Sharp energy self-determination of macroscopic quantum bodies in pure states, as a validation of the First Law of Thermodynamics

TL;DR

This paper demonstrates that macroscopic quantum pure states have sharply peaked energy distributions, supporting the idea that the First Law of Thermodynamics is fundamentally statistical for large systems.

Contribution

It shows that a broad class of pure states in macroscopic bodies naturally exhibit sharp energy distributions, validating the statistical nature of the First Law.

Findings

Energy distribution widths scale as N^{-1} to N^{-1/2} for large N.

Pure states are effectively microcanonical, with energy concentrated near the mean.

Supports the thermodynamic assumption that closed systems maintain constant energy.

Abstract

We argue that a very large class of quantum pure states of isolated macroscopic bodies have sharply peaked energy distributions, with their width relative to the average scaling between $\sim N^{-1}$ and $\sim N^{-1/2}$, with $N \gg 1$, the number of atoms conforming the body. Those states are dense superpositions of energy eigenstates within arbitrary finite or infinite energy intervals that decay sufficiently fast. The sharpness of the energy distribution implies that closed systems in those states are {\it microcanonical} in the sense that only energy eigenstates very near to the mean energy contribute to their thermodynamic evolution. Since thermodynamics accurately describes processes of macroscopic bodies and requires that closed systems have constant energy, our claim is that these pure states are typical of macroscopic systems. The main assumption beneath the energy sharpness is that the isolated body can reach thermal equilibrium if left unaltered. We argue that such a self-sharpness of the energy in macroscopic bodies indicates that the First Law of Thermodynamics is statistical in character.