1. Describe the Michelson Morley experiment and discuss the importance of its negative result. 2. Calculate the fringe shift in Michelson-Morley experiment. Given that: [pic], [pic], [pic], and [pic]. 3. State the fundamental postulates of Einstein special theory of relativity and deduce from them the Lorentz Transformation Equations . 4. Explain relativistic length contraction and time dilation in special theory of relativity? What are proper length and proper time interval? 5. A rod has length 100 cm. When the rod is in a satellite moving with velocity 0.9 c relative to the laboratory, what is the length of the rod as measured by an observer (i) in the satellite, and (ii) in the laboratory?.
6. A clock keeps correct time. With what speed should it be moved relative to an observer so that it may appear to lose 4 minutes in 24 hours? 7. In the laboratory the ‘life time’ of a particle moving with speed 2.8x108m/s, is found to be 2.5×10-7 sec. Calculate the proper life time of the particle. 8. Derive relativistic law of addition of velocities and prove that the velocity of light is the same in all inertial frame irrespective of their relative speed. 9. Two particles come towards each other with speed 0.9c with respect to laboratory.
Calculate their relative speeds. 10. Rockets A and B are observed from the earth to be traveling with velocities 0.8c and 0.7 c along the same line in the same direction. What is the velocity of B as seen by an observer on A? 11. Show that the relativistic invariance laws of conservation of momentum leads to the concept of variation of mass with speed and mass energy equivalence. 12. A proton of rest mass [pic] is moving with a velocity of 0.9c. Calculate its mass and momentum.
TUTORIAL SHEET: 1
(Module1: Special Theory of Relativity)
13. The speed of an electron is doubled from 0.2 c to 0.4 c. By what ratio does its momentum increase? 14. A particle has kinetic energy 20 times its rest energy. Find the speed of the particle in terms of ‘c’. 15. Dynamite liberates about 5.4×106 J/Kg when it explodes. What fraction of its total energy is in this amount? 16. A stationary body explodes into two fragments each of mass 1.0 Kg that move apart at speeds of 0.6 c relative to the original body. Find the mass of the original body. 17. At what speed does the kinetic energy of a particle equals its rest energy? 18. What should be the speed of an electron so that its mass becomes equal to the mass of proton? Given: mass of electron=9.1×10-31Kg and mass of Proton =1.67×10-27Kg.
19. An electron is moving with a speed 0.9c. Calculate (i) its total energy and (ii) the ratio of Newtonian kinetic energy to relativistic energy. Given: [pic] and[pic]. 20. (i) Derive a relativistic expression for kinetic energy of a particle in terms of momentum. (ii) Show that the momentum of a particle of rest mass [pic] and kinetic energy [pic], is given by[pic]. 21. Find the momentum (in MeV/c) of an electron whose speed is 0.60 c. Verify that v/c = pc/E TUTORIAL SHEET: 2(a)
(Module2: Wave Mechanics)
1. What do you understand by the wave nature of matter? Obtain an expression of de Broglie wavelength for matter waves. 2. Calculate the de-Broglie wavelength of an electron and a photon each of energy 2eV. 3. Calculate the de-Broglie wavelength associated with a proton moving with a velocity equal to 1/20 of the velocity of light. 4. Show that the wavelength of a 150 g rubber ball moving with a velocity of [pic] is short enough to be determined.
5. Energy of a particle at absolute temperature T is of the order of [pic]. Calculate the wavelength of thermal neutrons at[pic]. Given: [pic], [pic] and [pic]. 6. Can a photon and an electron of the same momentum have the same wavelengths? Calculate their wavelengths if the two have the same energy. 7. Two particles A and B are in motion. If the wavelength associated with particle A is [pic], calculate the wavelength of the particle B if its momentum is half that of A. 8. Show that when electrons are accelerated through a potential difference V, their wavelength taking relativistic correction into account is [pic] , where e and [pic] are charge and rest mass of electrons, respectively. 9. A particle of rest mass m0 has a kinetic energy K. Show that its de Broglie wavelength is given by [pic] TUTORIAL SHEET: 2(a)
(Module2: Wave Mechanics)
16. Explain Heisenberg uncertainty principle. Describe gamma ray microscope experiment to establish Heisenberg uncertainty principle. 17. How does the Heisenberg uncertainty principle hint about the absence of electron in an atomic nucleus? 18. Calculate the uncertainty in momentum of an electron confined in a one-dimensional box of length[pic]. Given:[pic]
TUTORIAL SHEET: 2(b)
(Module 2: Wave Mechanics)
1. Differentiate between Ψ and IΨI2. Discuss Born postulate regarding the probabilistic interpretation of a wave function. 2. Write down the set of conditions which a solution of Schrödinger wave equation satisfies to be called a wave function. 3. What do you mean by normalization and orthogonality of a wave function? 4. Show that if potential energy V(x) is changed everywhere by a constant, the time independent wave equation is unchanged. What is the effect on the energy Eigen values? 5. Show that[pic], where [pic]the reduced mass and B is the binding energy of the particles. 6. Show that [pic]is an acceptable eigen function, where k is some finite constant. Also normalize it over the region[pic].
7. Explain the meaning of expectation value of x. write down the Eigen operators for position, linear momentum and total energy. 8. Show that time independent Schrödinger equation is an example of Eigen value equation. 9. Derive the time independent Schrödinger equation from time dependent equation for free particle. 10. For a free particle, show that Schrödinger wave equation leads to the de-Broglie relation [pic]. 11. Derive expression for probability current density or particle flux. Also , show that the probability density ρ and probability current density [pic] satisfy the continuity equation[pic]
TUTORIAL SHEET: 2(b)
(Module 2: Wave Mechanics)
12. Write Schrödinger equation for a particle in a box and determine expression for energy Eigen value and Eigen function. Does this predict that the particle can possess zero energy? 13. Find the expectation values of the position and that of momentum of a particle trapped in a one dimensional rigid box of length L. 14. The potential function of a particle moving along positive x-axis is given by V(x) = 0for x < 0
V(x) = V0for x [pic] 0
Calculate the reflectance R and transmittance T at the potential discontinuity and show that R+T=1. 15. An electron is bounded by a potential which closely approaches an infinite square well of width[pic]. Calculate the lowest three permissible quantum energies the electron can have. 16. A particle is moving in one dimensional box and its wave function is given by [pic]. Find the expression for the normalized wave function.
17. Calculate the value of lowest energy of an electron moving in a one-dimensional force free region of length 4[pic]. 18. A particle of mass [pic]kg is moving with a speed of [pic] in a box of length[pic]. Assume this to be one dimensional square well problem, calculate the value of n. 19. A beam of electron impinges on an infinitely wide energy barrier of height 0.03 eV, find the fraction of electrons reflected at the barrier if the energy of the electrn is (a) 0.025 eV (b) 0.030 eV (c) 0.040 eV
TUTORIAL SHEET: 3(a)
(Module 3: Atomic Physics)
1. What are the essential features of Vector Atom model? Also discuss the quantum numbers associated with this model. 2. For an electron orbit with quantum number l = 2, state the possible values of the components of total angular momentum along a specified direction. 3. Differentiate between L-S coupling (Russel-Saunders Coupling) and j-j coupling schemes. 4. Find the possible value of J under L-S and j-j coupling scheme if the quantum number of the two electrons in a two valence electron atom are n1 = 5 l1 = 1 s1 =1/2 n2 = 6 l2 = 3 s2 = 1/2
5. Find the spectral terms for 3s 2d and 4p 4d configuration. 6. Applying the selection rule, show which of the following transitions are allowed and not allowed D5/2 [pic] P3/2; D3/2 [pic] P3/2 ; D3/2 [pic] P1/2 ; P3/2 [pic] S1/2 ; P1/2 [pic] S1/2
7. What is Paschen back effect? Show that in a strong magnetic field, anomalous Zeeman pattern changes to normal Zeeman pattern. 8. Why does in normal Zeeman effect a singlet line always splitted into three components only. 9. Illustrate Zeeman Effect with the example of Sodium D1 and D2 lines. 10. An element under spectroscopic examination is placed in a magnetic field of flux density 0.3 Web/m2. Calculate the Zeeman shift of a spectral line of wavelength 450 nm. 11. The Zeeman components of a 500 nm spectral line are 0.0116 nm apart when the magnetic field is 1.0 T. Find the ratio (e/m) for the electron. 12. Calculate wavelength separation between the two component lines which are observed in Normal Zeeman effect, where – the magnetic field used is 0.4 weber/m2 , the specific charge- 1.76x1011Coulomb/kg and λ=6000[pic]. TUTORIAL SHEET: 3(b)
(Module 3: Atomic Physics)
1. Distinguish between spontaneous and stimulated emission. Derive the relation between the transition probabilities of spontaneous and stimulated emission. 2. What are the characteristics of laser beams? Describe its important applications. 3. Calculate the number of photons emitted per second by 5 mW laser assuming that it emits light of wavelength 632.8 nm. 4. Explain (a) Atomic excitations (b) Transition process (c) Meta stable state and (d) Optical pumping. 5. Find the intensity of laser beam of 15 mW power and having a diameter of 1.25 mm. Assume the intensity to be uniform across the beam.
6. Calculate the energy difference in eV between the energy levels of Ne-atoms of a He-Ne laser, the transition between which results in the emission of a light of wavelength 632.8nm. 7. What is population inversion? How it is achieved in Ruby Laser? Describe the construction of Ruby Laser. 8. Explain the operation of a gas Laser with essential components. How stimulated emission takes place with exchange of energy between Helium and Neon atom? 9. What is the difference between the working principle of three level and four level lasers? Give an example of each type. 10. How a four level Laser is superior to a three level Laser? TUTORIAL SHEET: 3(c)
(Module 3: Atomic Physics)
1. Distinguish between continuous X-radiation and characteristic X-radiation spectra of the element. 2. An X ray tube operated at 100 kV emits a continuous X ray spectrum with short wavelength limit λmin = 0.125[pic]. Calculate the Planck’s constant. 3. State Bragg’s Law. Describe how Bragg’s Law can be used in determination of crystal structure? 4. Why the diffraction effect in crystal is not observed for visible light. 5. Electrons are accelerated by 344 volts and are reflected from a crystal. The first reflection maxima occurs when glancing angle is 300 . Determine the spacing of the crystal. (h = 6.62 x 10-34 Js , e = 1.6 x 10-19 C and m = 9.1 x10-31 Kg) 6. In Bragg’s reflection of X-rays, a reflection was found at 300 glancing angle with lattice planes of spacing 0.187nm. If this is a second order reflection. Calculate the wavelength of X-rays.
7. Explain the origin of characteristic X-radiation spectra of the element. How Mosley’s law can explained on the basis of Bohr’s model. 8. What is the importance of Mosley’s law? Give the important differences between X-ray spectra and optical spectra of an element? 9. Deduce the wavelength of [pic] line for an atom of Z = 92 by using Mosley’s Law. (R= 1.1 x 105 cm-1). 10. If the Kα radiation of Mo (Z= 42) has a wavelength of 0.71[pic], determine the wavelength of the corresponding radiation of Cu (Z= 29). 11. The wavelength of Lα X ray lines of Silver and Platinum are 4.154 [pic]and 1.321[pic], respectively. An unknown substance emits of Lα X rays of wavelength 0.966[pic]. The atomic numbers of Silver and Platinum are 47 and 78 respectively. Determine the atomic number of the unknown substance. TUTORIAL SHEET: 4(a)
(Module 4: Solid State Physics)
1. Discuss the basic assumptions of Sommerfeld’s theory for free electron gas model of metals? 2. Define the Fermi energy of the electron. Obtain the expression for energy of a three dimensional electron gas in a metal. 3. Prove that at absolute zero, the energy states below Fermi level are filled with electrons while above this level, the energy states are empty. 4. Show that the average energy of an electron in an electron gas at absolute zero temperature is 3/5[pic], where[pic], is Fermi energy at absolute zero. 5. Prove that Fermi level lies half way down between the conduction and valence band in intrinsic semiconductor. 6. Find the Fermi energy of electrons in copper on the assumption that each copper atom contributes one free electron to the electron gas. The density of copper is 8.94(103 kg/m3 and its atomic mass is 63.5 u.
7. Calculate the Fermi energy at 0 K for the electrons in a metal having electron density 8.4x1028m-3. 8. On the basis of Kronig – Penney model, show that the energy spectrum of electron in a linear crystalline lattice consists of alternate regions of allowed energy and forbidden energy. 9. Discuss the differences among the band structures of metals, insulators and semiconductors. How does the band structure model enable you to better understand the electrical properties of these materials?
10. Explain how the energy bands of metals, semiconductors and insulators account for the following general optical properties: (a) Metals are opaque to visible light, (b) Semiconductors are opaque to visible light but transparent to infrared, (c) Insulator such as diamond is transparent to visible light. 11. Discuss the position of Fermi energy and conduction mechanism in N and P-type extrinsic semiconductors. TUTORIAL SHEET: 4(b)
(Module 4: Solid State Physics)
1. What do you mean by superconductivity? Give the elementary properties of superconductors. 2. Discuss the effect of magnetic field on a superconductor. How a superconductor is different from a normal conductor. 3. Discuss the effect of the magnetic field on the superconducting state of type I and type II superconductors. 4. What are the elements of the BCS theory? Explain the formation of Cooper pairs. 5. Explain the phenomena of Meissner effect and zero resistivity with the help of BCS theory. 6. The metals like gold, silver, copper etc. do not show the superconducting properties, why?
7. Describe the V-I characteristics of p-n junction diode. What do you understand by drift and diffusion current in the case of a semiconductor? 8. Explain the working and characteristics of a photodiode by using I-V curve. 9. Describe the phenomena of carrier generation and recombination in a semiconductor. 10. Define the phenomenon of photoconduction in a semiconductor. Deduce the relation between the wavelength of photon required for intrinsic excitation and forbidden energy gap of semiconductor. 11. Establish the relation between load current and load voltage of a solar cell. Describe the applications of solar cell in brief.
University/College: University of Chicago
Type of paper: Thesis/Dissertation Chapter
Date: 14 November 2016
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