Thermal Equilibrium Distribution in Infinite-Dimensional Hilbert Spaces
Abstract
The thermal equilibrium distribution over quantum-mechanical wave functions is a so-called Gaussian adjusted projected (GAP) measure, GAP(ρβ), for a thermal density operator ρβ at inverse temperature β. More generally, GAP(ρ) is a probability measure on the unit sphere in Hilbert space for any density operator ρ (i.e. a positive operator with trace 1). In this note, we collect the mathematical details concerning the rigorous definition of GAP(ρ) in infinite-dimensional separable Hilbert spaces. Its existence and uniqueness follows from Prohorov's theorem on the existence and uniqueness of Gaussian measures in Hilbert spaces with given mean and covariance. We also give an alternative existence proof. Finally, we give a proof that GAP(ρ) depends continuously on ρ in the sense that convergence of ρ in the trace norm implies weak convergence of GAP(ρ).
- Publication:
-
Reports on Mathematical Physics
- Pub Date:
- December 2020
- DOI:
- 10.1016/S0034-4877(20)30085-9
- arXiv:
- arXiv:2004.14226
- Bibcode:
- 2020RpMP...86..303T
- Keywords:
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- Gaussian measure;
- GAP measure;
- Scrooge measure;
- canonical ensemble in quantum mechanics;
- Mathematical Physics;
- Quantum Physics
- E-Print:
- 12 pages LaTeX, no figures