Quantum Mechanics of Infinite Systems and Macroscopic Evolution Equations

John von Neumann

Alexey Yur'evich Samarin

The change with time of the system consisting of the quantum object and the macroscopic measuring instrument is described on the base of the uniform dynamic law, which is suitable both evolution and reduction processes description. It is the integral wave equation with kernel in the form of a path integral. It is shown, that wave function collapse is the specific transformation which is fundamentally differ from Shrödinger's evolution. Specifically, a formal cause of the collapse is a local time derivative (infinite large) of the potential energy. Such transformation can not be described using mathematical apparatus of conventional quantum mechanics.

N. G. van Kampen

Alexey Yur'evich Samarin

Macroscopic body is considered as a system, consisting of an infinite number of quantum particles. Mechanical motion of the system center of mass in physical space is described, using Feynman's representation of the quantum evolution. It is shown that the center of mass of the system moves accordingly with the minimum action principle if action functional is much higher then Plank's constant.

Yuuya Chiba, Yasushi Yoneta, Ryusuke Hamazaki +1 more

The second law of thermodynamics for adiabatic operations -- constraints on state transitions in closed systems under external control -- is one of the fundamental principles of thermodynamics. On the other hand, it is recently established that even pure quantum states can represent thermal equilibrium. However, pure quantum states do not satisfy the second law in that they are not passive, i.e., work can be extracted from them if arbitrary unitary operations are allowed. It therefore remains unresolved how quantum mechanics can be reconciled with thermodynamics. Here, based on our key quantum-mechanical notions of thermal equilibrium and adiabatic operations, we address the emergence of the second law for adiabatic operations in the thermodynamics limit. We first introduce infinite-observable macroscopic thermal equilibrium (iMATE); a quantum state, including pure states, is in iMATE if the expectation values of all additive observables agree with their equilibrium values. We also introduce a macroscopic operation as unitary evolution generated by a time-dependent additive Hamiltonian, which is regarded as corresponding to adiabatic operations. Employing these concepts, we show that no extensive work can be extracted from any quantum state in iMATE through any macroscopic operations. Furthermore, we introduce a quantum-mechanical form of entropy density such that it agrees with thermodynamic entropy density for any quantum state in iMATE. We then prove that for any initial state in iMATE, this entropy density cannot be decreased by any macroscopic operations, followed by a time-independent relaxation process. Our theory thus proves two different forms of the second law, by adopting macroscopically reasonable classes of observables, equilibrium states, and operations. We also discuss the time scales of macroscopic operations in these results.

A. J. Leggett