UNIT 1 Quantum Mechanics

PRANVEER SINGH INSTITUTE OF TECHNOLOGY
I-B.TECH (Session 2023-2024)

ENGINEERING PHYSICS (BAS-101)
QUANTUM MECHANICS (UNIT-I)
Short questions
1. What are the inadequacies of classical mechanics?
2. What is black body?
3. Write down the Planck’s formula for spectral energy density in blackbody radiation.
4. What is the Planck’s assumption of blackbody radiation?
5. What is de-Broglie hypothesis?
6. Why matter waves are not observed in our daily life?
7. Write the difference between electromagnetic waves and matter waves.
8. Compare the wavelength of an electron and a photon if two have the same momentum.
9. What is the objective of the Davisson and Germar experiment? Why nickel crystal is
used in Davisson and Germar experiment?
10. What is wave-packet?
11. What are phase velocity and group velocity?
12. What is the relation between phase velocity and group velocity in the dispersive
medium and non-dispersive medium?
13. Show that the phase velocity of de-Broglie wave is greater than the speed of light.
14. What is a wavefunction? Give essential characteristics of the acceptable wave function.
15. Write the condition of normalization of wave function.
16. What do you mean by Eigen function and Eigenvalues?
17. What is Compton shift? How does it support the photon nature of light?
18. What is Compton wavelength? Why Compton shift is not observed with visible light.
19. Define modified and unmodified x-rays.
Long questions
1. Discuss the shortcomings of classical wave theory.
2. Explain the wave-particle duality of matter and obtain an expression for de-Broglie
wavelength of photon.
3. Show that the wavelength ( λ ) associated with a particle of mass m and kinetic energy E
is given by:
h
λ=
√2 mE
4.
Give the brief description of the Davisson-Germer experiment and explain the outcome
in details.
5. Illustrate the relation between phase velocity and group velocity in a dispersive and
non-dispersive medium.
6.
Illustrate that vp⨯vg = c2, where vp= phase velocity and vg = group velocity.
7. What is a wave function? Give a physical interpretation of wave function (Ψ)? What
conditions must it fulfill?
8. Deduce time-independent Schrö dinger’s wave equation. What would happen to the
equation if the particle is free?
9. Deduce Schrö dinger’s time-dependent wave equation.
10. A particle is in motion along a line between x = 0 and x = a, with zero potential energy. At
points for which x < 0 and x >a, the potential energy is infinite. Show that the wave
function for the particle in the nth state is given by
2  nx 
 n ( x)  sin  
a  a 
Also, obtain the energy Eigenvalues.
11.
What is Compton Effect? Explain the physical significance of the Compton shift.
12.
Explain Compton Effect and show that Compton shift depends only on the angle of
scattering and is independent of the wavelength of the incident photon.

13.
Illustrate the expression of kinetic energy and direction of recoiled electron in Compton
scattering.
NUMERICALS
1) Calculate the velocity & kinetic energy of the neutron having de-Broglie wavelength 1 Å .
(v= 3.96 x 103 m/s& E= 8.2x 10-2ev)
2) Calculate the de- Broglie wavelength of an α-particle accelerated through 200 volts. (0.00716 Å)
3) Calculate the de-Broglie wavelength of an electron accelerated from rest through a potential
difference of 100 volts. (1.227Å )
4) Calculate the de-Broglie wavelength of neutron of energy 28.8 eV. (0.0533 Å)
5) Calculate the de-Broglie wavelength proton moving with velocity 2x10 m/s. 8
(1.47⨯10-5Å)
6) Calculate de-Broglie wavelength associated with nitrogen at 3.0 atmospheric pressure and 270 C .
mass of N2 -atom= 4.65⨯10-26 kg. (0.2754 Å)
7) Calculate the wavelength of thermal neutrons at 27 0 c, assuming energy of a particle at absolute
temperature T is of the order of KT. (1.779 Å)
8) Can a photon and an electron of same momentum have the same wavelength? Compare their
wavelengths if two have the same energy? (λph/λe=(2mc2/E )1/2)
9) Calculate the speed of a particle whose de-Broglie wavelength and Compton wavelength are
equal. Ans: c/√ 2
10)An electron has de- Broglie wave-length 2.0 x 10 -12m. Calculate its kinetic energy.(E=293.64 KeV)
11) The phase velocity of sea waves is given by
The wavelength is 580 m. Take g= 9.8 m/s2.

gλ
2π√( )
. Calculate the group velocity of these waves.
(15 m/s)
12) The phase velocity of ripple on a liquid surface is
√ 2 πσ
λρ
. Where σ =¿ surface tension and ρ =
3
density of liquid. Calculate the group velocity of the ripples. (vg= vp)
2
13) The angular velocity ω and the propagation constant k of a wave are related by the relation
ω =c k + ω0 . Where c and ω 0 are constant.Calculate the phase velocity and group velocity.
2 2 2 2
14) Calculate the lowest two permitted energy values of the electron which is bound in one-dimensional
potential box of width 2.5 x 10 - 10 m and having infinite height. (E1= 6.04 eV, E2= 24.16 eV )
15) Calculate the energy difference between the ground state and the first excited state for an
electron in a one-dimensional box of length 10-8 cm. (114 eV)
16) Calculate the least energy of an electron moving in one-dimension in an infinitely high potential
box of width 1Å . (5.68 eV)
17) A particle is confined to move along x- axis has the wave function Ψ=ax, between x=0 and x=1.0
and Ψ=0 elsewhere. Calculate the probability that particle can be found between x=0.35 to
x=0.45. (P=0.0161 a2)
18) A particle is moving in 1D potential box of infinite height and width 25 Å . Calculate the
probability of finding the particle within an interval of 5 Å at the center of the box when it is in its
state of least energy. ( P= 0.4)
19) Calculate the probabilities of finding a particle trapped in a box of length L in the region from
0.45L to 0.55L for the ground state and the first excited state. . (P1=19.8% & P2 =0.65%)
20) When X-ray of energy 100 keV strikes a target, they are scattered at an angle 60 o. Calculate the
energy and direction of the recoiled electrons.
21) An X-ray photon is found to have its wavelength doubled on being scattered through 90 o.
Calculate the wavelength and energy of the incident photon. (λ= 0.0245Å , E= 8.106 ⨯10-14
Joule)
22) X-rays with λ= 1 Å are scattered from a carbon block. The scattered radiation is viewed at 90 o to
the incident beam:
(a) Calculate the Compton shift Δ λ (0.0243 Å )
(b) Calculate how much kinetic energy is imparted to the recoiling electrons? ( 295
eV)
(c) Calculate how much energy is lost during collision. ( 2.3 % )
23) A beam of gamma radiation having photon energy of 510 keV is incident on a foil of Al . Calculate
the wavelength of radiation at 90oand also the energy and direction of the emission of the
corresponding electron. (4.83x10-12 m, 0.25 MeV, 26o 37’ )
24)For what wavelength photon does Compton scattering result in a photon whose energy is one-half of the
original photon, at scattering angle 45o ? (λ= 0.007Å)
25) X-ray of wavelength λ= 2 Å are scattered from a block of carbon. The scattered X-rays are
observed at an angle of 45 oto the incident beam. Calculate the fraction of energy lost by the
photon in this collision. (0.0035)

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PRANVEER SINGH INSTITUTE OF TECHNOLOGY

I-B.TECH (Session 2023-2024)

Also, obtain the energy Eigenvalues.

scattering and is independent of the wavelength of the incident photon.

11) The phase velocity of sea waves is given by

The wavelength is 580 m. Take g= 9.8 m/s2.

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