*Abraham Nitzan*

- Published in print:
- 2006
- Published Online:
- November 2020
- ISBN:
- 9780198529798
- eISBN:
- 9780191916649
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198529798.003.0015
- Subject:
- Chemistry, Physical Chemistry

The first question to ask about the phenomenon of relaxation is why it occurs at all. Both the Newton and the Schrödinger equations are symmetrical under time reversal: The Newton equation, dx/dt = ...
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The first question to ask about the phenomenon of relaxation is why it occurs at all. Both the Newton and the Schrödinger equations are symmetrical under time reversal: The Newton equation, dx/dt = v ; dv/dt = −∂V/∂x, implies that particles obeying this law of motion will retrace their trajectory back in time after changing the sign of both the time t and the particle velocities v. The Schrödinger equation, ∂ψ/∂t = −(i/.h) Ĥ ψ, implies that if (ψ (t) is a solution then ψ *(−t) is also one, so that observables which depend on |ψ|2 are symmetric in time. On the other hand, nature clearly evolves asymmetrically as asserted by the second law of thermodynamics. How does this asymmetry arise in a system that obeys temporal symmetry in its time evolution? Readers with background in thermodynamics and statistical mechanics have encountered the intuitive answer: Irreversibility in a system with many degrees of freedom is essentially a manifestation of the system “getting lost in phase space”:Asystem starts from a given state and evolves in time. If the number of accessible states is huge, the probability that the system will find its way back to the initial state in finite time is vanishingly small, so that an observer who monitors properties associated with the initial state will see an irreversible evolution. The question is how is this irreversible behavior manifested through the reversible equations of motion, and how does it show in the quantitative description of the time evolution. This chapter provides an introduction to this subject using the time-dependent Schrödinger equation as a starting point. Chapter 10 discusses more advanced aspects of this problem within the framework of the quantum Liouville equation and the density operator formalism.
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The first question to ask about the phenomenon of relaxation is why it occurs at all. Both the Newton and the Schrödinger equations are symmetrical under time reversal: The Newton equation, dx/dt = v ; dv/dt = −∂V/∂x, implies that particles obeying this law of motion will retrace their trajectory back in time after changing the sign of both the time t and the particle velocities v. The Schrödinger equation, ∂ψ/∂t = −(i/.h) Ĥ ψ, implies that if (ψ (t) is a solution then ψ *(−t) is also one, so that observables which depend on |ψ|2 are symmetric in time. On the other hand, nature clearly evolves asymmetrically as asserted by the second law of thermodynamics. How does this asymmetry arise in a system that obeys temporal symmetry in its time evolution? Readers with background in thermodynamics and statistical mechanics have encountered the intuitive answer: Irreversibility in a system with many degrees of freedom is essentially a manifestation of the system “getting lost in phase space”:Asystem starts from a given state and evolves in time. If the number of accessible states is huge, the probability that the system will find its way back to the initial state in finite time is vanishingly small, so that an observer who monitors properties associated with the initial state will see an irreversible evolution. The question is how is this irreversible behavior manifested through the reversible equations of motion, and how does it show in the quantitative description of the time evolution. This chapter provides an introduction to this subject using the time-dependent Schrödinger equation as a starting point. Chapter 10 discusses more advanced aspects of this problem within the framework of the quantum Liouville equation and the density operator formalism.