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Cambridge University Press eBooks

Quantum Rate Processes

Uri Peskin

Published May 11, 2023• 0 citations
Publisher

Abstract

The rates of charge transfer and energy transfer are essential for understanding nanoscale phenomena and processes. Here we introduce the general conditions for the emergence of rate processes in quantum mechanics. We refer to the generic scenario in which a transition is induced between eigenstates of a given Hamiltonian by a weak perturbation. Analysis reveals that when the initial and final states are pure states, the transition rate is time-dependent and fails to reach a finite constant value. If, however, the final state is a mixed ensemble that is sufficiently wide and dense in energy, a rate constant emerges, given by Fermi’s Golden Rule (FGR). When the initial state is also mixed, for example, a thermal equilibrium state, a thermal rate constant is formulated in terms of transition rates between specific eigenstates, summed over the final eigenstates, and averaging over the thermal distribution of the initial eigenstates.

DOI
10.1017/9781108877787.018

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The rates of charge transfer and energy transfer are essential for understanding nanoscale phenomena and processes. Here we introduce the general conditions for the emergence of rate processes in quantum mechanics. We refer to the generic scenario in which a transition is induced between eigenstates of a given Hamiltonian by a weak perturbation. Analysis reveals that when the initial and final states are pure states, the transition rate is time-dependent and fails to reach a finite constant value. If, however, the final state is a mixed ensemble that is sufficiently wide and dense in energy, a rate constant emerges, given by Fermi’s Golden Rule (FGR). When the initial state is also mixed, for example, a thermal equilibrium state, a thermal rate constant is formulated in terms of transition rates between specific eigenstates, summed over the final eigenstates, and averaging over the thermal distribution of the initial eigenstates.
Basic concepts of quantum mechanics: Schroedinger equation; Dirac notation; the energy representation; expectation value; Hermite operators; coherent superposition of states and motion in the quantum world; perturbation Hamiltonian. Time-dependent perturbation theory: harmonic perturbation. Transition rate: Fermi’s golden rule. The density matrix; pure and mixed states. Temporal dependence of the density operator: von Neuman equation. Randomizing Hamiltonian. Longitudinal and transverse relaxation times. Density matrix and entropy.
Controlling transition rates is a central objective across quantum optics, condensed-matter physics, and chemical physics, encompassing processes such as spontaneous emission, energy and charge transfer, spin relaxation, and chemical reactions. Although Fermi's golden rule provides a powerful framework for calculating such rates of weakly coupled transitions, conventional calculations require the interaction to be specified in advance; the range of rate responses attainable over a broader interaction space remains unknown. We formulate spectral-overlap golden-rule rates as quadratic forms of the interaction and derive rigorous bounds on rate modulation over a specified interaction space. As an example, application to energy transfer from a donor to a molecular vibration coupled to an optical cavity reveals that enhancement and suppression depend jointly on the interaction and cavity detuning, including continuous reversal of the rate response for a fixed interaction. The same approach may elucidate rate control for other golden-rule processes, including excitation-energy transfer, nonadiabatic electron transfer, nonradiative electronic transitions, vibrational and spin relaxation, and spontaneous emission.
When Fermi's golden rule (FGR) is studied in the Wigner representation, the transition rate from an initial pure state or from an initial thermal distribution into a quasicontinuum manifold of degenerate states is given by an overlap integral of Wigner functions in phase space. In the semiclassical limit the transition rate is obtained by integrating over the regions in phase space where the energy difference between the initial and final potential surfaces is equal to the available energy. The integral is weighted by the initial probability density to be at that phase-space region. The classical limit of FGR is thus both simple and intuitive. In one dimension a relation to the Landau–Zener–Stuckelberg formula is established. The multi-dimensional case is considered by induction, proving that for separable multi-dimensional systems deviations of the logarithm of the transition rate from its classical limit scale at worst linearly with the dimension.
The transition process in quantum mechanics has been studied with the probability ΓT proportional to the time interval T between the initial and final states, where Γ is the rate.Recently it was found that the constant term P (d) is added as P = ΓT + P (d) .P (d) has been ignored for a macroscopic T, but that is not ignoable in various systems.We find P (d) in such processes that Γ ≈ 0 and show that it has wave-like properties and leads unusual quantum phenomena.P (d) is large in magnitude and in spatial size for light particles and has universal properties in relativisticaly invariant systems.Implications to processes of neutrinos in beta decays and gamma rays in various radiative transitions are presented.
Fermi's golden rule is one of the most important tools of quantum mechanics. It gives the general formula for transition rates, the rates at which particles are ‘scattered’ from one state to another by a perturbation. ‘Scattered’ is in quotation marks because it is a much more general concept than one might guess. An obvious example is provided by impurities in a crystal, which scatter an electron from one Bloch state to another. They change its momentum but not its energy. Similarly, phonons (vibrations of the lattice) also scatter electrons, but in this case they change the energy of the electrons as well as their momentum. A less obvious example is the absorption of light, which can be viewed as a scattering process in which an electron collides with a photon. The converse process also occurs, where an electron loses energy to a photon, and gives rise to spontaneous and stimulated emission. Thus scattering is a remarkably general concept. The examples suggest that there are two broad classes of scattering processes that we should treat: (i) potentials that are constant in time, such as impurities in a crystal, which do not change the energy of the particle being scattered; (ii) potentials that vary harmonically in time as cos ω q t , such as phonons and photons, which change the energy of the particle by ± ħ ω q .
Lower and upper bounds are derived for the decay and transitions of quantum states, evolving under a time-dependent Hamiltonian, in terms of the energy uncertainty of the initial and final state. The bounds are simultaneously a rigorous version of Fermi's golden rule and of the time-energy uncertainty relation. They are sharp, refer to short times, and are compared with recent long-time results for time-independent Hamiltonians. Illustrations for tunneling systems, laser-driven processes, and neutron interferometry in time-dependent magnetic fields are given.