Abstract
For a quantum system, a density matrix ρ that is not pure can arise, via averaging, from a distribution μ of its wave function, a normalized vector belonging to its Hilbert space ℋ. While ρ itself does not determine a unique μ, additional facts, such as that the system has come to thermal equilibrium, might. It is thus not unreasonable to ask, which μ, if any, corresponds to a given thermodynamic ensemble? To answer this question we construct, for any given density matrix ρ, a natural measure on the unit sphere in ℋ, denoted GAP(ρ). We do this using a suitable projection of the Gaussian measure on ℋ with covariance ρ. We establish some nice properties of GAP(ρ) and show that this measure arises naturally when considering macroscopic systems. In particular, we argue that it is the most appropriate choice for systems in thermal equilibrium, described by the canonical ensemble density matrix ρβ = (1/Z) exp (−β H). GAP(ρ) may also be relevant to quantum chaos and to the stochastic evolution of open quantum systems, where distributions on ℋ are often used.
Similar content being viewed by others
References
M. Aizenman, S. Goldstein, and J. L. Lebowitz, On the stability of equilibrium states of finite classical systems. J. Math. Phys. 16, 1284–1287 (1975).
V. I. Arnol’d and A. Avez, Ergodic Problems of Classical Mechanics (Benjamin, New York, 1968).
M. Berry, Regular and irregular eigenfunctions. J. Phys. A: Math. Gen. 10, 2083–2091 (1977).
H.-P. Breuer and F. Petruccione, Open Quantum Systems (Oxford University Press, 2002).
D. C. Brody and L. P. Hughston, The quantum canonical ensemble. J. Math. Phys. 39, 6502–6508 (1998).
D. Dürr, S. Goldstein and N. Zanghì, Quantum equilibrium and the origin of absolute uncertainty. J. Statist. Phys. 67, 843–907 (1992).
S. Goldstein, Stochastic mechanics and quantum theory. J. Statist. Phys. 47, 645–667 (1987).
S. Goldstein, J. L. Lebowitz, R. Tumulka, and N. Zanghì, Canonical typicality. Phys. Rev. Lett. 96, 050403 (2006).
S. Goldstein, J. L. Lebowitz, R. Tumulka, and N. Zanghì, Typicality of the GAP measure. In preparation.
F. Guerra and M. I. Loffredo, Thermal mixtures in stochastic mechanics. Lett. Nuovo Cimento (2) 30, 81–87 (1981).
R. Haag, D. Kastler, and E. B. Trych-Pohlmeyer, Stability and equilibrium states. Commun. Math. Phys. 38, 173–193 (1974).
S. Hortikar and M. Srednicki, Correlations in chaotic eigenfunctions at large separation. Phys. Rev. Lett. 80(8), 1646–1649 (1998).
L. D. Landau and E. M. Lifshitz, Statistical Physics. Volume 5 of Course of Theoretical Physics. Translated from the Russian by E. Peierls and R. F. Peierls (Pergamon, London and Paris, 1959).
J. L. Lebowitz, Boltzmann’s entropy and time’s arrow. Phys. Today 46, 32–38 (1993).
J. L. Lebowitz, Microscopic reversibility and macroscopic behavior: physical explanations and mathematical derivations. In 25 Years of Non-Equilibrium Statistical Mechanics, Proceedings, Sitges Conference, Barcelona, Spain, 1994, in Lecture Notes in Physics, J.J. Brey, J. Marro, J.M. Rubí, and M. San Miguel (eds.) (Springer-Verlag, Berlin, 1995).
J. L. Lebowitz, Microscopic origins of irreversible macroscopic behavior. Physica (Amsterdam) 263A, 516–527 (1999).
A. Martin-Löf, Statistical Mechanics and the Foundations of Thermodynamics. Lecture Notes in Physics 101 (Springer, Berlin, 1979).
E. Nelson, Quantum Fluctuations (Princeton University Press, 1985).
S. Popescu, A. J. Short, and A. Winter, The foundations of statistical mechanics from entanglement: Individual states vs. averages. Preprint quant-ph/0511225 (2005).
S. Popescu, A. J. Short, and A. Winter, Entanglement and the foundations of statistical mechanics. Nature Physics 21(11), 754–758 (2006).
D. N. Page, Average entropy of a subsystem. Phys. Rev. Lett. 71(9), 1291–1294 (1993).
E. Schrödinger, The exchange of energy according to wave mechanics. Annalen der Physik (4) 83, 956–968 (1927).
E. Schrödinger, Statistical Thermodynamics., 2nd edn. (Cambridge University Press, 1952).
H. Tasaki, From Quantum dynamics to the canonical distribution: general picture and a rigorous example. Phys. Rev. Lett. 80, 1373–1376 (1998).
J. D. Urbina and K. Richter, Random wave models. Preprint (2005). To appear.
J. von Neumann, Beweis des Ergodensatzes und des H-Theorems in der neuen Mechanik. Z. Physik 57, 30–70 (1929).
J. von Neumann, Mathematical Foundations of Quantum Mechanics (Princeton University Press, Princeton, 1955). Translation of Mathematische Grundlagen der Quantenmechanik (Springer-Verlag, Berlin, 1932).
J. D. Walecka, Fundamentals of Statistical Mechanics. Manuscript and Notes of Felix Bloch (Stanford University Press, Stanford, CA, 1989).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Goldstein, S., Lebowitz, J.L., Tumulka, R. et al. On the Distribution of the Wave Function for Systems in Thermal Equilibrium. J Stat Phys 125, 1193–1221 (2006). https://doi.org/10.1007/s10955-006-9210-z
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-006-9210-z