CHAPTER 4
QUANTUM PHYSICS
INTRODUCTION
Newton’s corpuscular theory of light fails to explain the phenomena like interference, diffraction, polarization etc. The wave theory of light which was proposed by Huygen in 1679, explain these phenomena. However, new phenomena like Compton effect, photoelectric effect, Zeeman effect, emission of light, absorption of light etc., cannot be explained by the above theories. The failure of these theories leads to the discovery of a new theory, called quantum theory of radiation of light.
BLACK BODY RADIATION
Definition
A body which can absorb all wavelength of an electromagnetic radiation and also can emit all wavelength of radiation when it is heated to a suitable temperature is called a black body. The radiation emitted from black body is known as black body radiation or total radiation.
- Emissive power:Energy emitted per unit area per unit time.
- Absorptive power: Energy absorbed per unit area per unit time.
Characteristics of Black Body Radiation
Black body radiations are characterized by Stefan’s law, Wein’s law and Rayleigh-Jean’s law. According to these laws, black body radiation is stated as follows:
- Stefan-Boltzmann’s law
The radiation energy (E) emitted per unit time per unit area of a perfect black body is directly proportional to the fourth power of its absolute temperature T.
E = σ T 4
Where, σ is the proportionality constant known as Stefan’s constant and its value is 5.78 X 10-8 Wm-2K-4.
2. Wein’s displacement law
Wein’s displacement law states that the wavelength corresponding to the maximum energy is inversely proportional to absolute temperature T.
λm T = Constant
This law shows that as the temperature increases the wavelength corresponding to maximum energy decreases. This law holds good only for shorter wavelengths and not for Longer wavelengths.
3. Rayleigh-Jean’s law
According to this law, the energy distribution is directly proportional to the absolute temperature and is inversely proportional to the fourth power of the wavelength. It is governed by the equation.
This law holds good only for longer wavelength regions and not for shorter wavelengths.
Energy Distribution of a Black Body
The distribution of energy for different wavelength at various temperatures of the source is as shown in following fig.
From the above graph, the following observations are made:
- The distribution of the energy is not uniform.
- For a particular temperature, the intensity of radiation increases up to a particular wavelength and then it is found to decrease with increase in wavelength.
- As temperature increases the peak energy shifts towards shorter wavelengths.
Based on the applications of Wein and Rayleigh equation, the energy spectrum of black body radiation cannot be explained completely. In order to explain the distribution of energy in the spectrum of a black body, Planck suggested a new hypothesis in the year 1900 and therefore, he derived the new radiation law with some assumptions.
Planck’s Quantum theory of Black Body Radiation
Planck derived the expression for the energy distribution based on the following hypothesis:
Planck’s Hypothesis:
The black body radiation chamber is filled up not only with radiations but also with a large number of oscillating particles. The particles can vibrate in all possible frequencies.
The frequency of radiation emitted by an oscillator is the same as that of the frequency of the vibrating particles.
The oscillatory particles cannot emit energy continuously. Radiation is emitted (or) absorbed by a body in an integral multiple of a fundamental quantum of energy called “photon”.
The vibrating particles can radiate energy when the oscillators move from one state to another. The radiation of energy is not continuous, but discrete in nature. The values of the energy of the oscillators are like 0, hγ, 2hγ, 3hγ….nhγ
Planck’s Quantum Theory of radiation Law
Let us consider the number of vibrating particles in the body as N0,N1,N2,….Nn.
According to Planck’s hypothesis, the energy of the above particles can be written as
0, ε,2ε,3ε,4ε….nε.
Therefore, the total number of vibrating particles is given as,
N = N0+N1+N2+….+Nn…(1)
Similarly, the total energy of the body is given as,
E = 0+ ε+2ε+3ε+4ε+….+nε…(2)
Therefore, the average energy of the particle is given as,
…(3)
According to Maxwell’s distribution formula, the number of particles in the nth oscillatory system can be written as,
Nn = N0 e –nε/kT…(4)
Where,ε is the energy per oscillator, k is the Boltzmann’s constant and T is the absolute temperature.
According to Maxwell’s distribution function, the total number of particles N can be written as,
N = N0+N0e –ε/kT+N0e –2ε/kT+N0e –3ε/kT+….
OrN = N0[1+e –ε/kT +e –2ε/kT +e –3ε/kT+….] …(5)
We know, 1+x+x2+x3+…….+ =. Therefore, we can write equation (5) as,
…(6)
Similarly, the total energy of the body can be written as,
E= 0+εN0e –ε/kT +2εN0e –2ε/kT +3εN0e –3ε/kT+….
OrE= εN0e –ε/kT [1+2e –ε/kT +3e –2ε/kT +….] …(7)
We know, 1+2x+3x2+4x3+…….+nxn-1 =. Therefore, we can write equation (7) as,
…(8)
Or…(9)
Substituting the values of ε = hγ in equation(9), we have
…(10)
If γ and γ+dγ is the frequency range, the number of oscillations can be written as,
…(11)
Therefore, the total energy per unit volume for a particular frequency can be obtained by multiplying equations (11) and (10) as,
Or…(12)
Equation (12) is known as Planck’s equation for radiation law interms of frequencies.
It can also be written interms of wavelength as,
Or…(13)
Equation (13) represents the Plank’s radiation law interms of wavelength.
- Deduction of Wien’s displacement law
When λ is very small, γ is very large
Or …(14)
This is Wien’s displacement law.
- Deduction of Rayleigh-Jean’s law
When λ is very large, γ is very small [ hence, we neglect the higherorders]
Therefore,
Or
…(15)
Equation (15) represents Rayleigh-Jean’s law. Thus, Wien’s and Rayleigh-Jean’s formula are the special cases of Plank’s formula on black body radiation.
Planck’s formula has been found to agree remarkably well with experimental observations of Lammer, Pringsheim , Kurlbaum and others as shown in Fig and thus established one for all the validity of the quantum hypothesis.
Photon & Its Properties
Photon
The discrete energy values in the form of small packets or quanta’s of definite frequency or wavelength are called photons. Photons propagated like a particle with speed of light as 3 X 108 ms-1.
Properties of Photon
(i)The existence of photon and electron are same in nature.
(ii)Photons will not have any charge. So they are not affected by magnetic & electric field.
(iii)They donot ionize gases.
(iv)The energy of one photon is given by E=hγ.
(v)The mass of a photon is given by, m = h/Cλ.
(vi)Momentum of the photon is given by P= h/λ.
Derivation of De-Broglie wavelength
Energy of a photon can be written as E = h------(1)
Where is the frequency of the photon
is the wavelength
If the photon possesses some mass by virtue of its motion then according to the theory of relatively its energy is given by
E = (mass of a photon) c2
E = mc2------(2)
From (1) & (2)
Mass of the photon,
momentum of the photon
De-Broglie carried these considerations over to the dynamics of a particle and said that the wavelength of the wave associated with a moving mattered particle having a momentum mv is given by
------(3)
This equation is called De-Broglie wave equation
3. Expression for wavelength associated with the electron accelerated by
potential difference
Consider an electron of charge ‘e’ accelerated by a potential difference of ‘V’ volts. If the velocity acquired by the electron of mass ‘m’ is ‘v’ then
------(4)
de Broglie wavelength associated with the electron,
------(5)
Substituting eqn.(4) in eqn.(5)
Substituting
and
we get,
`Å
4.7 Schrödinger wave equation
The equation that describes the wave like behavior of electrons with the appropriate potential energy and boundary conditions is called the Schrödinger’s equations. It is a differential equation capable of describing the motion of an electron.
It forms the foundations for quantum mechanics without this equation if could not be possible to understand the principles and operations of many some conductor devices.
1. Derivation of time independent wave equation
A traveling wave in the positive x direction resulting from sinusoidal oscillations of a particle can be described by a traveling wave equation
where the displacement at time t and at distance x from the equation and the angular frequency.
Let x,y,z be the coordinates of the particle and , the wave displacement for the de Broglie waves at any time t.
The classical differential equation of a wave motion is given by
------(1)
Here, is the wave velocity. Then ------(2)
The solution of eqn. (1) gives as a periodic displacement in terms of time.
------(3)
Differentiating eqn. (2) twice with respect to t, we get
Substituting the value of in eqn. (1) , we get
------(4)
But,(Here, is the frequency)
Substituting the value of ineqn.(4)
------(5)
From de Broglie relation, if a particle is behaving as a wave then
------(6)
A particle can behave as a wave under motion only if its is kept in a potential field.
Let E = total energy of the particle
V = Potential energy of the particle
The kinetic energy of the particle is
or ------(7)
Substituting this in eqn.(6), we get
or------(8)
Eqn.(8) is called Schrödinger time independent wave equation.
The quantity is called wave function
Let us now substitute in eqn. (8)
Then the Schrödinger time-independent wave equation, in usually used form, may be written as
------(9)
In one dimension
For a free particle the potential energy, V=0
Then the Schrödinger time independent wave equation becomes
------(10)
2. Derivation of time dependent Schrödinger wave equation
The Schrödinger time dependent wave equation is obtained from Schrödinger time independent wave equation by eliminating E.
The classical differential equation of a wave motion is given by
------(11)
The solution of eqn. (11) gives as a periodic displacement in terms of time.
------(12)
Here, is the amplitude at the point considered. It is function of position i.e., of co-ordinates (x,y,z) and not of time t.
Differentiating eqn. (12) with respect to t, we get
(since )
( or )
------(13)
Schrödinger time independent wave equation (9) is
------(14)
Substituting the value of in eqn.(14) , we get
------(15)
Eqn. (15) is called time dependent Schrödinger equation.
Eqn. (15) can be written as
or------(16)
Here,
and
Eqn. (16) describes the motion of a non-relativistic material particle.
4.8 Physical significance of the wave function
1. is a wave function and used to identify state of the particle.
2. The wave function measures the variations of the matter wave. Thus it
connects the particle and its associated wave statistically.
3. is a measure of the probability of finding particle at a particular position
and does not give exact location of the particle.
4. is a complex quantity and we cannot measure it exactly.
5. The probability of finding out a particle in a particular volume element is
given by where is called the complex
conjugate of .
6., means where the particles presence is certain in space.
7. The wave function has no physical meaning, whereas the probability
density has physical me…a0ning.
4.9 Applications of Schrödinger equation
1. Particle in a box
Consider the behavior of an electron when it is confined to a certain region 0<x<L. The potential energy of the electron is zero inside that region and infinite outside as shown in fig.The electron cannot escape because it would need an infinite potential energy.
In terms of the boundary conditions imposed by the problem the potential function is
V(x) = 0,
andV(x) = ,
Schrodinger wave equation is,
Inside the potential well, equation becomes
------(1)
where------(2)
This is the wave equation for a free particle inside a potential well.
A possible solution to equation (2) is
------(3)
where A and B are constants.
Since the particle cannot penetrate an infinitely high potential barrier
for atx=0, B must be zero
for atx=L, kL must be an integral multiple of .
Therefore,where k = n/L, n = 1,2,4 ………
------(4)
which is called Eigen function or characteristic function. All for n = 1,2,4 ……. Constitute the Eigen functions of the system. Each Eigen function identifies a possible state for the electron. For each n value,there is one special k value.
Substituting for k from equation (2) and (4) we get
or
------(5)
The energies En defined by the above equation with n= 1,2,3,….. are called Eigen energies of the system.
We still have not completely solved the problem, because A has yet to be determined. To find A, we use what is called the normalization condition. The total probability of finding the electron in the whole region 0 < x < L is unity because we know the electron is somewhere in this region. Therefore summed between x=0 and x=L must be unity or
------(6)
------(7)
(since
i.e.i.e.
The normalized wave function is
------(8)
we can now summarize the behavior of an electron in one dimensional potential energy well. The wave functions and the corresponding energies En, which are often called Eigen function and Eigen values shown in fig.
Both depend on the quantum number n. The energy of the electron increases with n2, so the minimum energy of the electron corresponds to n=1. This is called ground state and the energy in the ground state is the lowest energy the electron can possess. The energy of the electron in the potential well cannot be zero even though the PE is zero. Thus the electron always has KE even when it is in the ground state. The node of a wave function is defined as the point where inside the well.
The energy is found to increase as the number of nodes increases in a wave function.
2. Three dimensional potential well
To examine properties of a particle confined to a region of space, we take three dimensional space with a volume marked by a,b,c along the x,y,and z axis respectively. The PE is zero inside and is infinite on the outside as shown in Fig. This is a three – dimensional potential well.
The solution for a one-dimensional well can be extended quite easily to study the behavior of the electron in such three dimensional well. If we assume that all the sides of the box are of the same length, the eigen functions are given by
------(1)
where,
we can write this energy in terms of as follows
------(2)
where,
conclusion is that in three dimension
(i)There are three integers n1,n2, and n3 called quantum number which are required to specify completely each energy state.
(ii)The energy E now depends on three quantum numbers.
(iii)Several combinations of three quantum numbers may give different wave functions, but of the same energy value. Such states and energy levels are said to be degenerate.
For example three Independent states having quantum numbers (1,1,2), (1,2,1) and (2,1,1) have the same energy
------(3)
These levels are three fold degenerate. That is the number of states that have the same energy is termed the degeneracy of that energy level. The second level E211 is thus three-fold degenerate. It must be noted that states such as (1,1,1), (2,2,2)etc, are non degenerate.
4.10 Degeneracy
It should be noted that the energy E depends only on the sum of the squares of nx, ny and nz.
Consequently there will be in general several different wave functions having the same energy. For example the three independent stationary states having quantum numbers (2,1,1), (1,2,1) and (1,1,2) for nx,ny,nz have the same energy value . Such states and energy levels are said to be degenerate and the corresponding wave functions are On other hand if there is only one wave function corresponding to a certain energy, the state and the energy level are said to be non-degenerate. For example the ground state with quantum numbers (1,1,1) has the energy and no other state has this energy. The degeneracy breaks down on applying a magnetic field or electric field to the system.
Electron Microscope:
Electron microscope is like the optical microscope. This is an instrument primarily used for magnifying small objects to such an extent that their minute parts may be observed and studied in detail. With the electron microscope, magnifications of 10 to 100 times that of finest optical microscope make some of their detailed structure, visible to the eye.
Principle:
A stream of electrons can be focused by suitable electric and magnetic fields and are possible through the object. These electrons which carries information’s about the object.
The resolving power of on optical instrument like a microscope is directly proportional to the aperture of the lens used and inversely proportional to the wavelength of the light used. So the smaller the wavelength, the greater shall be the magnifying power.
If we accelerate the electrons through a potential difference of say 60,000 V,
the velocity ‘v’ attained by them can be calculated from the relation,
eV = (1/2)mv2
Where m is the mass of electron = 9x10-31kg. calculating λ from the de-Broglie wave equation, λ= h/mv, the wavelength comes out to be about 5x 10-12m which is smaller than that of the visible light.
Various types of electron microscopes are
- Transmission Electron Microscope(TEM)
- Scanning Electron Microscope (SEM)
- Scanning Transmission Electron Microscope (STEM)
Applications
The electron microscope as a very large magnifying power of the order of 50Å in size has been made visible. An electron microscope is put to valuable uses in all the fields.
In medicine and Biology, it is used to study virus, the disease causing agent beyond reach of ordinary microscope. The bacteria are shown in greater detail. So that suitable means may be used for their destruction.
It has been used in the investigation of atomic structure and structure of crystals in details.
Small particles forming colloids have become open to study and analysis.
It has also been used in the study of the structure of Textile fibers, purification of lubricating oils, compassion of paper and paints, surfaces of metals and plastics.
Scanning Electron Microscope
Principle:
When the accelerated from the electron gun strike the sample, it produces secondary electrons. These secondary electrons are collected by an electron detector which in turn gives a 3- dimensional image of the sample.
Description:
It consists of an electron gun to produce high energy electron beam. A set of magnetic conducting lenses are used to condense the electron beam and a scanning coil is arranged between magnetic condensing lens and the sample. The sample is placed over a sample are collected by electron detector and are converted into electrical signals. These electrical signals can be fed into CRO through video amplifier.
Working:
The electrons produce by the electron gun called primary electrons are accelerated by the anode plate. These accelerated primary electrons are made to incident on the sample through magnetic condensing lenses and scanning coil.