntheta=1
Thus the transition in Bohr’s theory that appeared as a single lone can be a composition of many
transitions in Sommerfield theory.
Among all the possibilities or orbits the nr=0 defines a circular orbit or Bohr orbit.
N is called principal quantum number and ntheta is called azimuthal quantum number. In the above
diagrams only ntheta is shown because the semi major and semi minor axes of the ellipse are related
by the ratio of ntheta and n
What about the energy of electron in a given orbit in Sommerfield theory?
Energy of an electron in given orbit is determined by the principal quantum number “n”
According to sommerfiled theory the energy of the electron in an orbit is completely determined by n
For a given value of n the possible orbits that are elliptical and circular in shape have same energy this
is called degeneracy.
Degeneracy means that all the orbits for a given value of n have same energy.
Sommefield’s Second correction:
Relativistic treatment: Sommerfield fixed the degeneracy issue by considering the relativistic picture.
In the relativistic situation the electron moving in an orbit of hydrogen atom is considered to have
relativistic speeds
if we consider the relativistic speed of electrons then there will
be a change or correction in the expression for mass (relativistic mass) and hence there will be a
change in the expression for energy En. The change in energy due to the relativistic speed of electron
should be able to explain the splitting of energy levels of hydrogen which are observed as fine
structure. Here splitting means each line in the spectrum of hydrogen is composed of many lines .
Each level of hydrogen atom is split due to the relativistic motion of electron which accounts for the
splitting of lines or the fine structure. After a large calculation sommerfield introduced the expression
for energy taking into account the relativistic speed of electron
The quantity α is called the fine structure constant The value of α is
Α is the quantity that determines the extent of splitting in the energy levels or lines of the hydrogen
atom spectra.
The above diagram is showing Hydrogen atom energy levels according to Sommerfield’s theory or
correction. The transition form n=2 to n=1 in Bohr theory was a single line but in Sommerfield theory
the transition appears as two line s. Same is true for the other transitions. Similarly for n=3 which is
composed of three orbits there are three transitions in Sommerfield’s theory where there was only
one in Bohr’s theory. Solid lines show the transitions that were already observed in Bohr’s theory .
Dotted lines show the transtions that occur according to Sommerfield model and can be observed as
fine structure . The dotted line transtions not always occur but have certain conditions that must be
fulfilled for these transitions
The above equation is called the selection rule , the transitions where these conditions are fulfilled are
called allowed transtions,
Matter Waves and De-Broglie Hypothesis
Planck’s law states that energy is transmitted as electromagnetic radiation in the form of particles
called quanta. Quanta is the particle of energy. When we say particle this means that the object is
localized or found at a fixed point. On the other hand a wave is spread in a region. De-Broglie
proposed that if radiation can behave as particle then it is also possible that the particle behaves or
has a wave nature asscociated with it . De Broglie presented his hypotheis of wave particle duality.
According to De Broglie’s hypothesis particle and wave are related to each other. A moving body is
asscociated with awave similar to a radiation that has particle nature.
A body in motion carries a wave along with it. De Broglie considering Planck’s law of radiation used
the equation
The momentum can be related to the wavelength of the wave
Here p is the momentum of the particle and λ is the wavelength of the wave. So De Broglie
connected the particle to a wave using the pPlanck Constant. So the first conclusion we reach at is
that the wave nature of matter or particle is considerable only in the atomic world. So
Λ is called De-Broglie wave or matter wave . It is the wave length of the wave that is associated or
moving with a body whose momentum is p (mv). The situation is similar to that of wave optics
where diffraction is observable only when the obstacle is of the size of the wavelength of light . The
matter wave is observed only when the moving body has mass comparable to Planck Constant
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De Broglie’s hypothesis is called the wave particle duality . According to this wave and particle nature
are complimentary , which means at a particular situation or experimental condition only one aspect
is observable
The diagram above shows the experimental arrangement of Davisson Germer experiment This
experiment is a conformation of the wave particle duality .
The experiment contains a filament that emits electros which are accelerated by a potential V.The
electrons gain energy eV and fall on a single crystal of Nickel C
The scattered electrons are emitted in different directions that can be detected by the detector D.The
detector D is moving .
Observations
The detector D is set at a particular angle θ and the intensity of the scattered beam is plotted against
potential V . The observations of the experiment were plots of potential on x axis and the intensity of
beam on the Y axis and the plots were obtained as shown in the figure .The intensity of the scattered
beam of electrons was observed as current that was measured in milliampere . Kinetic energy eV is
the energy gained by the electrons due to the variation of accelerating potential V
In the first diagram a peak is obtained at θ=50 when the accelerating potential is about 55 V. This sort
of observation is obtained in the case of diffraction experiments which is totally due to wave aspect
but in this experiment there are electrons which are particles . Then how to explain the observation?
The explanation of the observation is obtained using De Broglies hypothesis if wave particle duality.
The electrons moving with some energy eV are asscociated with a De- Broglie wavelength . There are
waves moving with the electrons . The waves scattered by the crystal are interfering constructively at
a particular θ and thus a peak is observed . The periodic arrangement of the atoms inside the crystal
is causing the constructive interference of the waves associated with the electrons. The result cannot
be explained by considering the electrons as particles but it can be explained only by considering
waves that are moving with electrons. In the experiment the maximum intensity at a particular θ is
due to the waves of a single electron that is scattered by different regions of the crystal.
The periodic arrangement of the Nickel crystal is causing the scattering of a single electron in
different directions and the scattered wave of the electron interefers constructively in a particular
direction. The intensity of the electron beam is kept very low such that at a time only one electron
falls on the crystal . Since the condition of constructive interference is that the path difference
between the two waves must be an integral multiple of the wavelength so this condition must be
fulfilled here in this experiment also
This condition is mathematically expressed as
d is the interatomic spacing or the distance between two atomic planes inside the crystal.
The De-Broglie wavelength of the electrons that are incident of the crystal
Θ is the angle between the incident beam and the scattered beam on the other hand the angle φ is
90-50/2=65
Now using this angle if we evaluate λby putting d= 0.91 angstorm then we have
The agreement between the wavelength values obtained from the rwo results confirms the wave
nature and it also confirms that the maxima at θ=50 is due to the constructive interference of waves .
Next most important result is that the constructive interference or wave like behavior can be obtained
inly when the spacing between the atomic planes of the crystal are of a fixed value . The spacing
between atomic planes is close to the wavelength of the incident electrons thus the situation similar
to that of diffraction of light.
This experiment can be used to study the spacing between atomic planes in a crystal. If we know the
wavelength of the incident beam then by applying the equation
We can find out d. This result is called Bragg’s Law. Bragg’s law is the fundamental law or equation
that is used in the study of crystals by X ray diffraction. In X ray diffraction X rays of a particular
wavelength are made incident on a given crystal and using the Bragg law equation d ofg the crystal
can be calculated.
The above Bragg’equation is not only useful for the study of crytals but can be used for studying all
kind of materials
Principle of Complementarity
The principle of complementarity was proposed by Neils Bohr. The principle of complementarity
states that at a particular experimental condition only one aspect is observable , that is either particle
nature or wave nature . For example in the Davisson Germer experiment only wave aspect is
observable and the particle nature is not observed similary if we are doing the photoelectric effect
experiment than only particle nature is observed and wave nature cannot be4 observed.
According to this principle matter or radiation are not only matter or radiation but they can behave
as radiation and matter . So matter and wave properties are interchangeable .
The link between the wave and particle aspects is established by a probability of wave –particle
duality . Two interpretations of the wave particle duality are known.
1. Einstein’s theory of combining wave and particle aspects of matter
2. Max Born’s theory of wave particle duality in terms of probability
In the case of wave we define intensity of a wave as the energy incident per unit area . here we are
considering electromagnetic wave or radiation. In the case of electromagnetic wave the intensity is
proportional to the average of the square of electric field component of the wave.
when an
electromagnetic radiation fall on any material the electric field interacts with the outer electrons of
the atom of the material.
The electrons experience a force which is equal to
F=Eesinωt this makes the electron to vibrate with the same frequency as the frequency of the
incident electric field .An oscillating electron emits radiation of the same frequency as the incident
electric field. This process is called classical scattering.
In the photon (particle of radiation) picture the intensity is defined as the product of the number of
photons or particles incident on a unit area multiplied by the energy of each photon. One photon of
radiation has energy E=hν if N number of photons are incident then the intensity is I=Nhν
Einstein suggested that the intensity of em radiation can be measured by the average number of
photons that are incident on the surface .
Einstein’s interpretation is called statistical interpretation of the radiation . Einstein showed that t] a
source of light emits photons in all directions and the number of photoins reaching a surface decrease
as the distance between the source and surface increases .
Einstein related the definition of intensity in wave theory ( square of average of the electric field) and
the definition of Intensity in photon picture as the number of photons fallin on the surface
Comparing the left hand side and right hand side of the above equation we conclude that the quantity
=N
The intensity in the wave picture is equal to the number of photons in the particle picture .
Max Born’s interpretation of matter waves and wave particle duality
Max Born developed a similar theory of wave particle duality to explain the De-Broglie waves that are
associated with a moving body. The De-Broglie wave like any other wave must have an equation,
wavelength, frequency etc. The most important thing is that the De-Broglie wave is moving with the
particle so it must be such that it contains all the information regarding the particle. This is the reason
that the De-Broglie wave is different from classical waves .
Max Born represented the matter waves (De-Broglie) waves by a symbol ψ ,this representation was
used to differentiate the matter waves form ordinary classical waves .This symbol was called wave
function
We consider a particle moving in x direction with some velocity and momentum .There is a wave
associated with it for which the equation has to be written
The wave function (wave equation or wave which is De Broglie wave)
If the above equation is compared to the electric field component of an electromagnetic wave
As we have studied that the square of the electric field gives information regarding intensity so the
square of ψ(x,t) must also give some information similar to intensity
From the above consideration the square of ψ(x,t) gives information regarding the probability of
finding the particle in a given volume of space at a particular time
According to Born’s interpretation the whole course of study is related to the laws of probability.
According to Born from the wave function or wave equation we are trrying to find the probability of
finding the particle in a given volume of space at a particular time .
Both Max Born and Einstein gave a probabilistic interpretation of the wave function.
Now we need to know some important aspects of the wave function ψ(x,t)
The wave function ψ(x,t) has no physical meaning itself .
The quantity of interest or physical significance is the square of the wave function |ψ(x,t)|2 this
quantity gives the probability of finding a particle in a given region of space at a given time t.
Since |ψ(x,t)|2 is the probability of finding the particle in a region of space so it is actually a number,
which lies between 0 and 1.
This means if in a region |ψ(x,t)|2 has a high value then there is a strong probability of finding the
particle at that point
Similiary if in aregion |ψ(x,t)|2 is small then it means that the probability of finding the particle at
that point is small NOT ZERO.
Unless |ψ(x,t)|2 is zerothere is finite probability of finding the particle in that region
Wave function ψ is al;ways related to the particle
A particle is alocalized body which means it is found at a fixed point , but a wave is spread . The
concept of probability means that at a certain region of space there is a finite probability of the
particle being found at agiven point of time . For example if in a measurement the value of |ψ(x,t)|2 is
20% then it means that there is 20% chance that the particle will be found at that point . And the
particle will be found as a whole .
|ψ(x,t)|2 is also called probability density . Because in three dimensions we find |ψ(x,t)| 2 in a unit
volume .
If in an experiment there are alrge number of particle that are identical then again the probability of
finding those particles is given by |ψ(x,t)|2
Now we need to find out what is the speed of the De-Broglie waves ? What is the correct form of the
equation describing a matter wave or De-Broglie wave ?
At a first instance it appears that since the wave is moving with the particle its speed will be same as
that of the particle . We will check whether this happens or not .
Let us assume that the speed of the De-Broglie wave is denoted by VD . According to the general
formula to find the speed of the wave we have
VD=νλ . the wavelength of the De-Broglie wave is given by
gamma is due to
relativistic correction as we are considering the electromagnetic wave situation which moves with
speed c. The frequency can be found from the expression E=hν which gives
This gives the speed of the De-Broglie wave as
Is this result acceptable?????
This means that the wave is moving much faster (faster than speed of light ) than the particle which is
physically unacceptable .
To get a physically acceptable result we must review the wave equation for the De-Broglie wave. For
our ease we consider a wave equation which is similar to a wave moving on a stretched string . For
such a system the wave equation can be written as
At t=0 the amplitude is maximum y=A
The above equation tells the value of displacement y as a function of time t . We need information
regarding the x axis also or the x coordinate also because the wave is moving along the x axis . So the
complete wave equation should contain the parameter x also.
From the above diagram we notice that the peaks of thewave are moving along the x axis with
passage of time . So in the complete wave equation we require a parameter for x also.
The wave equation described above shows the displacement of a single point on the wave with
passage of time .
The equation is not describing the speed of the complete wave it is describing the speed of the points
with same phase .
Let us consider that the string is plucked at x=0 at t=0 . If the speed of the wave in the string is v p then
in time t the wave travels a distance x= vp t This information should be contained in the wave
equation so the correct wave equation is
The above equation can be expressed in terms of wavelength as
Where angular frequency is
And wave number is
In the most simple form the wave equation becomes
This shows the wave moving along + x axis
The amplitude of the wave function gives the probability of finding the particle atr a given point. To
represent
In the case of De Broglie waves we need an equation that shows the variation of amplitude ir
probability of finding the particles at different points along the x axis (or in any direction ) with time .
So we need to describe the wave moving with a particle by a wave packet or group of waves so that
we can get amplitude at different points at different times .
Wave packet contains many waves whose amplitudes are related to the probability of finding the
particle or the amplitude of the De Broglie wave.
Let us consider two waves with slightly different frequencies and wave numbers that interfere to
produce a resultant wave
Since the difference in the frequencies and wave numbers are vesy small we have
The equation thus obtained is
The important aspect of this equation is that the amplitude is itself a wave . The wave equation
Represents a a large of waves that have same amplitude A but in the case of matter waves the
amplitude is changing with the probability of finding the pqarticle in different regions of space. So we
need to have a wave equation that shows a varying or oscillating amplitude which we can get by the
superposition of two waves with slightly different frequencies and wavenumbers. The result of
superposition of two such waves give
This equation when compared with
Comparison of the two equations shows that the result of superposition of two waves gives a wave
with oscillating amplitude, such a wave is called wave packet or a wave group The shape of the wave
group or packet is determined by the variation of amplitude .
(i)
(ii)
If the velocities of the superposing or interfering waves are same then the group travels
with common phase velocity
In the next situation the phase velocity varies with wavelength. This occurs when the
phase velocity is different for different waves and is called dispersion.
The velocity of wave group as a whole is different from the waves that make the group . In the
equation of the resultant wave
We have a wave moving with frequency ω wave number k. This wave is modulated by a wave whose
frequency is
and wave number is
that have a varying amp-litude
Here we get two velocities
(i)
Phase velocity
. Effect of modulation is to produce a group of waves
(ii)
Group velocity
The group velocity is also called the dispersion relation . Group velocity is the velocity with which
the group of waves is moving . In the xpression of group velocity the del sign indicates that the
frequency and the wave number are spread and are not exact values. For this reason the group
velocity is expressed as
To calculate group velocity from a given wave equation we use the above form of group velocity
The angular frequency of the De-Broglie waves is given by
Using the factor ϒ for relativistic speed and the De-Broglie relation for wave length we have
In case of De-Broglie waves the frequency and wavenumber are both functions of the velocity of
the body
After the differentiation we get the result
so we conclude that the velocity of
the De-Broglie wave for a particle moving with velocity v is the group velocity.
In the other hand the phase velocity of the De-Broglie waves is
Because of this the phase velocity has no physical significance .
The application or the result of the wave particle duality is shown as an important principle called the
Uncertainty Principle.
When we consider that a moving particle is associated with a wave with it called the matte wave or
De-Broglie and the particle can be considered as a group of waves then there comes a limitation in
determining the speed and position of the particle.
For example we consider the wave group as shown in the figure
From the above diagram we conclude that the particle can be any where in the group.(wave group).
Since the probability of finding the particle at ant time is related to the amplitude of the wave or
so the particle is most likely to be found at the middle of the group as the amplitude is
maximum but there are finite probabilities of the particle being found at other regions so we cannot
say that the particle will be surely found at a given point but we say that the particle is most likely to
be found at a given point. If the wave group shrink or becomes narrower then we the following
picture
If the wave is made narrow then it will not be possible to measure accurately the wavelength because
in a narrow region the complete wave will not fit since wavelength De-Broglie wavelength is
determined by the momentum of the particle so we cannot determine the momentum exactly. In the
diagram since there is only one peak or max amplitude so we can say that there is only one position
where the particle has the maximum probability of being found so the pposition of the particle is
accurate .
In this case we can find out or locate the particle exactly but we cannot determine the momentum
with precision. This is the difference from classical world .
In the next situation we consider a wave which is elongated or a large wave. The diagrammatic
representation is shown below
In this case the wave group is large and the wavelength is well defined so is the momentum . But since
the wave group is large there are many peaks or amplitudes which imply that there are many points
where there is finite probability of finding the particle . Thus the position of the particle is now
inaccurate or uncertain. Thus when the momentum can be determined exactly or precisely then the
position cannot be determined with same accuracy or precision.
From both situations we conclude that at a given time if we make a measurement then we cannot
determine momentum and position simultaneously with exact accuracy. This principle is called the
Heisenberg’s uncertainty principle.
In case of De-Broglie waves the particle which is moving with a velocity V is associated with a single
wave packet or wave group that has a definite shape.
Now we denote the wave group which is moving with the particle by ψ(x)
this wave group is represented as the Fourier integral of the form
The function g(k) represents how the amplitudes of the waves that form the wave packet ψ(x) vary
with wave number k. g(k) is the Fourier Transform of ψ(x). It shows how different waves combine to
form the wave packet ψ(x). Since x denotes position and k denotes momentum so we are making a
transition from position to momentum space and this is why fourier transform is used. The limit of
integration shows that a wave group can contain waves whose wave numbers vary from 0 to infinity ,
when we are considering De-Broglie waves then we have a wave packet in which ther are waves
whose wave numbers vary within a limit
So when we are measuring the position or momentum
of a particle then in that case
shows the spread in the value of k and similarly the
represents the spread in the value of x. The spread is called uncertainty or error in the language of
experiment. The relation between
and
depends on the shape of the wave packet . From
the previous formalism we can say that the spread or uncertainty in the values of x and k must be
related so that the product is minimum. The value of the product determines the shape of wave
group. The product
shape . now if we consider
has a minimum value when the shape of the wave packet is Gaussian
and
as the deviations of ψ(x) and g)K) then from
mathematical consideration the product of
and
is minimum when
applicable for the Gaussian shape. Thus the general formof product of
and
is
is
The greater than equal to sign is used because the wave packets deviate slightly from actual Gaussian
shape .
The product shown above physically means that if the uncertainty in x is large ( longer wave group
where the position is highly uncertain) then
the position in momentum is less (as in a longer
wave group where the wavelength or momentum can be determined with certainty) and vice versa.
This the uncertainty principle . In the diagram below it is shown how ψ and g are related
The figure (a) shows a pulse where in the position space (x) the position of the particle is certain or
exact corresponding to this ψ the momentum space show g(k) as a curve which shows that
momentum is uncertain.
The figure (b) shows that the position is uncertain and the momentum in k space is certain. The
Gaussian shape is shown as
Since the De- Broglie wavelength and the wave number are related by the following relations
And
Expressing the momentum of the particle in terms of the wave number we have
since uncertainty in k comes from the uncertainty in momentum p so we have
then using the relation
uncertainty principle which is
we get the exact form of
Uncertainty and measurement:
When we are measuring or performing some experiment then in that case we or the device interacts
with the system. The interaction causes a disturbance which is negligible in the classical world.
However in the atomic world any measurement disturbs(deviates the system from its original state )
the system .This disturbance in the experimental verification of the Uncertainty principle . The
diagram below shows the experimental proof of the Uncertainty principle and is commonly known as
the Heisenberg’s microscope
The incident photon when strikes the electron or any atomic particle then the photon disturbs the
electron. Disturbs means the interaction of photon and electron causes the electron to change its
position and momentum and from the reflected photon we “see” a pattern which is the spread in
position or momentum
The fact that we cannot “see” an electron or an atomic particle is due to the uncertainty principle.
The momentum change caused due to the interaction of photon with electron is
The Schrodinger Equation (one dimension)
In the atomic or subatomic world where the Planck constant is not negligible and the wave properties
of the particle moving with some velocity are significant , we talk of probabilities of occurrence of
certain events or values. For example we say that this is the most probable position of the particle, or
most probable path of the particle. The probability is a number that is related to the square of
modulus of the wave function ψ. We find out the most probable values of the observable quantities
like energy, momentum , angular momentum etc.
So in this world of atomic particles we need a separate equation or theory to deal with the situation.
The certainty of occurrence of some events or values in classical mechanics is not present in the
quantum mechanical world. Here we reach at most probable values or events.
The probability is given by
so in any situation we need to find out
conditions, which is a number that gives information about probability.
So rthe quantity of interest is
under given
and ψ alone has no physical significance .
Some important aspects of wave function
(i)
Wave functions are related to matter wave so in order to include the phase factor the
wave function is expressed as complex quantity.
(ii)
The square of the modulus of the wave function
is areal number .
There are certain conditions that the wave function ψ must satisfy in order to explain a physical
situation and to yield acceptable values . Such conditions are
1. Normalization
Since the
is related to the probability of finding the particle in space (all space) so at any point
in space the particle has to be found so if we take the integral of
over all space then the
integral must be finite. This is the condition of normalization. In other words the integral of
over all space must be such that it is equal to the probability of finding the particle. Thus the
mathematical condition of normalization is
In other form the above equation can be expressed as
dV is the volume .
The above equation is called to normalization equation. Any wave function that follows the
normalization condition is said to be normalizable in that region.
Ψ must satisfy some other conditions to be physical acceptable under certain situations.
(i)
Ψ must be single valued which means at a particular point in space the value of
must give a single value which is consistent with normalization condition . ψ must be
continuous also to give physically acceptable .
(ii)
The derivatives of ψ
must also be continuous and single
valued this result follows from momentum conservation
Ψ must be normalizable this means that ψ must be zero at infinite distances i.e
(iii)
When all these conditions are satisfied then the wave function ψ is said to be well behaved
In general we consider the wave functions in some finite regions or space with defined boundaries
in that case the normalization equation becomes . In one dimensional situation if we are along the
x axis in aregion bounded between x1 and x2 then the normalization equations
This equation gives the probability of finding the particle in aregion with boundaries at x1 and x2.
In order to explain different situations in the quantum mechanical world Schrodinger formulated
an equation which became the fundamental equation in quantum mechanics.
Schrodinger equation is not a derived equation it is an equation that contains some important
physical laws.
Schrodinger equation is a wave equation.
The general form of wave equation is expressed as
The solution of the wave equation above can give manyresults standing wave, progressive wave
etc.
If we consider a particle moving on x axis and is not acted upon by any force such a particle is
called free particle the particle is not bounded and is moving along the x axis . From De-Broglie
hypothesis this particle is associated with a wave which is also moving with the particle at
constant speed along the x axis
How to describe such a wave?
In any kind of wave whether on a string or like the situation above the general equation is
Plus minus sign denotes the direction of propagation of wave F is a function that can be
differentiated . For the awave moving with the particle we need an equation of the formsimilar to
that shown above .
Since the particle is free and no force acts on it so its speed ,energy and amplitude are constant.
Amplitude is constant since we assume that there are no damping factor
To represent such a wave the equation is
Schrodinger equation (time dependent)
Since ψ in this case represents a wave which is moving with a particle in x axis and since ψ itself
has no physical significane so it can be represented as a complex quatntity similar to that shown
above and the most general form is
Schrodinger equation is NOT A DERIVED EQUATION it only contains some basic laws like
So in order to include these laws we replace ω by 2πν and velocity v by λν then we have
Now using the De-Broglie equation and the equation for wave number we have
h bar is h/2π
with all these substitutions we have the wave equation or wave function for a free particle in the
form
This equation describes a free particle moving along positive x axis with energy E
The equation
describes only a free particle . However in order
to describe bounded systems like an electron moving around a nucleus we need some thing more
in the equation. How to proceed for a more general wave equation or wave function ?
To proceed we use the formalism of double space derivative and double time derivative of ψ.
Differentiating the wave function of free particle twice wrt x we have
Interchanging sides
Differentiating the equation
wrt t once
We have
For a particle moving with a speed v (v<<c) the total energy of the particle is the sum of potential
energy and kinetic energy
Multiplying the above equation by ψ we have
Now in the above equation we put the values of
and
Putting these values we have
This is the Schrodinger equation in one dimension. The equation is called time dependent
Schrodinger equation because we have considered that the potential is a function of time .
The general problem of quantum mechanics lies in the solution of Schrodinger for different
situations and finding out ψ
Schrodinger equation is not a derived equation but it is itself a basic equation.
Some important aspects of Schrodinger equation
Linearity and Superposition
Linearity means the highest power of the independent variable is 1. The important property of
Schrodinger equation is that it is linear in ψ (highest power of ψ is 1) . This means that the SE will
not contain any term with ψ having power 0 or more than 1. This physically means that if ψ1 and
ψ2 are two wave functions then their sum will also be a wave function that can be described as
From the equation above it is clear that the wave function ψ follows superposition principle same
as that of sound or electromagnetic waves.
The diagram shows electron diffraction experiment. Electrons from a source are incident on a
double slit arrangement with two slits S1 and S2 similar to YDSE. There is a screen placed art some
distance from the slits .
(i)
(ii)
(iii)
At first situation only one slit is open suppose s1 is open and s2 is closed
Next S2 is open and S1 is closed
The next situation is both slits are open
When the slit s1 is open and s2 is closed then the pattern observed on the screen is shown by figure
(B)
When the slit S1 is closed and S2 is open then the pattern is shown in figure c
When slit s1 is open then the electrons are coming from s1 and striking the screen and the quantity
ψ12 is the probability of finding the electrons on the screen. The screen shows a pattern which is due
to the wave nature of the electrons.
In case S1 is open the screen shows an intensity pattern which is seen on the screen is actually the
probability of finding the electrons when S1 is open and this intensity
Where ψ1 is the wave function that describes the electrons coming from slit S1. Ψ* is bthe complex
conjugate of ψ.
Similarly when S2 is open then the pattern is described by c and the probability os given by
Where ψ2 is the wave function due to the electrons emitted from S2
The figure d shows the sum of P1 and P2 described above . This is the situation if we consider the sum
of P1 and P2 however the actualpattern that is observed is shown in figure e.
Figure shows that the pattern is due to sum of ψ1 and ψ2 and NOT DUE TO P1 AND P2 SO WE
CONCLUDE THAT THE PATTERN IS DUE TO WAVE FUNCTIONS SUPERPOSITRON AND NOT DUE TO
ADDITION OF PROBABILITIES
THIS IS CALLED ELECTRON DIFFRACTION EXPERIMENT .
In figure e the pattern observed is due to the linearity condition of wave functions and there is
resultant wave or wave function whih is
The probability density due to the resultant wave function is
=
=
=
The terms
are responsible for the pattern observed in e and show the oscillation
of the electron density on the screen
Expectation values
The solution of Shcrodinger equation for a given situation gives ψ(x,t) that can be used to find the
probability density. Except for quantized quantities we get probabilities from the sloution of
Schrodinger equation.
Suppose we have a particlemoving along the positive x direction and the wave asscociated with that
particle is ψ. If we measure the value of x or position of the moving particle many number of times then
we get the result which is called the expectation value of x denoted by
Suppose in another experiment we have a large number of identical particles that are moving along
the x axis such that there are N1 particles at position x1, N2 particles at position x2 and so on then the
question is how to find out the average position of the particles ?
The average position is found from the equation
Suppose we have a single particle moving in the x direction then in that situation we replace the
number Ni by the probability of finding the particle at xi. The probability of finding the particle at xi is
given by
I denotes a particul;ar position along the x axis .
Thus for a single particle all along the axis the expectation value of position is given by
By normalization the denominator becomes 1 and thus
is the expectation value of position
P(x,t) ??????? this cannot be written due to uncertainty principle as x and p cannot be determined
simultaneously
How to find out the expectation value of quantities like momentum and energy?
To answer this question we have the concept of operators
To obtain the expectation value of momentum we differentiate wrt space the wave function
Differentiating we get
From this equation we can get the momentum as
P=
Similarl y differentiating
thus we get the momentum operator as
Schrodinger equation in steady state form
wrt time w e get the value of energy as
and energy operator as
In situations where the potential energy is not dependent on time then the force acting on the
particle and the potential energy change only with position and not with time . In this case the
Schrodinger equation takes a different form and is called the steady satte equation
In the steady state form the spacce and time parts are separated.We have the wave function of the
form
Opening the bracket we get
This equation can be written in a more compact form as
On the above equation ψ represents the space part of the wave function. Thus here we have
separated the space part ψ and time part . If we multiply
We get this form of equation by putting
by E then we get
into the time dependent ofr m of SE
Removing the I factor we get
this is the steady form of SE.
The steady state form of SE is used in solving one dimesional problems . One important property of
the Schrodinger equation in steady state form is that it can have more than one solution for a given
situation. The interesting aspect is that all these solutions have a specific energy E. Thus the
quantization of energy will appear in the solution of steady state SE.
(iii)