Download PH-102 (Modern Physics) (Maxwell-Boltzmann distribution, Bose

PH-102 (Modern Physics)
(Maxwell-Boltzmann distribution, Bose-Einstein and Fermi-Dirac Statistics and applications)
Tutorial –2
1
For an ideal gas of classical non-interacting atoms in thermal equilibrium, the Cartesian
components of the velocity of are statistically independent. In three dimensions, the probability
density distribution for the velocity is
P  vx , v y , vz    2
Where  2 

2 3/2
 vx2  v y2  vz2 
exp  


2 2


kT
1
. The energy of a given atom is E  m v 2 .
m
2
Find and sketch the probability density distribution for the energy of an atom in (i) three
dimension, (ii) two dimension and (iii) one dimension.
2
consider a particle undergoing simple harmonic motion, x = x0sin(ωt+Φ), where the phase Φ is
completely unknown. The amount of time this particle spends between x and x+dx is inversely
proportional to the magnitude of its velocity at x. if one thinks in terms of large number of
similarly prepared oscillators, one comes to conclusion that the probability density for finding an
oscillator at x, P(x) is proportional to the time a given oscillation spends near x.
(a) Find the speed of x as a function of x, ω and the fixed maximum displacement x0.
(b) Find P(x) and sketch. Try to compare this classical result with the result for a quantum
harmonic oscillator in an energy eigen-state with a high value of the quantum number ‘n’ and
the same total energy.
(c ) Classically, what are the most probable values of x? What is the least probable? What is the
mean?
3
Given a gas of particles with a Maxwellian distribution for velocity:
(a) What is the most probable velocity?
(b) What is the average velocity?
(c ) What is the root-mean-square speed?
(d) What is the most probable speed?
(e) What is the average speed?
(f) What is the average energy of a particle?
4.
Consider the Maxwellian distribution for the molecules in the atmosphere at the temperature T.
(a) What is the ratio of the number of molecules at the earth’s surface to the number at the
height ‘h’?
(b) What is the ratio of the density of the gas at the height h to the density at the surface?
5.
Three identical particles with total energy 6ε are distributed among four energy levels with
energies, ε, 2ε, 3ε and 4ε of which the second level has a degeneracy of 4. What are the
possible distributions if the particles are (i) distinguishable, (ii) bosons and (iii) fermions? Which
is the most probable distribution in each case?
6.
A system has three energy states with energies ε1, ε2 and ε3 with respective degeneracies of 2,4
and 8. In how many ways a distribution of six fermions can be obtained, where 1 particle has
energy ε1, 2 particles have energy ε2 and 3 particles have energy ε3 (call it 1,2,3). Is this
distribution more probable than (1,3,2). What are the ratios of the probability?
7.
Consider a system of 5 fermions (spin ½) occupying the energy levels of ‘a particle in a three
dimensional cubic box’.
h2
(i) Show that the minimum energy (E0) of the system is 24C, where C 
.
8mL2
(ii) Find the energy of the next three excited states of the system, say E1, E2 and E3 in units of C.
(iii) Find the number of macrostates corresponding to E1 and E3.
(iv) Find the number of microstates for each macrostates obtained in part (iii).
8.
Consider a gas of seven fermions (spin degeneracy =2), each of which can occupy a state of
energy nε (n= 0,1,2,….).
(a) What is the minimum total energy of the gas?
(b) What is the Fermi energy of this gas when the total energy is a minimum?
(c ) What is the minimum average energy?
(d) Suppose the energy is E = 13ε, identify the possible states of the systems and calculate the
‘discrete distribution function’ for this gas- that is, determine average number of fermions.
 n W

W
s
i
ni
s
s
s
s
s
Where Ws is the number of microstates in the macrostates s and ni
in the energy level Ei for that macrostates ‘s’.
is the number of fermions
Plot ni vs. Ei. From the graphs, can we
comment on the Fermi energy and how does the answer compare with (b) part of this problem?
9.
Derive the Plank’s radiation assuming
(i) The Maxwell-Boltzmann distribution for electromagnetic radiation carrying discrete amount
of energy nhv (n=0,1,2,…).
(ii) The Bose-Einstein statistics for the photons.
10.
Calculate the electronic contribution to the specific heat of tungsten. Let εF = 9 eV and T =
3000K.
11.
Prove that the specific heat per molecule is given by cv 


2
1
 2   , where  ,  2 is the
2
kT
average energy and mean square energy, respectively, of a molecule.
12
Consider a cubical box of sides L. what is the lowest energy of a system consisting of 10 free
particles in this box if the particles obey (a) Fermi-Dirac statisitics, (b) Bose-Einstein statistics.
(consider the spin degeneracy of the particles).
13
A collection of non-interacting bosons is kept at a temperature T. When the temperature is
0.4k, a fraction 26/27 of the total number of particles is in the ground state. If the temperature
is increased to 0.9 K, what fraction of particles will be in the ground state?
14
Prove that for a system obeying Fermi-Dirac statistics the probability that a level lying ∆E below
the Fermi level is not occupied is same as the probability that a level above the Fermi level is
occupied.
15
Consider a system of non-interacting fermions at temperature T =0. Calculate the average
energy of each particle in terms of the Fermi energy EF of the system.
16
Assume that the nucleons in a nucleus obey Fermi-Dirac statistics. Find the Fermi energy and
the average energy of the protons or neutrons in the nucleus of a calcium atom, which contains
20 protons and 20 neutrons and has a radius of 4.1x10-15 m. Are these values reasonable?
17
Show that the kinetic energy of a three dimensional gas of N free electrons at 0K is (3/5)NεF.
18
Using the data given and any other constants, evaluate the Fermi energy of the alkali metals.
Density (g/cc)
Atomic Weight
Li
0.534
6.939
Na
K
0.971 0.869
22.99 39.102
Rb
Cs
1.530 1.870
85.47 132.905
19
Show that the fraction of electrons within kT of the Fermi energy is 1.5 kT/ε F, under the
assumption that the temperature is so low that the probability of occupancy of levels is not
altered from the one at oK. Calculate numerically the value of this fraction for copper at (ε F =
7.04 eV) 300 K and 1360 K (approximate melting point of Cu). This fraction is of interest because
it is a rough measure of the percentage of electrons excited to higher energy states at a
temperature T. Find roughly the electronic contribution to specific heat of Cu using this
expression.
20
(a) Using the Fermi-Dirac statistics, find the probability that a state is occupied if its energy is
higher than εF by 0.1 kT, 1.0 kT, 2.0 kT and 10.0 kT, where εF is the Fermi energy. How good is
the approximation of neglecting 1 in the denominator for energy equal to 10kT.
(b) In the Fermi-Dirac distribution substitute    F   . Compute δ for the probability of
occupancy equal to 0.25 and 0.75.
21
In certain quantum mechanical system the x component of the angular momentum, Lx is
quantized and can take on only the three values –ħ, 0 or ħ. For a given state of the system it is
given that Lx 
1
2
and L2x 
3
3
2
. Find the probability density of the x component of the
angular momentum, P(Lx) for the given values of Lx.
22
A copper wire of cross – sectional area 3.3 x 10-6 m2 is carrying a current of 25 A. One meter
length of this wire has a resistance of 5.8 x 10-3 Ω. Calculate the conductivity, average drift
velocity of the electrons, electron mobility and mean free path of the electrons in Drude and
Sommerfield models.
23
For sodium the conductivity at 300K is 2.17 x 107 Ω-1m-1 and the effective mass of the electron is
1.2 times the mass of the free electron. Calculate the relaxation time and the mean free path.
Calculate the drift velocity of the electrons in an electric field of 100 V/m. (density of Na = 970
km/m3, Atomic weight 23).
24
The Fermi energy for Cu is 7.04 eV. Calculate the velocity and de Broglie wavelength of
electrons at the Fermi energy of Cu. Can these electrons be diffracted by a crystal?
25
Assuming that Silver is a monovalent metal obeying Somerfield model, calculate the following
quantities (a) Fermi energy and Fermi temperature (b) radius of Fermi sphere, (c ) Fermi velocity
(d) the average energy of free electrons at 0K, (e) the temperature at which the average
molecular energy in the ideal gas will have the same value as the average energy of free
electrons at 0K, (f) the speed of electron with this energy, (g) mean free path of electrons at
room temperature and near absolute zero., (h) the ratio of Fermi Velocity to drift velocity at
room temperature in a field of 1 V/cm.
{density of Ag : 10.5 g/cc, Atomic wt.: 107.87
gm/mol, Resistivity of Ag at 295 K: 1.61 x 10-6 Ω-cm and at 20 K : 3.8 x 10-9 Ω-cm}.
26
Consider a gas of electrons (mass m) confined to a dimensional square box of size a. find an
expression of density of state and Fermi energy at 0K.
27
Find an expression of density of state for free electrons confined within a distance L in one
dimension.