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Quantum Mechanics and the Arrow of Time

Richard Joseph Oldani

Published Jan 27, 2026• 0 citations
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Abstract

Matrix mechanics is not considered important for understanding quantum mechanics; however, it is the key to understanding time reversal asymmetry. We use a single, solitary atom to derive the diagonalized Hamiltonian matrix in continuous time and use it to visualize the irreversibility of time in a quantum system. Graphic evidence in support of the model is obtained from experiments performed with the simplest quantum system, an electron cyclotron. Theoretical arguments supporting time asymmetry in statistical mechanics that were presented by Feynman and Einstein are also cited. It is hypothesized that equations describing time reversal asymmetry must be derived in energy-time coordinates.

DOI
10.20944/preprints202601.2089.v1

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Matrix mechanics is not considered important for understanding quantum mechanics; however, it is the key to understanding time reversal asymmetry. We use a single, solitary atom to derive the diagonalized Hamiltonian matrix in continuous time and use it to visualize the irreversibility of time in a quantum system. Graphic evidence in support of the model is obtained from experiments performed with the simplest quantum system, an electron cyclotron. Theoretical arguments supporting time asymmetry in statistical mechanics that were presented by Feynman and Einstein are also cited. It is hypothesized that equations describing time reversal asymmetry must be derived in energy-time coordinates.
We show time reversal asymmetry in quantum mechanics by isolating a single, solitary hydrogen atom in a black body, deriving from it the diagonalized Hamiltonian matrix in continuous time, and using it to visualize the irreversibility of the emission and absorption of radiation. Graphic evidence in support of the model is obtained from experiments performed with the simplest quantum system, an electron cyclotron. The experimental evidence is used to obtain equations of motion to describe time asymmetry with respect to the energy-time conjugate variables in linear time which complement the time symmetric equations of motion in position-momentum coordinates given with respect to cyclical time. Theoretical arguments by Feynman and Einstein are shown to support time reversal asymmetry.
We reformulate matrix mechanics by incrementally heating a single hydrogen atom in order to derive the complete, diagonalized Hamiltonian matrix. The spectral lines and transition probabilities cannot be generated by reversing time thereby demonstrating asymmetry. Experimental evidence in support of the theoretical model is obtained from experiments performed with the simplest quantum system, an electron cyclotron. Wave mechanics is also reformulated by altering the original Schrödinger equation describing a time symmetric, conservative system to a system of two independent equations describing a time asymmetric, non-conservative system.
We reformulate matrix mechanics by incrementally heating a single hydrogen atom in order to derive the complete, diagonalized Hamiltonian matrix. The spectral lines and transition probabilities cannot be generated by reversing time thereby demonstrating asymmetry. Experimental evidence in support of the theoretical model is obtained from experiments performed with the simplest quantum system, an electron cyclotron. Wave mechanics is also reformulated by altering the original Schrödinger equation describing a time symmetric, conservative system to a system of two independent equations describing a time asymmetric, non-conservative system.
It is one of the most important and long-standing issues of physics to derive the irreversibility out of a time-reversal symmetric equation of motion. The present paper considers the breaking of the time-reversal symmetry in open quantum systems and the emergence of an arrow of time. We claim that the time-reversal symmetric Schrödinger equation can have eigenstates that break the time-reversal symmetry if the system is open in the sense that it has at least a countably infinite number of states. Such eigenstates, namely the resonant and anti-resonant states, have complex eigenvalues. We show that, although these states are often called "unphysical", they observe the probability conservation in a particular way. We also comment that the seemingly Hermitian Hamiltonian is non-Hermitian in the functional space of the resonant and anti-resonant states, and hence there is no contradiction in the fact that it has complex eigenvalues. We finally show how the existence of the states that break the time-reversal symmetry affects the quantum dynamics. The dynamics that starts from a time-reversal symmetric initial state is dominated by the resonant states for t > 0 ; this explains the phenomenon of the arrow of time, in which the decay excels the growth. The time-reversal symmetry holds in that the dynamic ending at a time-reversal symmetric final state is dominated by the anti-resonant states for t < 0 .
While the microscopic laws of physics are often symmetric under time reversal, most natural processes that we observe are not. The emergent asymmetry between typical and time-reversed processes is referred to as the arrow of time. In quantum physics, an arrow of time emerges when a sequence of measurements is performed on a system. We introduce quantum control tools that can yield dynamics more consistent with time flowing backward than forward. The control tools are based on the explicit construction of a Hamiltonian that can replicate the stochastic trajectories of a monitored quantum system. Such a Hamiltonian can reverse the effect of monitoring and, via a feedback process, generate trajectories consistent with a reversed arrow of time. It can also be used to simulate the backward-in-time dynamics of an open quantum system. Finally, we design a feedback-driven continuous measurement engine powered by the energy pumped into the system by the monitoring process. We show that the engine can operate under experimentally realizable conditions with feedback delay and finite-efficiency measurements.
Deriving an arrow of time from time-reversal symmetric microscopic dynamics is a fundamental open problem in many areas of physics, ranging from cosmology, to particle physics, to thermodynamics and statistical mechanics. Here we focus on the derivation of the arrow of time in open quantum systems and study precisely how time-reversal symmetry is broken. This derivation involves the Markov approximation applied to a system interacting with an infinite heat bath. We find that the Markov approximation does not imply a violation of time-reversal symmetry. Our results show instead that the time-reversal symmetry is maintained in the derived equations of motion. This imposes a time-symmetric formulation of quantum Brownian motion, Lindblad and Pauli master equations, which hence describe thermalisation that may occur into two opposing time directions. As a consequence, we argue that these dynamics are better described by a time-symmetric definition of Markovianity. Our results may reflect on the formulations of the arrow of time in thermodynamics, cosmology, and quantum mechanics.