1000 Solved Problems in Modern Physics

92 2 Quantum Mechanics – I

2.2 Problems..................................................

2.2.1 deBroglieWaves

2.1 (a) Write down the equation relating the energyEof a photon to its frequency
f. Hence determine the equation relating the energyEof a photon to its
wavelength.
(b) Aπ^0 meson at rest decays into two photons of equal energy. What is the
wavelength (in m) of the photons? (The mass of theπ^0 is 135 MeV/c)
[University of London 2006]

2.2 Calculate the wavelength in nm of electrons which have been accelerated from
rest through a potential difference of 54 V.
[University of London 2006]

2.3 Show that the deBroglie wavelength for neutrons is given byλ= 0. 286 A ̊/

√

E,

whereEis in electron-volts.

[Adapted from the University of New Castle upon Tyne 1966]

2.4 Show that if an electron is accelerated through V volts then the deBroglie wave-

length in angstroms is given byλ=

( 150

V

) 1 / 2

2.5 A thermal neutron has a speed v at temperatureT=300 K and kinetic energy
mnv^2
2 =

3 kT

2. Calculate its deBroglie wavelength. State whether a beam of these

neutrons could be diffracted by a crystal, and why?

(b) Use Heisenberg’s Uncertainty principle to estimate the kinetic energy (in

MeV) of a nucleon bound within a nucleus of radius 10−^15 m.

2.6 The relation for total energy (E) and momentum (p) for a relativistic particle
isE^2 =c^2 p^2 +m^2 c^4 , wheremis the rest mass andcis the velocity of light.
Using the relativistic relationsE =ωandp=k, whereωis the angular
frequency andkis the wave number, show that the product of group velocity
(vg) and the phase velocity (vp) is equal toc^2 , that isvpvg=c^2

2.2.2 Hydrogen Atom...................................

2.7 In the Bohr model of the hydrogen-like atom of atomic numberZthe atomic

energy levels of a single-electron are quantized with values given by

En=

Z^2 mee^4

8 ε^20 h^2 n^2

wheremis the mass of the electron,eis the electronic charge andnis an

integer greater than zero (principal quantum number)