92
2 Quantum Mechanics – I
2.2 Problems
2.2.1 de Broglie Waves
2.1 (a) Write down the equation relating the energy E of a photon to its frequency
f . Hence determine the equation relating the energy E of a photon to its
wavelength.
(b) A π
0 meson at rest decays into two photons of equal energy. What is the
wavelength (in m) of the photons? (The mass of the π
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 by λ = 0.286 ˚
A/
√
E,
where E is 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 wavelength in angstroms is given by λ =
150
V
1/2
2.5 A thermal neutron has a speed v at temperature T = 300 K and kinetic energy
mn v
2
2
=
3kT
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
is E
2
= c
2 p
2
+ m
2 c
4 , where m is the rest mass and c is the velocity of light.
Using the relativistic relations E = ω and p = k, where ω is the angular
frequency and k is the wave number, show that the product of group velocity
(v g ) and the phase velocity (v p ) is equal to c
2 , that is v p v g = c
2
2.2.2 Hydrogen Atom
2.7 In the Bohr model of the hydrogen-like atom of atomic number Z the atomic
energy levels of a single-electron are quantized with values given by
E n =
Z
2 m e e
4
8ε
2
0 h 2 n 2
where m is the mass of the electron, e is the electronic charge and n is an
integer greater than zero (principal quantum number)
2 Quantum Mechanics – I
2.2 Problems
2.2.1 de Broglie Waves
2.1 (a) Write down the equation relating the energy E of a photon to its frequency
f . Hence determine the equation relating the energy E of a photon to its
wavelength.
(b) A π
0 meson at rest decays into two photons of equal energy. What is the
wavelength (in m) of the photons? (The mass of the π
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 by λ = 0.286 ˚
A/
√
E,
where E is 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 wavelength in angstroms is given by λ =
150
V
1/2
2.5 A thermal neutron has a speed v at temperature T = 300 K and kinetic energy
mn v
2
2
=
3kT
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
is E
2
= c
2 p
2
+ m
2 c
4 , where m is the rest mass and c is the velocity of light.
Using the relativistic relations E = ω and p = k, where ω is the angular
frequency and k is the wave number, show that the product of group velocity
(v g ) and the phase velocity (v p ) is equal to c
2 , that is v p v g = c
2
2.2.2 Hydrogen Atom
2.7 In the Bohr model of the hydrogen-like atom of atomic number Z the atomic
energy levels of a single-electron are quantized with values given by
E n =
Z
2 m e e
4
8ε
2
0 h 2 n 2
where m is the mass of the electron, e is the electronic charge and n is an
integer greater than zero (principal quantum number)