The Density Matrix, Entropy, and Coupled Harmonic Oscillators Revisited
- 1Department of Physics, Middle East Technical University, Ankara 06800, Türkiye
- 2Department of Radiology, New York University, New York, NY 10016, USA
Abstract
In this review article, we first introduce the density matrix, along with its properties and time evolution. We note that one of the principal uses of the density matrix is to calculate measurement probabilities in quantum mechanics. When only some of the variables can be measured, it is necessary to integrate out the variables that are inaccessible to measurement. The resulting subsystem is then analogous to an open quantum system, where the unmeasured variables belong to an unspecified environment. Coupled and Lorentz-covariant harmonic oscillators are described and their differences are explained. We introduce the density matrix for two harmonic oscillators and demonstrate how they become entangled as well as entangled excited states of the Lorentz-covariant harmonic oscillator. We elaborate on the concept of von Neumann entropy and show how entropy increases when not all the variables of the system can be measured. We use the timeseparation between quarks as an example of an unmeasurable quantity and then present a discussion of von Neumann entropy and entanglement. We include “boiling quarksgluons”, whose most interesting feature is the transition from the confinement phase to the plasma phase, to illustrate the application of the density function and entropy. In this context, we discuss the quark-parton puzzle and a possible explanation.
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BibTeX
@article{baskal2026density,
title = {{The Density Matrix, Entropy, and Coupled Harmonic Oscillators Revisited}},
author = {Sibel Başkal and Marilyn E. Noz},
journal = {Axioms},
year = {2026},
month = oct,
doi = {10.3390/axioms15100742},
url = {https://www.mdpi.com/2075-1680/15/10/742},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - The Density Matrix, Entropy, and Coupled Harmonic Oscillators Revisited
AU - Sibel Başkal
AU - Marilyn E. Noz
PY - 2026
DA - 2026/10/03
JO - Axioms
DO - 10.3390/axioms15100742
UR - https://www.mdpi.com/2075-1680/15/10/742
AB - In this review article, we first introduce the density matrix, along with its properties and time evolution. We note that one of the principal uses of the density matrix is to calculate measurement probabilities in quantum mechanics. When only some of the variables can be measured, it is necessary to integrate out the variables that are inaccessible to measurement. The resulting subsystem is then analogous to an open quantum system, where the unmeasured variables belong to an unspecified environment. Coupled and Lorentz-covariant harmonic oscillators are described and their differences are explained. We introduce the density matrix for two harmonic oscillators and demonstrate how they become entangled as well as entangled excited states of the Lorentz-covariant harmonic oscillator. We elaborate on the concept of von Neumann entropy and show how entropy increases when not all the variables of the system can be measured. We use the timeseparation between quarks as an example of an unmeasurable quantity and then present a discussion of von Neumann entropy and entanglement. We include “boiling quarksgluons”, whose most interesting feature is the transition from the confinement phase to the plasma phase, to illustrate the application of the density function and entropy. In this context, we discuss the quark-parton puzzle and a possible explanation.
SN - 2075-1680
ER -