1 Elements Of Wave Mechanics 1
1.1 Introduction . . . 1
1.2 Black body radiation . . . 1
1.2.1 Experimental observation of black body radiation . . . 2
1.2.2 Laws of black body radiation . . . 3
1.2.3 Stefan Boltzmann radiation law . . . 3
1.2.4 Wien’s Laws . . . 3
1.2.5 Rayleigh – Jean’s law . . . 4
1.2.6 Planck’s radiation law . . . 4
1.2.7 Derivation of Wien’s law from Planck’s law . . . 5
1.2.8 Derivation of Rayleigh – Jean’s law from Planck’s law . . . . 6
1.3 Photoelectric effect . . . 6
1.4 Compton effect . . . 9
1.5 Matter waves and de Broglie’s hypothesis . . . 10
1.5.1 Davisson-Germer experiment . . . 11
1.5.2 G.P. Thomson experiment . . . 13
1.5.3 Wave packet and de Broglie waves . . . 14
1.5.4 Characteristics of matter waves . . . 14
1.6 Phase and group velocities . . . 15
1.6.1 Relation between phase velocity and group velocity . . . 16
1.6.2 Relation between group velocity and particle velocity . . . 17
1.6.3 Derivation of de Broglie relation . . . 18
1.7 Heisenberg’s Uncertainty principle . . . 19
1.7.1 Origin and nature of the Principle . . . 19
1.7.2 An illustration of uncertainty principle . . . 21
1.7.3 Physical significance of uncertainty principle . . . 22
1.8.1 Characteristics of wave function . . . 25
1.8.2 Physical significance of wave function . . . 25
1.8.3 Schrodinger’s wave equation . . . 26
1.8.4 Eigen values and eigen functions . . . 27
1.9 Applications Of Schrodinger’s Equation . . . 28
1.9.1 Case of a free particle . . . 28
1.9.2 Particle in a box . . . 28
1.9.3 Finite Potential well . . . 32
1.9.4 Tunnel effect . . . 35
1.9.5 Examples of tunneling across a finite barrier . . . 37
1.9.6 Theoretical interpretation of tunneling . . . 39
1.9.7 Harmonic oscillator . . . 40
1.9.8 Practical applications of Schrodinger’s wave equation . . . 42
Numerical Examples . . . 43
Exercise . . . 51
2 Crystallography and X-rays 55 2.1 Crystal Structure . . . 55
2.1.1 Unit cell and space lattice . . . 55
2.1.2 Crystal systems . . . 57
2.1.3 Bravais lattices . . . 59
2.1.4 Miller indices and their uses . . . 60
2.1.5 Interplanar spacing in cubic crystals . . . 63
2.1.6 Atomic packing factors . . . 65
2.1.7 Some crystal structures . . . 68
2.2 X-Rays . . . 72
2.2.1 Origin of x-rays . . . 72
2.2.2 Continuous x-ray spectrum . . . 73
2.2.3 Characteristic x-ray spectrum . . . 75
2.3.1 Bragg’s law . . . 78 2.3.2 Bragg’s spectrometer . . . 79 2.3.3 Structure determination . . . 80 2.4 Electron diffraction . . . 81 2.5 Neutron diffraction . . . 82 Numerical Examples . . . 83 Exercise . . . 87
3 Electrical Conductivity In Metals 89 3.1 Introduction . . . 89
3.2 Classical Free Electron Theory Of Metals (Drude - Lorentz Theory) . . . 90
3.2.1 Expression for electrical conductivity . . . 90
3.2.2 Electron - lattice interaction and consequences . . . 95
3.2.3 Failure of classical free electron theory . . . 96
3.3 Quantum free electron theory of metals . . . 97
3.3.1 Density of energy states in a metal . . . 100
3.3.2 Metal as a Fermi gas . . . 103
3.3.3 Band theory of metals . . . 104
3.3.4 Merits of quantum free electron theory . . . 105
3.4 Electron Scattering Mechanisms . . . 106
3.4.1 Effect of temperature . . . 107 3.4.2 Effect of impurities . . . 108 3.5 Thermionic emission . . . 110 Numerical Examples . . . 112 Exercise . . . 115 4 Superconductivity 117 4.1 Introduction . . . 117
4.2 Characteristic features of superconductors . . . 118
4.2.1 Isotope effect . . . 119
4.4 Applications of superconductors . . . 123
4.5 Theoretical interpretation of superconductivity . . . 128
4.6 High Temperature Superconductors . . . 129
Numerical Examples . . . 130
Exercise . . . 131
5 Semiconductors 132 5.1 Band Structure of Solids . . . 132
5.2 Intrinsic semiconductors . . . 134
5.2.1 Carrier generation in intrinsic semiconductors . . . 135
5.2.2 Fermi factor and Fermi energy . . . 136
5.2.3 Conductivity of an intrinsic semiconductor . . . 137
5.2.4 Effect of temperature on conductivity . . . 138
5.3 Extrinsic semiconductors . . . 138
5.3.1 Conductivity of an extrinsic semiconductor . . . 141
5.3.2 Effect of temperature on the conductivity of extrinsic semicon-ductors . . . 142
5.3.3 Concentration and mobility of current carriers . . . 143
5.4 Generation and recombination of carriers . . . 146
5.5 Direct and indirect band gap semiconductors . . . 146
5.5.1 Semiconductor materials . . . 149
5.6 Hall effect . . . 151
5.6.1 Experimental determination of carrier concentration . . . 153
5.6.2 Hall effect in intrinsic semiconductors . . . 154
5.6.3 Applications of Hall effect . . . 155
5.7 p-n Junction . . . 155
5.7.1 Unbiased p-n junction . . . 156
5.7.2 Semiconductor junction with applied bias . . . 160
5.7.3 Incremental junction capacitance . . . 163
5.8.1 Zener breakdown mechanism . . . 165
5.8.2 Identification of breakdown mechanism in a p-n junction . . . 166
5.9 Applications of p-n junctions . . . 167
5.9.1 Junction diode as rectifier . . . 167
5.9.2 Zener diode . . . 168
5.9.3 Photo diode . . . 169
5.9.4 Photovoltaic effect and solar cell . . . 171
5.9.5 Light emitting diode . . . 174
Numerical Examples . . . 175
Exercise . . . 180
6 Dielectric Properties Of Materials 182 6.1 Introduction . . . 182
6.2 Polarization . . . 184
6.2.1 Mechanisms of polarization . . . 186
6.2.2 Temperature dependence of polarization . . . 189
6.2.3 Effect of frequency on polarization . . . 190
6.3 Dielectric Constant . . . 191
6.3.1 Dielectric constant of monoatomic gases . . . 191
6.3.2 Dielectric constant of polyatomic gases . . . 193
6.3.3 Internal field in solids and liquids . . . 193
6.3.4 Dielectric constant of elemental solids . . . 196
6.3.5 Dielectric constant of ionic solids without permanent dipoles . 197 6.3.6 Dielectric constant of polar materials . . . 198
6.4 Ferroelectric materials . . . 199
6.5 Piezoelectric Effect . . . 202
6.6 Dielectric losses . . . 203
Numerical Examples . . . 204
7.1 Introduction . . . 209
7.2 Classification of magnetic materials . . . 210
7.3 Origin of permanent dipoles . . . 212
7.4 Magnetic hysteresis . . . 216
7.5 Hard and soft magnetic materials . . . 218
7.6 Metallic and ceramic magnetic materials . . . 219
7.7 Ferrites . . . 220
7.8 Applications of magnetic materials . . . 221
Numerical Examples . . . 222
Exercise . . . 224
8 Applied Optics 225 8.1 Absorption and emission of radiation . . . 225
8.1.1 Luminescence . . . 225
8.1.2 Induced absorption . . . 226
8.1.3 Spontaneous emission . . . 227
8.1.4 Stimulated emission . . . 227
8.2 Lasers - basic principles . . . 228
8.2.1 Einstein’s theory of stimulated emission . . . 228
8.2.2 Conditions for laser action . . . 230
8.2.3 Methods of achieving population inversion . . . 231
8.2.4 Requirements of a laser system . . . 231
8.3 Types Of Lasers . . . 234
8.3.1 Ruby Laser . . . 234
8.3.2 Helium-Neon laser . . . 236
8.3.3 Semiconductor diode laser . . . 237
8.4 Applications of lasers . . . 239
8.4.1 Industrial applications . . . 239
8.4.2 Medical applications . . . 240
8.4.3 Estimation of atmospheric pollution . . . 240
8.5.1 Recording of holograms . . . 241
8.5.2 Reconstruction of images . . . 242
8.5.3 Applications of holography . . . 243
8.6 Optical fibers . . . 243
8.6.1 Materials for optical fibers . . . 243
8.6.2 Propagation of light through an optical fiber . . . 244
8.6.3 Modes of propagation in a fiber . . . 247
8.6.4 Signal distortion in optical fibers . . . 249
8.6.5 Signal attenuation in optical fibers . . . 249
8.7 Applications of optical fibers . . . 250
8.7.1 Fiber optic communication . . . 250
8.7.2 Applications in medicine and industry . . . 252
Numerical Examples . . . 252
Exercise . . . 254
9 Modern Materials And Methods 256 9.1 Ceramics . . . 256
9.1.1 Glasses . . . 256
9.1.2 Clay products . . . 257
9.1.3 Refractories and abrasives . . . 257
9.1.4 Cements . . . 258
9.1.5 Cermets . . . 258
9.2 Composite Materials . . . 259
9.3 Smart Materials . . . 260
9.4 Shape Memory Alloys . . . 261
9.5 Microelectromechanical Systems . . . 263
9.5.1 Sensors . . . 263
9.5.2 Actuators . . . 265
9.6 Nano Materials . . . 265
9.6.1 Synthesis of nano materials . . . 267
9.6.4 Carbon nano clusters . . . 273
9.6.5 Carbon nano tubes . . . 274
9.7 Liquid Crystals . . . 274
9.7.1 Classification of liquid crystals . . . 275
9.7.2 Applications of Liquid Crystals . . . 278
9.8 Non Destructive Testing Of Materials . . . 279
9.8.1 Radiographic methods . . . 279 9.8.2 Ultrasonic methods . . . 280 9.8.3 Magnetic methods . . . 283 9.8.4 Electrical methods . . . 284 9.8.5 Optical methods . . . 284 9.8.6 Thermal methods . . . 284 9.9 Quantum Computation . . . 285
9.9.1 Properties of quantum bits . . . 285
9.9.2 Quantum gates . . . 287
9.9.3 Multiple qubits . . . 288
Exercise . . . 289
10 Special Theory of Relativity 291 10.1 Introduction . . . 291
10.1.1 Frames of Reference . . . 291
10.1.2 Galilean transformation . . . 292
10.1.3 Michelson-Morley experiment . . . 293
10.2 Postulates of Special Theory of Relativity . . . 294
10.3 Time Dilation . . . 296
10.4 Length Contraction . . . 299
10.5 Twin Paradox . . . 301
10.6 Relativity of Mass . . . 301
Elements Of Wave Mechanics
1.1
Introduction
The nineteenth century was a very eventful period as far as Physics is concerned. The pioneering work on dynamics by Newton, on electromagnetic theory by Maxwell, laws of thermodynamics and kinetic theory were successful in explaining a wide variety of phenomena. Even though a majority of experimental evidence agreed with the classi-cal physics, a few experiments gave results that could not be explained satisfactorily. These few experiments led to the development of modern physics. Modern physics refers to the development of the theory of relativity and the quantum theory. Inabil-ity of the classical concepts to explain certain experimental observations, especially those involving subatomic particles, led to the formulation and development of mod-ern physics. Early twentieth century saw the development of modmod-ern physics. The pioneering work of Einstein, Planck, Compton, Roentgen, Born and others formed the basis of modern physics. The dual nature of matter proposed by de Broglie was confirmed by experiments. The wave mechanics and quantum mechanics were later shown to be identical in their mathematical formulation. The validity of classical con-cepts was explained to be the result of an extrapolation of modern theories to classical situations. In the present chapter, experimental observations of three important phe-nomena – black body radiation, photoelectric effect and Compton effect – considered as the beginning of modern physics, are briefly described.
1.2
Black body radiation
When radiation is incident on material objects, it is either absorbed, reflected or trans-mitted. These processes are dependent on the radiation and the object involved. An object that is capable of absorbing all radiation incident on it is called a black body. Practically, we cannot have a perfect black body but can have objects that are only close to a black body.
For example, a black body can be approximated by a hollow object with a very small hole leading to the inside of the object. Any radiation that enters the object through the hole gets trapped inside and will be reflected by the walls of the cavity till it is absorbed. Objects that absorb a particular wavelength of radiation are also
found to be a good emitter of radiation of that particular wavelength. Hence, a black body is also a good emitter of all radiations it has absorbed.
Emissions from objects depend on the temperature of the object. It has been ob-served that the energy emitted from objects increases as the temperature of the object is increased. Laws of radiation have been formulated to explain the emission of energy by objects maintained at specific temperatures.
1.2.1 Experimental observation of black body radiation
Experiments have been carried out to study the distribution of energy emitted by a practical blackbody as a function of wavelength and temperature.
E λ λ T T T T T T 1 2 3 1> 2> 3
Figure 1.1 Distribution of emitted energy as a function of wavelength and temperature for a black body.
Figure 1.1 shows the distribution curves in which the energy density Eλ is plotted
as a function of wavelength at different temperatures of the black body. Energy density is defined as the energy emitted by the black body per unit area of the surface. The important features of these distribution curves may be summarized as follows:
(i) The energy vs wavelength curve at a given temperature shows a peak indicating that the emitted intensity is maximum at a particular wavelength and decreases as we move away from the peak.
(ii) An increase in temperature results in an increase in the total energy emitted and also the energy emitted at all wavelengths.
(iii) As the temperature increases, the peak shifts to lower wavelengths. In other words, at higher temperatures, maximum energy is emitted at lower wavelengths.
1.2.2 Laws of black body radiation
The initial attempts to explain black body radiation were based on classical theories and were found to be limited in application. They could not explain the entire spectrum of the radiation satisfactorily.
1.2.3 Stefan Boltzmann radiation law
It states that the total energy density E◦ of radiation emitted from a black body is directly proportional to the fourth power of its absolute temperature T . Energy density E0is defined as the total of all the energy emitted at all wavelengths per unit area of the emitter surface.
E◦α T4
or E◦= σT4 (1.1)
where σ is a constant called Stefan’s constant. It has a numerical value equal to 5.67 × 10−8 watt m−2K−4. This law was suggested empirically by Stefan and later derived by Boltzmann on thermodynamic considerations. The law agrees well with the experimental results.
1.2.4 Wien’s Laws
Wien’s displacement law states that the wavelength λm corresponding to the maxi-mum emissive energy decreases with increasing temperature.
i.e., λm∝ 1/T or λmT = b (1.2)
where b is called the Wien’s constant and is equal to 2.9 × 10−3 mK. The energy
density emitted by a black body in the wavelength range λ and λ + d λ is given by Eλdλ = c1λ−5exp (−c2/λT )dλ (1.3) where c1 and c2 are constants. This is known as Wien’s distribution law. This law holds good for smaller values of λ but does not fit the experimental curves for higher values of λ (fig 1.2).
λ Eλ
Wien′s law
Figure 1.2 Comparison of experimental distribution curve with Wien’s law.
1.2.5 Rayleigh – Jean’s law
According to this law, the energy density emitted by a black body in the wavelength range λ And λ + dλ is given by
Eλdλ =
8πkT
λ4 dλ (1.4)
This equation does not show any peak in the energy value but the energy goes on increasing with decrease in wavelength. The total energy emitted is infinite for all temperatures above 0K.
This is not at all in agreement with the experimental observation. The law holds good only for large values of wavelength (fig 1.3). At lower wavelengths, the energy density increases and becomes very large for wavelengths in the ultra violet region. Such a large increase in the energy emitted at low wavelength does not occur experi-mentally. This discrepancy is known as “Ultraviolet catastrophe” of classical physics.
All the above laws are based on classical thermodynamics and statistics. They are insufficient to explain the black body radiation satisfactorily.
1.2.6 Planck’s radiation law
This law is based on quantum theory. Max Planck proposed that atoms or molecules absorb or emit radiation in quanta or small energy packets called photons. Energy of
λ Eλ
T R J law
Figure 1.3 Comparison of experimental distribution curve with Rayleigh-Jean’s law.
each photon can be expressed as
E = hν
where ν is the frequency of the radiation corresponding to the energy E, h is a constant called Planck’s constant and is equal to 6.63 × 10−34 Js. Light quanta are
indistin-guishable from each other and there is no restriction on the number of quanta having the same energy. In other words, Pauli’s exclusion principle is not applicable to them. The quantum statistics applicable to photons is Bose-Einstein statistics. Considering all the energy emitted by the black body in the form of photons of different energy, Planck applied Bose - Einstein statistics to obtain the energy distribution of photons. Accordingly, the energy density emitted in the wavelength range λ and (λ + dλ) is given by Eλdλ = 8πhc λ5 1 (ehc/λkT − 1)dλ (1.5) This distribution agrees well with the experimental observation of black body ra-diation and is valid for all wavelengths. Further, it reduces to Wien’s law for lower wavelength region and to Rayleigh – Jean’s law for higher wavelength region.
1.2.7 Derivation of Wien’s law from Planck’s law When λ is small, we can consider
∴ h
ehc/λkT − 1i≈ ehc/λkT Substituting in equation (1.5), we get
Eλ dλ = 8πhc λ5 · 1 ehc/λkT · dλ i.e., Eλ dλ = c1λ−5. exp (−c2/λT )dλ (1.6) where c1 = 8πhc and c2 = hc/k
Equation (1.6) is the Wien’s law.
1.2.8 Derivation of Rayleigh – Jean’s law from Planck’s law When λ is large, λkThc < 1.
h
ehc/λkT − 1i≈ hc/λkT Substituting in equation (1.5), we get
Eλ dλ = 8πhc λ5 · λkT hc · dλ i.e., Eλ dλ = 8πkT λ4 dλ (1.7)
Equation (1.7) is the Rayleigh – Jean’s law.
1.3
Photoelectric e
ffect
Emission of electrons from a metal surface when light of suitable energy falls on it is called Photoelectric effect. The experimental setup for observing photoelectric effect consists of a pair of metal plate electrodes in an evacuated tube connected to a source of variable voltage as shown in fig.1.4.
When light of suitable energy is incident on the cathode, electrons are emitted and a current flows across the tube. The characteristic curves for the photoelectric emission as shown in fig. 1.5.
The important properties of the emission are as follows:
(i) There is no time interval between the incidence of light and the emission of pho-toelectrons.
A
V
Figure 1.4 Experimental set up to study photoelectric effect.
L3
L2
L1
I
V
Figure 1.5 Current - voltage characteristics of photocell. The Intensity of illumination increases from L1to L3.
(ii) There is a minimum frequency for the incident light below which no photoelec-tron emission occurs. This minimum frequency, called threshold frequency, depends on the material of the emitter surface. The energy corresponding to this threshold frequency is the minimum energy required to release an electron from the emitter surface. This energy is characteristic of the material of the emitter and is called the work function of the material of the emitter.
(iii) For a given constant frequency of incident light, the number of photoelectrons emitted or the photo current is directly proportional to the intensity of incident light.
(iv) The photoelectron emission can be stopped by applying a reverse voltage to the phototube, i.e. by making the emitter electrode positive and the collector nega-tive. This reverse voltage is independent of the intensity of incident radiation but increases with increase in the frequency of incident light. The negative collec-tor potential required to stop the photo electron emission is called the stopping potential.
These characteristics of photoelectron emission can not be explained on the basis of classical theory of light but can be explained using the quantum theory of light. Ac-cording to this theory, emission of electrons from the metal surface occurs when the energy of the incident photon is used to liberate the electrons from their bound state. The threshold frequency corresponds to the minimum energy required for the emission. This minimum energy is called the work function of the metal. When the incident pho-ton carries an energy in excess of the work function, the extra energy appears as the kinetic energy of the emitted electron. When the intensity of light increases, the num-ber of photoelectrons emitted also increases but their kinetic energy remains unaltered. The reverse potential required to stop the photoelectron emission, i.e. the stopping po-tential, depends on the energy of the incident photon and is numerically equivalent to the maximum kinetic energy of the photoelectrons.
When a photon of frequency ν is incident on a metal surface of work function Φ, then, hν = Φ + 1 2mv 2 ! max (1.8) where (12mv2)
maxis the maximum kinetic energy of the emitted photoelectrons. This is known as Einstein’s photoelectric equation. Since Φ = hv◦, it can also be written as
1 2mv 2 ! max = hv − Φ = h(v − v◦) (1.9)
If V◦is the stopping potential corresponding to the incident photon frequency v, then, 1 2mv 2 ! max = hv − Φ = eV◦ (1.10)
Then, by experimental determination of V◦, it is possible to find out the work func-tion of the metal.
The experimental observation of photoelectric effect leads to the conclusion that the energy in light is not spread out over wavefronts but is concentrated in small packets called photons. All photons of a particular frequency have the same energy. A change
in the intensity of the incident light will change the number of photoelectrons emitted but not their energies. Higher the frequency of the incident light, higher will be the kinetic energy of the photoelectrons. These observations confirm the particle properties of light waves.
1.4
Compton e
ffect
When x-rays are scattered by a solid medium, the scattered x-rays will normally have the same frequency or energy. This is a case of elastic scattering or coherent scatter-ing. However, Compton observed that in addition to the scattered x-rays of same fre-quency, there existed some scattered x-rays of a slightly higher wavelength (i.e., lower frequency or lower energy). This phenomenon in which the wavelength of x-rays show an increase after scattering is called Compton effect.
Compton explained the effect on the basis of the quantum theory of radiation. Con-sidering radiation to be made up of photons, he applied the laws of conservation of energy and momentum for the interaction of photon with electron. Consider an x-ray photon of energy hν incident on an electron at rest (fig. 1.6.) After the interaction, the x-ray photon gets scattered at an angle θ with its energy changed to a value hν′ and
the electron which was initially at rest recoils at an angle Φ. It can be shown that the increase in wavelength is given by
△λ = h
m◦c(1 − cos θ) (1.11)
where m◦ is the rest mass of the electron.
E = hν p = hν/c E′ = hν′ p′ = hν′/c θ Φ
Figure 1.6 Schematic diagram of the scattering of a photon by a stationary electron.
When θ = 90◦, △λ = h
m◦c = 0.0242A
◦.
This constant value is called Compton wavelength. When θ = 180◦, △λ = 2h
Experimental observation indicate that the change in the wavelength of the scat-tered x-rays is indeed in agreement with equation (1.11), thus providing further confir-mation to the photon model.
Thus, Planck’s theory of radiation, photoelectric effect and Compton effect are experimental evidences in favour of the quantum theory of radiation.
1.5
Matter waves and de Broglie’s hypothesis
Quantum theory and the theory of relativity are the two important concepts that led to the development of modern physics. The quantum theory was first proposed by Planck to explain and overcome the inadequacies of classical theories of black body radiation. The consequences were very spectacular. Louis de Broglie made the suggestion that particles of matter, like electrons, might possess wave properties and hence exhibit dual nature. His hypothesis was based on the following arguments:
The Planck’s theory of radiation suggests that energy is quantized and is given by
E = hv (1.12)
where ν is the frequency associated with the radiation. Einstein’s mass-energy relation states that
E = mc2 (1.13)
Combining the two equations, it can be written as E = hv = mc2
Hence, the momentum associated with the photon is given by p = mc = hv/c = h/λ
Extending this to particles, he suggested that any particle having a momentum p is associated with a wave of wavelength λ given by
λ = h/p (1.14)
This is called de Broglie’s hypothesis of matter waves and λ is called the de Broglie wavelength.
The de Broglie wavelength can be calculated for any particle using the above re-lation. In case of charged particles like electrons, a beam of high energy particles can be obtained by accelerating them in an electric field. For example, an electron starting
from rest when accelerated with a potential difference V, the kinetic energy acquired by the electron is given by
(1/2)mv2 = eV
where v is the velocity of the electron. The momentum may be calculated as p = mv = (2meV)1/2
Using the de Broglie equation, the wavelength associated with the accelerated elec-tron can be calculated as
λ = h/p = h/(2meV)1/2 (1.15)
This equation suggests that, at a given speed, the de Broglie wavelength associated with the particle varies inversely as the mass of the particle. This concept of mat-ter waves aroused great inmat-terest and several physicists launched experiments designed to test the hypothesis. Heisenberg and Schrodinger proceeded on to develop mathe-matical theories whereas Davisson and Germer, G.P. Thomson and Kikuchi attempted experimental verification.
1.5.1 Davisson-Germer experiment
The hypothesis of de Broglie was verified by the electron diffraction experiment con-ducted by Davisson and Germer in the United States. The experimental set up used by them is shown in the figure 1.7.
Figure 1.7 Experimental arrangement for Davisson-Germer experiment.
The apparatus consists of a filament heated with a small a.c power supply to pro-duce thermionic emission of electrons. These electrons are attracted towards an anode
in the form of a cylinder with a small aperture maintained at a finite positive potential with respect to the filament. They pass through the narrow aperture forming a fine beam of accelerated electrons. This electron beam was made to incident on a single crystalline sample of nickel. The electrons scattered at different angles were counted using an ionization counter as a detector. The experiment was repeated by record-ing the scattered electron intensities at various positions of the detector for different accelerating potentials (Fig.1.8).
44ev φ
0 48ev 54ev
50◦
64ev 68ev
Figure 1.8 Scattered electron intensity maps at different accelerating potentials.The vertical axis represents the direction of the incident electron beam and Φ is the scat-tering angle.The radial distance from the origin at any angle represents the intensity of scattered electrons.
When a beam of electrons accelerated with a potential of 54 V was directed per-pendicular to the nickel target, a sharp maximum occurred in the electron density at an angle of 50◦ with the incident beam. When the angle Φ between the direction of the incident beam and the direction of the scattered beam is 500, the angle of incidence will be 250 and the corresponding angle of diffraction θ will be 650. The spacing of the planes responsible for diffraction was found to be 0.091nm from x-ray diffraction experiment. Assuming first order diffraction, the wavelength of the electron beam can be calculated as
λ = 2d sin θ = 2 × 0.091 × sin 650= 0.165nm.
The wavelength of the electrons can also be calculated using the de Broglie’s relation as λ = h/ (2meV)1/2
= 6.63 × 10−34/(2 × 9.1 × 10−31 × 1.6 × 10−19× 54)1/2 = 0.166nm.
Thus, the Davisson-Germer experiment directly verifies the de Broglie’s hypothe-sis.
1.5.2 G.P. Thomson experiment
At almost the same time as the Davisson-Germer experiment, G.P.Thomson of England carried out electron diffraction experiments independently using a thin polycrystalline foil of aluminium metal. The experimental set up is shown in fig. 1.9.
Electron Gun
Aluminium Foil
Screen
θ
Figure 1.9 Experimental arrangement for G.P.Thomson experiment.
He allowed a beam of accelerated electrons to fall on the aluminium foil and ob-served a diffraction pattern consisting of a series of concentric rings around the di-rection of the incident beam. This pattern was similar to the Debye-Scherrer pattern obtained for aluminium using x-ray diffraction. Using the data available on aluminium, he calculated the wavelength of the electrons using the Bragg’s equation,
nλ = 2d sin θ
He also calculated the de Broglie wavelength of the electrons with the knowledge of accelerating potential using the relation,
λ = h/(2meV)1/2
The value of wavelength calculated from the two equations matched well thereby ex-perimentally proving the de Broglie’s relation.
A similar experiment was conducted by Kikuchi in Japan in which he obtained electron diffraction pattern by passing an electron beam through a thin foil of mica to confirm the validity of de Broglie’s relation.
The wave nature of particles is not restricted to electrons. Any particle with a mo-mentum p has a de Broglie wavelength equal to (h/p). Neutrons produced in nuclear reactors possess energies corresponding to wavelength of the order of 0.1nm. These particles also should be suitable for diffraction by crystals. Neutrons from a nuclear reactor are slowed down to thermal energy of the order of kT and used for diffrac-tion and interference experiments. The results agree well with the de Broglie reladiffrac-tion. Since neutrons are uncharged particles, they are particularly useful in certain situations for diffraction studies. Neutron beams have also been used as probes to investigate the magnetic properties of nuclei.
1.5.3 Wave packet and de Broglie waves
We have seen that moving particles may be represented by de Broglie waves. The am-plitude of these de Broglie waves does not represent any parameter directly describing the particle but is related to the probability of finding the particle at a particular place at a particular time. Hence, we cannot describe de Broglie waves with a simple wave equation of the type,
y = A cos(ωt − kx) (1.16) Instead, we have to use an equation representing a group of waves. In other words, a wave packet consisting of waves of slightly differing wavelengths may represent the moving particle. Superposition of these waves constituting the wave packet results in the net amplitude being modified, thereby defining the shape of the wave group. The phase velocity of individual waves depends on the wavelength. Since the wave group consists of waves with different wavelengths, all the waves do not proceed together and the wave group has a velocity different from the phase velocities of the individual waves. Hence, de Broglie waves may be associated with group velocity rather than the phase velocity.
1.5.4 Characteristics of matter waves
1. Matter waves are associated with a moving body.
2. The wavelength of matter waves is inversely proportional to the velocity with which the body is moving. Hence, a body at rest has an infinite wavelength and the one traveling with a high velocity has a lower wavelength.
3. Wavelength of matter waves also depends on the mass of the body and decreases with increase in mass. Due to this reason, the wavelike behaviour of heavier bodies is not very evident whereas wave nature of subatomic bodies could be observed experimentally.
4. A wave is normally associated with some quantity that varies periodically with the frequency of the wave. For example, in a water wave, it is the height of the water surface; in a sound wave it is the pressure and in an electromagnetic wave, it is the electric and magnetic fields that vary periodically. But in matter waves, there is no physical quantity that varies periodically. We use a wave function to define matter waves and this wave function is related to the probability of finding the particle at any place at any instant, which varies periodically.
5. Matter waves are represented by a wave packet made up of a group of waves of slightly differing wavelengths. Hence, we talk of group velocity of matter waves rather than the phase velocity. The group velocity can be shown to be equal to the particle velocity.
6. Matter waves show properties similar to other waves. For example, a beam of accelerated electrons produces interference and diffraction effects similar to an electromagnetic wave of same wavelength.
1.6
Phase and group velocities
A wave is represented by the formula
y = A cos(ωt − kx) (1.16) where y is the displacement at any instant t, A is the amplitude of vibration, ω is the angular frequency equal to 2πν and k is the wave vector, equal to (2π/λ). The phase velocity of such a wave is the velocity with which a particular phase point of the wave travels. This corresponds to the phase being constant.
i.e., (ωt − kx) = constant
or x = constant + ωt/k
Phase velocity vp = dx/dt = ω/k
= 2πν/(2π/λ) = λν (1.17) vpis called the ‘wave velocity’ or ‘phase velocity’.
The de Broglie waves are represented by a wave packet and hence we have ‘group velocity’ associated with them. Group velocity is the velocity with which the wave packet travels. In order to understand the concept of group velocity, consider the com-bination of two waves represented by the formula
y1= A cos(ωt − kx)
y2= A cos{(ω + △ω)t − (k + △k)x} The resultant displacement is given by
y = y1+ y2 = 2A cos{(ω + ω + △ω)t − (k + k + △k)x} 2 cos (△ωt − △kx) 2 ≈ 2A cos(ωt − kx). cos △ωt 2 − △kx 2 ! (1.18)
The velocity of the resultant wave is given by the speed with which a reference point, say the maximum amplitude point, moves. Taking the amplitude of the resultant wave as constant, we have
2A cos △ωt 2 − △kx 2 ! = constant or △ωt 2 − △kx 2 ! = constant or x = constant + (△ωt/△k) Group velocity vg = dx/dt = (△ω/△k) (1.19) Instead of two discrete values for and k, if the group of waves has a continuous spread from ω to (ω + △ω) and k to (k + △k), then, the group velocity is given by
vg= dω
dk (1.20)
It can be shown that the group velocity of the wave packet is equal to the velocity of the particle with which the wave packet is associated.
1.6.1 Relation between phase velocity and group velocity We have the mathematical relation for phase velocity given by
The group velocity vgis given by vg = dω dk = d(k · vp) dk = vp+ k · dvp dk = vp+ (2π/λ) · dvp d(2π/λ) = vp+ (2π/λ) · (−λ2/2π).dvp dλ = vp− λ · dvp dλ (1.21)
In the above expression, if (dvp/dλ) = 0, i.e., if the phase velocity does not depend on wavelength, then the group velocity and phase velocity are equal. Such a medium is called a non-dispersive medium. In a dispersive medium, (dvp/dλ) is positive and hence the group velocity is less than the phase velocity.
1.6.2 Relation between group velocity and particle velocity
(Velocity of de Broglie waves)
The phase velocity of waves depend on the wavelength. This is responsible for the well known phenomenon of dispersion. In the case of light waves in vacuum, the phase velocity is same for all wavelengths. In the case of de Broglie waves, we have,
ω = 2πν = 2πmc2/h = 2πm0c 2
h(1 − v2/c2)1/2 (1.22) and k = 2π/λ = 2πmv/h = 2πm0v
h(1 − v2/c2)1/2 (1.23) The group velocity of de Broglie waves is given by
vg = dω/dk = dω/dv dk/dv dω/dv = (2πm0c2/h) · d dv(1 − v 2/c2)1/2= 2πm0v h(1 − v2/c2)3/2 (1.24) dk/dv = 2πm0 h(1 − v2/c2)3/2 (1.25)
From equations (1.24) and (1.25) we get, vg = v
Thus, the group velocity associated with de Broglie waves is just equal to the velocity with which the particle is moving. If we try to calculate the phase velocity,
vp= ω/k = c2/v = c2/vg (1.26) Since the group velocity or the particle velocity is always less than c, the phase velocity of de Broglie waves turn out to be greater than c. This only indicates that we cannot talk of phase velocity of de Broglie waves since they are made up of a group of waves. Phase velocity has no physical significance for de Broglie waves.
1.6.3 Derivation of de Broglie relation
The de Broglie relation may be derived as follows. If we assume a particle having a kinetic energy equal to mv2/2 to have a de Broglie wavelength λ, we can write
hν = mv2/2 (assuming the energy of the particle to be purely kinetic)
or ν = m
2h· v
2 (1.27)
Differentiating with respect to λ, dν dλ = m 2h· 2v · dv dλ (1.28) But we have vg = v = dω dk = 2πdν 2πd(1/λ) = −λ 2dν dλ ∴ dν dλ = − v λ2 (1.29)
Substituting in equation(1.28), we get mv h · dv dλ = − v dλ2 Rewriting this, we have
dv dλ = −
h
Integrating with respect to λ,
v= h mλ + c
By applying the boundary condition that the wavelength tends to infinity as the velocity tends to zero, we find that the constant of integration has to be zero. Hence, we get
λ = h
mv (1.31)
which is the de Broglie relation.
1.7
Heisenberg’s Uncertainty principle
1.7.1 Origin and nature of the Principle
When we assign wave properties to particles there is a limitation to the accuracy with which we can measure the properties like position and momentum.
∆x
Figure 1.10 A wave packet with an extension △x along x-axis.
Consider a wave packet as shown in fig.1.10. The particle to which this wave packet corresponds to may be located anywhere within the wave packet at any instant. The probability density suggests that it is most likely to be found in the middle of the wave packet. However, there is a finite probability of finding the particle anywhere within the wave packet. If the wave packet is smaller in extension, the position of the particle can be specified more precisely. But the wavelength of the waves will not be well defined in a narrow wave packet. Since wavelength is related to momentum through
de Broglie’s relation, the momentum is not precisely known. On the otherhand, a wave packet with large extension can have a more clearly defined wavelength and hence momentum at the cost of the knowledge about the position. This leads to the conclusion that it is impossible to know both the position and momentum of an object precisely at the same time. This is known as Uncertainty principle.
For a wave packet of extension △x with an uncertainty in the wave number △k assuming the uncertainties to be the standard deviation in the respective quantities, it may be shown that a minimum value of the product of such deviations is given by
△x · △k = 1
2 (1.32)
This minimum value of the product of uncertainties is for the case of a gaussian distri-bution of the wave functions. Since the wave packets in general do not have gaussian forms, the uncertainty relation becomes
△x · △k ≥ 1 2 (1.33) But we have k = 2π/λ (1.34) Also λ = h/p (1.35) Hence, k = 2πp/h △k = 2π h · △p (1.36)
Substituting in equation (1.33), we get △x · △p ≥ h
4π
or △x · △p ≥ ~
2 (1.37)
This equation states that the product of uncertainty △x in the position of an object at some instant and the uncertainty in the momentum in the x-direction at the same instant is equal to or greater than ~/2.
Another form of uncertainty principle relates energy and time. In the atomic pro-cess, if energy E is emitted as an electromagnetic wave during an interval of time △t, then, the uncertainty △E in the measured value of E depends on the duration of the time interval △t according to the equation,
It may be mentioned that these uncertainties are not due to the limitations of the precision of the measuring methods or measuring instruments but due to the nature of the quantities involved.
1.7.2 An illustration of uncertainty principle
We have the following ‘Thought experiment’ to illustrate the uncertainty principle. Imagine an electron being observed using a microscope (fig.1.11).
Microscope
LightSource
Electron
Figure 1.11 Schematic diagram of experimental set up to study uncertainty principle.
The process of observation involves a photon of wavelength λ incident on the elec-tron and getting scattered into the microscope. The event may be considered as a two-body problem in which a photon interacts with an electron. The change in the velocity of the photon during the interaction may be anything between zero( for grazing angle of incidence) and 2c (for head-on collision and reflection). The average change in the momentum of the photon may be written as equal to (hν/c) or (h/λ). This difference in momentum is carried by the recoiling electron which was initially at rest. The change or uncertainty in the momentum of the electron may thus be written as (h/λ). At the same time, the position of the electron can be determined to an accuracy limited by the resolving power of the microscope, which is of the order of λ. Hence, the product of the uncertainties in position and momentum is of the order of h. This argument implies that the uncertainties are associated with the measuring process. This illustration only estimates the accuracy of measurement, the uncertainty being inherent in the nature of the moving particles involved.
1.7.3 Physical significance of uncertainty principle
Uncertainty principle is a consequence of the wave particle duality. It states that it is impossible to know both the position and momentum of an object exactly and at the same time. Mathematically, it can be shown that the product of uncertainties in the position and momentum measured simultaneously will have a value greater that ~/2, i.e., (h/4π). If △x is the uncertainty in the measurement of the position x of an object and △px is the uncertainty in the measurement of momentum px , then, at any instant,
△x · △px > ~/2
We can try to estimate the product of the uncertainties with the help of illustrations as the one mentioned above. The principle is based on the assumption that a moving par-ticle is associated with a wave packet, the extension of which in space accounts for the uncertainty in the position of the particle. The uncertainty in the momentum arises due to the indeterminacy of the wavelength because of the finite size of the wave packet. Thus, the uncertainty principle is not due to the limited accuracy of measurement but due to the inherent uncertainties in determining the quantities involved. But we can still define the position where the probability of finding the particle is maximum and also the most probable momentum of the particle.
1.7.4 Applications of uncertainty principle
The uncertainty principle has far reaching implications. In fact, it has been very useful in explaining many observations which cannot be explained otherwise. A few of the applications of the uncertainty principle are worth mentioning.
(a) Diffraction of a beam of electrons Diffraction of a beam of electrons at a slit is the effect of uncertainty principle. As the slit is made narrower, thereby reducing the uncertainty in the position of the electrons in the beam, the beam spreads even more indicating a larger uncertainty in its velocity or momentum.
Figure 1.12 shows the diffraction of an electron beam by a narrow slit of width △x. The beam traveling along OX is diffracted along OY through an angle θ. Due to the wave nature of the electron, we observe Fraunhoffer diffraction on the screen placed along XY. The accuracy with which the position of the electron is known is △x since it is uncertain from which place in the slit the electron passes. According to the theory of diffraction, we have
X Y
θ O
Figure 1.12 Diffraction at a single slit.
Further, the initial momentum of the electron along XY was zero and after diffrac-tion, the momentum of the electron is p · sin θ where p is the momentum of the electron along the incidence direction. Hence, the change in momentum of the electron along XY is p · sinθ or (h/λ) · sin θ. Assuming the change in the momen-tum as representative of the uncertainty in momenmomen-tum, we get
△x · △px = λ sin θ·
h · sin θ
λ = h
(b) Nuclear beta decay: In beta decay, electrons are emitted from the nucleus of the radioactive element. Assuming the diameter of the nucleus to represent the uncertainty in the position of electron inside the nucleus, the uncertainty in the momentum can be calculated as follows:
Radius of the nucleus = r = 5 × 10−15m △x = 2r = 10−14m.
△p = h/2π△x = 6.62 × 10−34/(2 × 3.14 × 10−14) = 1.055 × 10−20kgms−1
Assuming that the electron was at rest before its emission, the change in momen-tum can be taken as equal to its momenmomen-tum. This magnitude of change in mo-mentum indicates large velocity for the electron. Hence, the energy of the emitted electron will be
E = pc = 1.055 × 10−20× 3 × 108 = 3.165 × 10−12J = 19.8MeV.
This indicates that the electrons inside the nucleus must have kinetic energy of 19.8 MeV. But the electrons emitted during beta decay have kinetic energy of the order of 1 MeV. This indicates that electrons do not exist in the nucleus of the atom but are ‘manufactured’ by the nucleus at the time of decay.
(c) Binding energy of an electron in an atom: In a hydrogen atom, the electron revolves round the nucleus in an orbit of radius 5 × 10−11m. Assuming this as the maximum uncertainty in position, we can calculate the minimum uncertainty in the momentum as
(△p)min = h/2π(△x)max = 2.1 × 10−24kgms−1.
Assuming this as the momentum of electron, the kinetic energy of the electron will be equal to
K.E. = p2/2m = 2.45 × 10−18J = 15.3eV.
Thus, the binding energy of an electron in hydrogen atom is nearly 15 eV which is found to be correct experimentally.
(d) Nitrogen doping of silicon: The laws of conservation of energy and momentum restrict the generation and recombination processes in semiconductors. Silicon, which is an indirect band gap semiconductor, has low efficiency as a material for photo diode or light emitting diode. Nitrogen doping of silicon will bind the free electrons to the lattice thereby restricting the value of uncertainty in position. This results in a larger uncertainty in momentum thereby increasing the probability for generation or recombination process.
1.8
Wave mechanics
Quantum theory is based on the quantization of energies. It deals with the particle nature of radiation. It implies that addition or liberation of energy will be between discrete energy levels. It assigns particle status to a packet of energy by calling it ‘quantum of energy’ or ‘photon’ and treats the interaction of radiation with matter as a two-body problem. On the other hand, de Broglie’s hypothesis and the concept of matter waves led to the development of a different formulation called ‘Wave mechan-ics’. This deals with the wave properties of material particles. It was shown later that the quantum mechanics and the wave mechanics are mathematically identical and lead to the same conclusion.
1.8.1 Characteristics of wave function
Waves in general are associated with quantities that vary periodically. For example, water waves involve the periodic variation of the height of the water surface at a point. Similarly, sound waves are associated with periodic variations of the pressure. In the case of matter waves, the quantity that varies periodically is called ‘wave function’. The wave function, represented by Ψ, associated with matter waves has no direct phys-ical significance. It is not an observable quantity. But the value of the wave function is related to the probability of finding the body at a given place at a given time. The square of the absolute magnitude of the wave function of a body evaluated at a partic-ular time at a particpartic-ular place is proportional to the probability of finding the body at that place at that instant.
The wave functions are usually complex. The probability in such a case is taken as ψ ∗ ψ, i.e. the product of the wave function with its complex conjugate. Since the probability of finding the body somewhere is finite, we have the total probability over all space equal to certainty.
i.e., Z
ψ ∗ ψ dV = 1 (1.39)
Equation (1.39) is called the normalization condition and a wave function that obeys the equation is said to be normalized. Further, Ψ must be single valued since the probability can have only one value at a particular place and time. Since the probability can have any value between zero and one, the wave function must be continuous. Mo-mentum being related to the space derivatives of the wave function, the partial deriva-tives ∂Ψ/∂x, ∂Ψ/∂y and ∂Ψ/∂z must also be continuous and single valued everywhere. Thus, the important characteristics of wave function are as follows:
1. Ψ must be finite, continuous and single valued everywhere.
2. ∂Ψ/∂x, ∂Ψ/∂y and ∂Ψ/∂z must be finite, continuous and single valued every-where.
3. Ψ must be normalizable.
1.8.2 Physical significance of wave function
We have already seen that the wave function has no direct physical significance. How-ever, it contains information about the system it represents and this can be extracted
by appropriate methods. Even though the wave function itself is not directly an ob-servable quantity, the square of the absolute value of the wave function is intimately related to the moving body and is known as the probability density. This probability density is the quantum mechanical method of finding the body at a particular position at a particular time. The wave function carries information about the particle’s wave-like behaviour. It also provides information about the momentum and energy of the particle at any instant of time.
1.8.3 Schrodinger’s wave equation
The motion of a free particle can be described by the wave equation.
Ψ = Aexp{−i(ωt − kx)} (1.40)
But ω = 2πν = 2π(E/h) = (E/~) and k = 2π/λ = 2π(p/h) = (p/~)
where E is the total energy and p is the momentum of the particle. Substituting in the equation (1.40), we get, Ψ = A exp −i ~(Et − px) (1.41) Differentiating equation (1.41) with respect to x twice, we get,
∂2Ψ ∂x2 = −p2 ~2 Ψ or p 2 Ψ = −~2· ∂ 2Ψ ∂x2 (1.42)
Differentiating equation (1.41) with respect to t, we get, ∂Ψ ∂t = − iE ~ ·Ψ or EΨ = − ~ i · ∂Ψ ∂t (1.43)
The total energy of the particle can be written as
E = p
2
2m + U (1.44)
where U is the potential energy of the particle. Multiplying both sides of the equation by Ψ
E Ψ = p
2Ψ
2m + U Ψ (1.45)
Substituting for EΨ and p2Ψ from equation (1.42) and (1.43) −~ i ∂Ψ ∂t = − ~2 2m ∂2ψ ∂x2 + UΨ (1.46)
This is known as Schrodinger’s time dependent equation in one dimension. The wave function Ψ in equation (1.41) may also be written as
Ψ = A exp i ~(Et − px) = A exp(−iEt) ~ · exp (ipx) ~ Ψ = Φ exp(−iEt) ~ (1.47)
where Φ is a position dependent function. Substituting this form of Ψ in equation (1.45),
EΦ exp(−iEt)
~ = p2 2mΦ exp (−iEt) ~ + UΦexp (−iEt) ~ or EΦexp(−iEt) ~ = − ~2 2m · ∂2Φ ∂x2 · exp (−iEt) ~ + UΦ exp (−iEt) ~ or ∂ 2Φ ∂x2exp (−iEt) ~ + 2m ~2(E − U)Φexp (−iEt) ~ = 0 or ∂ 2Ψ ∂x2 + 2m ~2 (E − U)Ψ = 0 (1.48)
This is the Schrodinger’s wave equation in one dimension. In three dimensions, the above equation may be written as
∂2Ψ ∂x2 + ∂2Ψ ∂y2 + ∂2Ψ ∂z2 + 2m(E − U)Ψ ~2 = 0 or ∇2Ψ +2m(E − U)Ψ ~2 = 0
This equation is known as the steady state or time independent Schrodinger wave equation in three dimensions.
1.8.4 Eigen values and eigen functions
These terms come from the German words and mean proper or characteristic values or functions respectively. The values of energy for which the Schrodinger’s equation can be solved are called ‘Eigen values’ and the corresponding wave functions are called ‘Eigen functions’. The eigen functions possess all the characteristics properties of wave functions in general (see section 1.8.1).
1.9
Applications Of Schrodinger’s Equation
1.9.1 Case of a free particle
A free particle is defined as one which is not acted upon by any external force that modifies its motion. Hence, the potential energy U in the Schrodinger’s equation is a constant and does not depend on position or time. For convenience, the potential energy may be assumed to be zero. Then, the Schrodinger’s equation for the particle becomes
∂2Ψ ∂x2 +
2m
~2EΨ = 0 (1.49) where E is the total energy of the particle which is purely kinetic. This is of the form,
∂2Ψ ∂x2 + k
2Ψ = 0
where k2 = 2mE/~2. The solution of this equation may be written as Ψ = A sin kx + B cos kx
Solving for the constants A and B pose some difficulties because we cannot apply any boundary conditions on the wave function as it represents a single wave which is not localized and not normalizable. Since the solution has not imposed any restriction on the value of k, the free particle is permitted to have any value of energy given by the equation,
E = ~2k2/2m
Since the total energy is purely kinetic, the momentum of the particle would be p = ~k or h/λ. This is just what we would expect, since we have constructed the Schrodinger equation to yield the solution for the free particle corresponding to a de Broglie wave.
1.9.2 Particle in a box
The simplest problem for which Schrodinger’s time independent equation can be ap-plied and solved is the case of a particle trapped in a box with impenetrable walls.
Consider a particle of mass m and energy E travelling along x-axis inside a box of width L. The particle is thus restricted to move inside the box by reflections at x = 0 and x = L (Fig. 1.13).
The particle does not lose any energy when it collides with the walls and hence the total energy of the particle remains constant. The potential energy of the particle is
considered to be zero inside the box and infinite outside. Since the total energy of the particle cannot be infinite, it is restricted to move within the box. The example is an oversimplified case of an electron acted upon by the electrostatic potential of the ion cores in a crystal lattice.
X = O X = L U
Figure 1.13 Schematic for a particle in a box. The height of the wall extends to infinity.
Since the particle cannot exist outside the box,
Ψ = 0 for x ≤ 0 and x ≥ L (1.50)
We have to evaluate the wave function inside the box. The Schrodinger’s equation (1.48) becomes ∂2Ψ ∂x2 + 2m ~2EΨ = 0 for 0 ≤ x ≤ L (1.51) Ψ = A sin 2mE ~2 !1/2 x + B cos 2mE ~2 !1/2 x (1.52)
where A and B are constants.
Applying the boundary condition that Ψ = 0 at x = 0, equation (1.52) becomes A sin 0 + B cos 0 = 0 or B = 0.
Again, we have Ψ = 0 at x = L. Then,
A. sin 2mE ~2
!1/2
If A = 0, the wavefunction will become zero irrespective of the value of x. Hence, A cannot be zero.
Therefore, sin 2mE ~2 !1/2 · L = 0 or 2mE ~2 !1/2 L = nπ where n = 1, 2, 3, . . . (1.53)
From (1.53), the energy eigen values may be written as
En=
n2π2~2
2mL2 where n = 1, 2, 3, . . . (1.54) From this equation, we infer that the energy of the particle is discrete as n can have integer values. In other words, the energy is quantized. We also note that n cannot be zero because in that case, the wave function as well as the probability of finding the particle becomes zero for all values of x. Hence, n = 0 is forbidden. The lowest energy the particle can possess is corresponding to n = 1 and is equal to
E1= π2~2 2mL2
This is called ‘ground state energy’ or ‘zero point energy’. The higher excited states will have energies like 4E1, 9E1, 16E1, etc. This indicates that the energy levels are not equally spaced.
The wave functions or the eigen functions are given by
Ψn = A · sin 2mEn ~2 !1/2 x or Ψn = A · sinnπ L x (1.55)
Applying the normalization condition,
i.e. Z
A2sin2nπx
L · dx = 1 (1.56)
Since the wave function is non-vanishing only for 0 ≤ x ≤ L, it can be shown that Z sin2nπx L dx = L 2 (1.57)
Substituting in equation (1.56), we have A2 L 2 = 1 or A = 2 L !1/2 (1.58) The eigen function or wave functions in equation (1.55) becomes
Ψn = 2 L !1/2 sin 2mEn ~2 !1/2 x Ψn = 2 L !1/2 sinnπx L (1.59) x = 0 x = L n = 1 ψ n = 2 n = 3
Figure 1.14 Variation of wave function associated with an electron confined to a box in its ground state and excited states.
Figure 1.14 shows the variation of the wave function inside the box for different values of n and Fig.1.15 shows the probability densities of finding the particle at dif-ferent places inside the box for difdif-ferent values of n. Thus, wave mechanics suggests that the probability of finding any particle at the lowest energy level is maximum at the centre of the box which is in agreement with the classical picture. However, the probability of finding the particle in higher energy states is predicted differently by the two formulations.
n = 1 n = 2 n = 3 |Ψ|2
x = 0 x = L
Figure 1.15 Probability density as a function of position.
1.9.3 Finite Potential well
In real life situations, the potential energy is never infinite. The box with impenetrable walls has no physical significance. However, we come across situations where the potential energy is finite. Let us try to solve the case of an electron in a finite potential well. We can consider two different cases corresponding to the following situations: (i) the total energy E being greater than the potential energy U, and
(ii) the total energy E being less than the potential energy U.
The first case may be represented by the figure 1.16. Consider the particle with total energy E inside a potential well of height U. In the region II, where the particle is not influenced by the potential (U = 0), the solution of the Schrodinger’s equation is of the form,
Ψ = A sin kx + B cos kx
where k = (2mE/~2)1/2. This particle may be represented by a wave of wavelength λ = 2π/k. When the particle is in region I and III, its wavelength changes to λ′ = 2π/k′
where k′ = [2m(E − U)/~2]1/2. In other words, the effect of the potential energy step is to reduce the kinetic energy of the particle as evident from an increase in the value of the wavelength.
In the second case, the total energy of the particle is less than the potential energy. Under this condition, classically, the particle cannot propagate beyond the step since this amounts to the kinetic energy being negative. But, wave mechanically, a different
E
U
λ
λ
x = 0
Figure 1.16Schematic for a particle in a potential well of finite depth (E greater than U).
solution results. Let U be greater than the total energy E of the electron but finite. To analyze this case, we have to consider the three regions separately.
In region II, since U = 0, the electron is free and the Schrodinger’s equation is d2Ψ
dx2 + 2m
~2 EΨ = 0 (1.60) In regions I and III, we have
d2Ψ dx2 +
2m
~2(E − U)Ψ = 0 (1.61)
The solutions for these equations can be assumed to be
ΨI = Aeiβx+ Be−iβxin region I (1.62) ΨII = Ceiαx+ De−iαxin region II (1.63) and ΨIII= Feiβx+ Ge−iβxin region III (1.64)
where α = [(2mE)/~2]1/2 (1.65)
β = [2m(E − U)/~2]1/2 (1.66)
since E is less than U, (E − U) is (−)ve and β is imaginary. Let us define a new constant
Then the equations (1.62) and (1.64) become
ΨI = Ae−γx+ Beγx (1.68)
ΨIII = Fe−γx+ Geγx (1.69)
To evaluate the constants, we consider the boundary condition in the region I where the wave function should reduce to zero as x → −∞. Then equation (1.68) becomes
0 = A · ∞ + B · 0 or A = 0.
∴ΨI = Beγx (1.70)
Similarly, in region III, since the wave function should reduce to zero as x → ∞ , equation (1.69) becomes
0 = F · 0 + G · ∞ or G = 0.
∴ΨIII = Fe−γx (1.71)
This indicates that the wave function decreases exponentially as we move away from the potential well on either sides. Inside the potential well the wave function repre-sented by the equation (1.63) varies sinusoidally. Further, since the wave function and its derivative are continuous at the boundaries corresponding to x = 0 and x = L, the wave functions are non-zero at these boundaries. The plots of the wave functions and the probability densities are shown in Fig. 1.17 and 1.18 respectively.
ψ
x = 0 x = L
n = 3
n = 2
n = 1
n = 1 n = 2 n = 3 |Ψ|2
x = 0 x = L
Figure 1.18 Probability density as a function of position.
Thus, we observe that in case of a particle in a potential well of finite height, the particle has a finite probability of penetrating into the wall. However, if the walls of the well are infinitely thick, the particle will be confined to the well and performs oscillatory motion inside the well.
1.9.4 Tunnel effect
In the previous case of a finite potential well, even though the height of the wall was finite, the thickness of the wall was assumed to be infinite. As a result, the particle was trapped in the well in spite of penetrating into the wall. Under the same condition of the total energy being less than the potential energy, if the thickness of the wall is reduced and made finite, the solution of the Schrodinger’s equation predicts a finite probability of the particle passing through the barrier and finding itself on the other side. Thus, a particle without the necessary energy to pass over the barrier can still penetrate through the barrier. This phenomenon is called “Quantum mechanical tunneling”.
Consider a particle with energy E incident on a potential barrier of height U and width L as shown in Fig. 1.19. The potential energy is zero in the regions I and III, but is finite and equal to U in region II. The Schrodinger’s equation for the three regions will be
d2Ψ dx2 +
2m
x = 0 x = L
λ
λ
III II I UFigure 1.19 Electron tunneling across a finite potential barrier.
d2Ψ dx2 + 2m ~2 (E − U)Ψ = 0 in region II (1.73) d2Ψ dx2 + 2m
~2 EΨ = 0 in region III (1.74) The solutions of these equations can be written as
ΨI = Aeiαx+ Be−iαx in region I (1.75) ΨII = Ce−γx+ Deγx in region II (1.76) ΨIII = Feiαx+ Ge−iαx in region III (1.77)
where α = [(2mE)/~2]1/2 (1.65)
β = [2m(E − U)/~2]1/2 (1.66)
and γ = −iβ (1.67)
The wavefunction in the region I is made up of two terms as evident from equation (1.75). The first term with a positive exponent represents an incoming or incident wave moving in the positive x-direction and the second term represents a wave reflected by the barrier moving in the negative x-direction. Similarly, the first term in equation (1.77) represents the transmitted wave moving in region III in the positive x-direction. The wavefunction in the region II is given by equation (1.76). Here, the exponents are real quantities and hence the wavefunction does not oscillate. The probability density |ΨII|2 is finite and represent the probability of finding the particle within the barrier. Such a particle may emerge into region III. This is called tunneling.
The transmission probability T for a particle to pass through the barrier is given by T = |ΨIII| 2 |ΨI|2 = FF ∗ AA∗ e −2γL (1.78)
The above equation represents the dependence of tunneling probability on the width of the barrier and the energy of the particle.
1.9.5 Examples of tunneling across a finite barrier
There are a few examples of tunneling across a thin finite potential barrier in nature. These observations are in fact proof in favour of the theory of quantum mechanical tunneling. Let us consider a few of them.
(a) Alpha decay: Alpha particles are made up of two protons and two neutrons. In radioactive decay, the alpha particle must free itself from the attractive nuclear force and penetrate through a barrier of repulsive coulombic potential to be emitted out of the nucleus(Fig.1.20). A calculation of the energy of the particle inside the nucleus and the measurement of the energy of the emitted alpha particle indicate that it is not possible that the particle has surmounted the barrier of coulombic potential but must have penetrated through it.
Nuclear surface
Energy of α − particle Repulsive coulomb potential
Attractive nuclear potential
d E
