Bd91etutorial+Sheet+Final

Amity School of Engineering and Technology (NOIDA)Applied Physics II: Modern Physics

TUTORIAL SHEET: 1(Module1: Special Theory of Relativity)

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:

, , , and .

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 2.5x10-7 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 is moving with a velocity of 0.9c.

Calculate its mass and momentum.

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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 5.4x106 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=9.1x10-31Kg and mass of

Proton =1.67x10-27Kg.

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:

and .

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 and

kinetic energy , is given by .

21.Find the momentum (in MeV/c) of an electron whose speed is 0.60 c. Verify

that v/c = pc/E

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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

is short enough to be determined.

5. Energy of a particle at absolute temperature T is of the order of .

Calculate the wavelength of thermal neutrons at . Given:

, and .

6. Can a proton 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 , 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

,

where e and 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

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TUTORIAL SHEET: 2(a)(Module2: Wave Mechanics)

10.Show that the phase velocity of de-Broglie waves associated with a moving

particle having a rest mass is given by

,

where the symbols have their usual meanings.

11.Discuss wave particle duality and describe briefly Davisson and Germer

experiment for qualitative verification of matter wave.

12.Define group velocity. Show that group velocity of a wave packet equals the

particle velocity.

13.Distinguish between phase and group velocity. Prove that product of phase

and group velocity is square of velocity of light.

14.Derive an expression for phase velocity of wave in terms of angular

frequency and propagation constant. Show that the phase velocity of wave

associated with a material particle is not equal to particle velocity.

15.An electron has de-Broglie wavelength of 1.0 pm. Calculate its kinetic

energy and the phase and group velocities of its de-Broglie waves. Given:

Planck’s constant, , and rest energy of electron,

.

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 . Given:

.

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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 is an acceptable eigen function, where k is some finite

constant. Also normalize it over the region .

5. The wave function of a particle is given is given as for

and elsewhere. Find (i) the value of A, and (ii) the probability of finding

the particle between to .

6. Explain the meaning of expectation value of x. write down the Eigen operators

for position, linear momentum and total energy.

7. Show that time independent Schrödinger equation is an example of Eigen

value equation.

8. Derive the time independent Schrödinger equation from time dependent

equation for free particle.

9. For a free particle, show that Schrödinger wave equation leads to the de-

Broglie relation .

10. Find the expectation values <x>, <p> and of a particle trapped in a one

dimensional box of length L.

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TUTORIAL SHEET: 2(b)(Module 2: Wave Mechanics)

11.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?

12.An electron is bounded by a potential which closely approaches an infinite

square well of width . Calculate the lowest three permissible

quantum energies the electron can have.

13.A particle is moving in one dimensional box and its wave function is given by

. Find the expression for the normalized wave function.

14.Calculate the value of lowest energy of an electron moving in a one-

dimensional force free region of length 4 .

15.A particle of mass kg is moving with a speed of in a box of

length . Assume this to be one dimensional square well problem,

calculate the value of n.

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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 P3/2; D3/2 P3/2 ; D3/2 P1/2 ; P3/2 S1/2 ; P1/2 S1/2

7. Why does in normal Zeeman effect a singlet line always splitted into three

components only.

8. Illustrate Zeeman Effect with the example of Sodium D1 and D2 lines.

9. 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.

10. 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.

11.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 .

TUTORIAL SHEET: 3(b)(Module 3: Atomic Physics)

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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. 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.

6. Ruby laser gives a laser beam of wavelength 694.3 nm. Calculate the

energy difference between the two energy levels involved in the transition.

7. Explain the operation of a Laser with essential components.

8. In a ruby laser, the total number of Cr3+ ions is . If the laser emits

a radiation of wavelength 700 nm, calculate the energy of laser pulse.

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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 . 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 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 , 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 and

1.321 , respectively. An unknown substance emits of Lα X rays of

wavelength 0.966 . The atomic numbers of Silver and Platinum are 47

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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 , where , 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. Consider silver in metallic state with one free electron per atom. Calculate

the Fermi energy. Given that density of silver is 10.5 g/cm3 and atomic

weight is 108.

7. There are free electrons per cubic meter of sodium. Calculate the

Fermi energy.

8. Determine the temperature at which there is one percent probability that a

state with energy 0.25 eV above the Fermi energy will be occupied by an

electron.

9. Calculate the Fermi energy at 0 K for the electrons in a metal having

electron density 8.4x1028m-3.

10.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?

11.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

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transparent to infrared, (c) Insulator such as diamond is transparent to

visible light.

12.Discuss the position of Fermi energy and conduction mechanism in N and

P-type extrinsic semiconductors.

13.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?

14.Describe the phenomena of carrier generation and recombination in a

semiconductor.

15.What do you mean by superconductivity? Give the elementary properties

of superconductors.

16.Discuss the effect of magnetic field on a superconductor. How a

superconductor is different from a normal conductor.

17. A superconducting tin has a critical temperature of 3.7 K at zero magnetic

field and a critical magnetic field 0.0306 Tesla at 0 K. Find the critical

magnetic field at 2K.

18.The metals like gold, silver, copper etc. do not show the superconducting

properties, why?

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