# 4: Irreversible Relaxation

$$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

At a fundamental level, the basic laws governing the time evolution of isolated quantum mechanical systems are invariant under time reversal. That is, there is no preferred direction to the arrow of time. The Time Dependent Schrödinger Equation is reversible, meaning that one can find solutions for propagating either forward or backward in time. If one reverses the sign of time and thereby momenta of objects, we should be able to go back where the system was at an earlier time. We can see this in the exact solution to the two-level problem, where amplitude oscillates between the two states with a frequency that depends on the coupling. If we reverse the sign of the time, the motion is reversed. In contrast, when a quantum system is in contact with another system having many degrees of freedom, a definite direction emerges to the arrow of time, and the system’s dynamics is no longer reversible. Such irreversible systems are dissipative, meaning they decay in time from a prepared state to a state where phase relationships between the basis states are lost.

• 4.1: Introduction to Dissipative Dynamics
How does irreversible behavior, a hallmark of chemical systems, arise from the deterministic Time Dependent Schrödinger Equation? We will answer this question specifically in the context of quantum transitions from a given energy state of the system to energy states its surroundings. Qualitatively, such behavior can be expected to arise from destructive interference between oscillatory solutions of the system and the set of closely packed manifold of energy states of the bath.

4: Irreversible Relaxation is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Andrei Tokmakoff via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.