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Avijit Lahiri - Statistical Mechanics

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Avijit Lahiri Statistical Mechanics
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    Statistical Mechanics
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Universities Press (India) Private Limited

Registered Office
3-6-747/1/A & 3-6-754/1, Himayatnagar, Hyderabad 500 029 (A.P.), INDIA
e-mail:


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Orient Blackswan Private Limited


Registered Office
3-6-752 Himayatnagar, Hyderabad 500 029 (A.P.), INDIA
e-mail:

Other Offices
Bangalore, Bhopal, Bhubaneshwar, Chennai,
Ernakulam, Guwahati, Hyderabad, Jaipur, Kolkata,
Lucknow, Mumbai, New Delhi, Noida, Patna

Universities Press (India) Private Limited 2002, 2008
First published 2002
Revised edition 2008

eISBN 978 81 7371 843 4

e-edition:First Published 2013

ePUB Conversion: .

All rights reserved. No part of this publication may be reproduced, distributed, or transmitted in any form or by any means, including photocopying, recording, or other electronic or mechanical methods, without the prior written permission of the publisher, except in the case of brief quotations embodied in critical reviews and certain other noncommercial uses permitted by copyright law. For permission requests write to the publisher.


To my parents and my students

Contents

1-1
1-1.1
1-1.2
1-1.3
1-1.4
1-1.5
1-2
1-2.1
1-2.2
1-2.3
1-2.4
1-2.5
1-2.6
1-2.7
1-2.8
1-2.9
1-2.10
1-2.11
1-2.12
1-2.13
1-2.14
1-3
1-3.1
1-3.2
1-3.3
1-3.4
1-3.5
1-3.6
1-4
1-4.1
1-4.2
1-4.3
1-4.4
1-4.5
1-4.6
1-4.7
1-5
1-6
2-1
2-2
2-2.1
2-2.2
2-3
2-4
2-4.1
2-5
2-5.1
2-5.2
2-5.3
2-6
2-6.1
2-6.2
2-6.3
2-6.4
2-7
2-7.1
2-7.2
2-7.3
2-7.4
2-7.5
2-7.6
2-7.7
2-7.8
3-1
3-2
3-3
3-3.1
3-3.2
3-4
3-5
3-6
3-7
3-8
3-8.1
3-8.2
3-8.3
3-8.4
3-9
3-10
3A1
3A2
3A3
3A4
3A5
4-1
4-2
4-2.1
4-2.2
4-2.3
4-2.4
4-3
4-3.1
4-3.2
4-3.3
4-3.4
4-3.5
4-3.6
4-4
4-4.1
4-4.2
4-4.3
4-5
4-5.1
4-5.2
4-5.3
4-6
4-6.1
4-6.2
4-7
4-8
4-9
4-9.1
4-9.2
4-9.3
4-9.4
4-10
4-10.1
4-10.2
4-10.3
4-10.4
4-11
4-12
4-12.1
4-12.2

List of Titled Problems

Quantum mechanics of a free particle
The fundamental commutators
The Uncertainty Principle
Stationary states and energies of a composite system
Entropy of a composite system
A pair of non-interacting harmonic oscillators
An assembly of harmonic oscillators
The coherent state
Spins with volume-dependent energies
The Maxwell-Boltzmann distribution
The single-particle partition function
Classical ideal gas with internal degrees of freedom
Weakly degenerate Bose and Fermi gases
Entropy of a mixture of classical ideal gases
Maxwell's velocity distribution
A two-level system
An assembly of sites that are either occupied or vacant
Two species competing for a site
The grand canonical probability distribution as a distribution with maximal disorder
Energy dispersion of the harmonic oscillator at temperature T
Non-interacting constituents: Working in the microcanonical ensemble
A system of harmonic oscillators, all of the same frequency
The Einstein crystal revisited
A system of two-state spins at temperature T
The assembly of spins in the -T specification
A system consisting of two species
Lattice sites filled with two atomic species entropy of mixing
Free energy and entropy of the classical ideal gas
Classical ideal gas in two dimensions
Internal energy and specific heat of the classical ideal gas
Specific heat of the classical crystalline solid
Equipartition of energy: anharmonic correction
'Equipartition' for additive quartic terms in the Hamiltonian
Rotational partition function for diatomic molecules
Dilute diatomic gas with translational and rotational degrees of freedom
Ortho- and para-hydrogen
Paramagnetism of two-state spins
Mean magnetic moment of a single ion
Molar susceptibility
Distribution formula for the photon gas: canonical ensemble
The F-D distribution function re-derived
Non-occupancy probability
The B-E distribution function re-derived
The Sommerfeld expansion
Chemical potential and internal energy of the degenerate electron gas
Two instances of degenerate Fermi gas
Degenerate Fermi gas in two dimensions
Chemical potential of a Bose gas
An estimate for T0
Degenerate Bose gas in two dimensions: absence of B-E condensation.

Preface to the First Revised Edition

In this first revision of the book, I have included a few sections so as to make the presentation more precise and solidly founded while at the same time retaining its informal flavour as also the elementary level of approach to the subject. In contrast to the previous edition, I have made use of classical mechanics as the launching pad for developing the basic notions of statistical mechanics and have then completed the development by underlining the essential role of quantum theory as the founding ground of the subject. I have corrected for a certain lack of balance in the earlier presentation by dwelling at length on the concept of the mixed state and of the reduced state of a subsystem, have added a few instructive problems to those in the previous edition, and have corrected a few other errors brought to my notice. I have also included an elementary exposition of the virial expansion, marking the boundary of the material I set out to present.

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