We study how the grand-canonical density matrix arises in macroscopic quantum systems. ``Canonical typicality'' is the known statement that for a typical wave function
Ψ from a micro-canonical energy shell of a quantum system
S weakly coupled to a large but finite quantum system
B, the reduced density matrix
ρ^ΨS=trB∣Ψ⟩⟨Ψ∣ is approximately equal to the canonical density matrix
ρ^can=Zcan−1exp(−βH^S). Here, we discuss the analogous statement and related questions for the \emph{grand-canonical} density matrix
ρ^gc=Zgc−1exp(−β(H^S−μ1N^1S−…−μrN^rS)) with
N^iS the number operator for molecules of type
i in the system
S. This includes (i) the case of chemical reactions (which requires some novel considerations) and (ii) that of systems
S defined by a spatial region which particles may enter or leave. It includes statements about how
ρ^gc arises from the density matrix of the appropriate (generalized micro-canonical) Hilbert subspace
Hgmc⊂HS⊗HB (defined by a micro-canonical interval of total energy and suitable particle number sectors) or from typical
Ψ in
Hgmc, as well as statements about the distribution of the (conditional) wave function
ψS of
S, which turns out to be a so-called GAP or Scrooge measure. That is, we discuss the foundation and justification of both the density matrix and the distribution of the wave function in the grand-canonical case. To this end (particularly for the chemical reactions), we also need to extend these considerations to the so-called generalized Gibbs ensembles, which apply to systems for which some macroscopic observables are conserved.
For a macroscopic, isolated quantum system in an unknown pure state, the expectation value of any given observable is shown to hardly deviate from the ensemble average with extremely high probability under generic equilibrium and nonequilibrium conditions. Special care is devoted to the uncontrollable microscopic details of the system state. For a subsystem weakly coupled to a large heat bath, the canonical ensemble is recovered under much more general and realistic assumptions than those implicit in the usual microcanonical description of the composite system at equilibrium.
The role of the system-bath coupling is studied for the generalized canonical thermalization of a system S surrounded by a heat bath B, which was most recently shown [S. Popescu, et al., Nature Physics 2,754(2006) and S. Goldstein et al., Phys. Rev. Lett. 96, 050403(2006)] as a kinematic reduction to a canonical equilibrium state from almost all the pure states of the universe formed by S plus B. This kinematic process is revisited by explicitly considering the energy shell deformation due to the interaction between S and B with an exactly solvable model. By calculating the state numbers of the universe and its subsystems S and B in the deformed energy shells, it is found that, for the overwhelming majority of the universe states (they are entangled at least), the diagonal canonical typicality remains robust with respect to finite interactions between S and B. Particularly, the kinematic decoherence is utilized here to account for the vanishing of the off-diagonal elements of the reduced density matrix of S. It is pointed out that the non-vanishing off-diagonal elements due to the finiteness of bath and the stronger system-bath interaction might offer novelties of the quantum thermalization.
It is typically assumed, without justification, that a weak coupling between a system and a bath is a necessary condition for the equivalence of a canonical ensemble and a microcanonical ensemble. For instance, in a canonical ensemble, temperature emerges if the system and the bath are uncoupled or weakly coupled. We investigate the validity region of this weak-coupling approximation, using a coupled composite-spin system. Our results show that the spin coupling strength can be as large as the level spacing of the system, indicating that the weak-coupling approximation has a much wider region of validity than usually expected.
The phase space Γ of quantum mechanics can be viewed as the complex projective space CPn endowed with a Kählerian structure given by the Fubini-Study metric and an associated symplectic form. We can then interpret the Schrödinger equation as generating a Hamiltonian dynamics on Γ. Based upon the geometric structure of the quantum phase space we introduce the corresponding natural microcanonical and canonical ensembles. The resulting density matrix for the canonical Γ-ensemble differs from the density matrix of the conventional approach. As an illustration, the results are applied to the case of a spin one-half particle in a heat bath with an applied magnetic field.
The phase space of quantum mechanics can be viewed as the complex projective space CP^n endowed with a Kaehlerian structure given by the Fubini-Study metric and an associated symplectic form. Therefore, we can interpret the Schrodinger equation as generating a natural Hamiltonian dynamics on CP^n. Based upon the geometric structure of the quantum phase space we introduce the corresponding natural microcanonical and canonical ensembles. The resulting distribution implied by the quantum canonical ensemble differs from distribution of the conventional density matrix approach, which in some respects can be viewed as semiclassical. The results are applied in detail to the case of a spin one-half particle in a heat bath with an applied magnetic field.