Abstract
This Lecture applies the general quantum-mechanical principles developed in Aside C to analyze the statistical mechanics of quantum-mechanical systems. The analysis is in two parts. The first, more conventional, part will show why a sum over energy eigenstates is an appropriate procedure for evaluating ensemble averages, and why your decision to use a particular set of basis vectors has no effect on the number you get when you calculate the ensemble average for a quantum system. The second, less conventional, part of the discussion will elaborate on the distinction between a complete set of all possible states of the system and a complete set of basis vectors. An ensemble average over all quantum states of a system agrees with an ensemble average over a complete set of basis vectors; however, the latter average is less tedious to calculate.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
P. A. M. Dirac, Quantum Mechanics, Oxford University Press, Oxford (1958).
There is the significant difficulty that in most cases the number N of basis vectors needed to form a complete set is not finite. If N is infinite, the integrals of (12.14) diverge; the proposed cancellation is meaningless. This difficulty can apparently be circumvented by taking N to be the number of basis vectors in a finite subspace of the system’s Hilbert space, demonstrating the cancellation, and then taking the limit. As N goes to infinity, the error in approximating any state vector ∣S as a sum of a subset of the set of all basis vectors willvanish.
The alert student will remember that Planck’s derivation of the black-body radiation formula, which appears to use energy quantization, dates to 1901. T. Kuhn, The Black Body Paradox and the Quantum Discontinuity, University of Chicago Press, Chicago (1978) discusses in careful detail what Planck appears to have thought that he was doing. In particular, Planck in 1901 appears to have viewed both the field and the oscillator sources as having continuous values for their energy, the use of a sum over quanta being a clever device (borrowed from Boltzmann’s use of the same combinatorial arguments) for avoiding a phase-space integral. Only after Jeans’ 1905 paper could it easily have been recognized that Planck did not obtain the expected classical result. The “ultraviolet catastrophe” described in many undergraduate texts as a motivating force for Planck’s work was unknown until Jeans’ calculation was published, and Jeans published his work after Planck did. [Rayleigh’s work refers to sound and ether waves, not to electromagnetic waves described by Maxwell’s equations.] Only in 1906 did Einstein point out that Planck’s 1901 calculation was incorrect—or at least did not match Planck’s description of it—in that Planck’s energy quantization condition was not a clever mathematical trick for approximating an integral, but instead a physical assumption that changed the result of the calculation.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2000 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Phillies, G.D. (2000). Formal Quantum Statistical Mechanics. In: Elementary Lectures in Statistical Mechanics. Graduate Texts in Contemporary Physics. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1264-5_16
Download citation
DOI: https://doi.org/10.1007/978-1-4612-1264-5_16
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4612-7068-3
Online ISBN: 978-1-4612-1264-5
eBook Packages: Springer Book Archive