Fermi's golden rule

In quantum physics, Fermi's golden rule is a simple formula for the constant transition rate (probability of transition per unit time) from one energy eigenstate of a quantum system into other energy eigenstates in a continuum, effected by a perturbation. This rate is effectively constant.

General

Although named after Enrico Fermi, most of the work leading to the Golden Rule is due to Paul Dirac who formulated 20 years earlier a virtually identical equation, including the three components of a constant, the matrix element of the perturbation and an energy difference.^[1]^[2] It was given this name because, on account of its importance, Fermi dubbed it "Golden Rule No. 2." ^[3]

The rate and its derivation

Consider the system to begin in an eigenstate, $|i\rangle$ , of a given Hamiltonian, $H$ ₀ . Consider the effect of a (possibly time-dependent) perturbing Hamiltonian, $H'$ . If $H'$ is time-independent, the system goes only into those states in the continuum that have the same energy as the initial state. If $H'$ is oscillating as a function of time with an angular frequency $ω$ , the transition is into states with energies that differ by $ħω$ from the energy of the initial state.

In both cases, the one-to-many transition probability per unit of time from the state $|i\rangle$ to a set of final states $|f\rangle$ is essentially constant. It is given, to first order in the perturbation, by

\Gamma _{{i\rightarrow f}}={\frac {2\pi }{\hbar }}\left|\langle f|H'|i\rangle \right|^{{2}}\rho

where $ρ$ is the density of final states (number of continuum states per unit of energy) and $\langle f|H'|i\rangle$ is the matrix element (in bra–ket notation) of the perturbation $H'$ between the final and initial states.

This transition probability is also called "decay probability" and is related to the inverse of the mean lifetime. Fermi's golden rule is valid when the initial state has not been significantly depleted by scattering into the final states.

The standard way to derive the equation is to start with time-dependent perturbation theory and to take the limit for absorption under the assumption that the time of the measurement is much larger than the time needed for the transition.^[4]^[5]

Derivation in time-dependent perturbation theory

Main article: Perturbation theory (quantum mechanics) § Time-dependent perturbation theory

The golden rule is a straightforward consequence of the Schrödinger equation, solved to lowest order in the perturbation $H'$ of the Hamiltonian,

\left(H_{0}+H'-\mathrm {i} \hbar {\frac {\partial }{\partial t}}\right)\sum _{n}a_{n}(t)|n\rangle \mathrm {e} ^{-\mathrm {i} tE_{n}/\hbar }=0,

where $E n$ and $| n ⟩$ are the stationary eigenvalues and eigenfunctions of $H$ ₀ .

To lowest order in a constant perturbation $H'$ which turns on at $t$ =0, then,

\mathrm {i} \hbar {\frac {\mathrm {d} a_{k}(t)}{\mathrm {d} t}}=\sum _{n}\langle k|H'|n\rangle a_{n}(t)\mathrm {e} ^{\mathrm {i} t(E_{k}-E_{n})/\hbar },

which integrates to

\mathrm {i} \hbar a_{k}(t)=2\langle k|H'|i\rangle \mathrm {e} ^{\mathrm {i} \omega t/2}{\frac {\sin \omega t/2}{\omega }}

for $\omega \equiv (E_{k}-E_{i})/\hbar$ , for a state with $a i (0)$ =1, $a k (0)$ =0, transitioning to a state with $a k (t)$ .

The transition rate is then

\Gamma _{i\rightarrow k}={\frac {\mathrm {d} }{\mathrm {d} t}}\left|a_{k}(t)\right|^{2}={\frac {2|\langle k|H'|i\rangle |^{2}}{\hbar ^{2}}}{\frac {\sin \omega t}{\omega }},

a sinc function peaking sharply for small $ω$ , where the variation of the rate with $t$ is linear, for an isolated state $|k\rangle$ !

By dramatic contrast, for states of energy $E$ embedded in a continuum, they must be all accounted for collectively. For a density of states per unit energy interval $ρ (E)$ , they must be integrated over their energies, and whence the corresponding $ω$ s,

\Gamma _{i\rightarrow f}={\frac {2}{\hbar }}\int _{-\infty }^{\infty }\mathrm {d} \omega \rho (\omega )|\langle f|H'|i\rangle |^{2}\,{\frac {\sin \omega t}{\omega }}~.

For large $t$ , the sinc function is sharply peaked at $ω$ ≈ 0, and negligible outside $[- 2π/t, 2π/t]$ ; the density and transition element can be taken out of the integral, so that the rate

\Gamma _{i\rightarrow f}={\frac {2\rho |\langle f|H'|i\rangle |^{2}}{\hbar }}\int _{-\infty }^{\infty }\mathrm {d} \omega {\frac {\sin \omega t}{\omega }}

is now merely proportional to a constant Dirichlet integral, $π$ .

The time dependence has vanished, and the constant decay rate of the golden rule follows.^[6] As a constant, it underlies the exponential particle decay laws of radioactivity. (For excessively long times, however, the secular growth of the $a k (t)$ s invalidates lowest-order perturbation theory, which requires $a k ≪ a i$ .)

Only the magnitude of the matrix element $\langle f|H'|i\rangle$ enters the Fermi's Golden Rule. The phase of this matrix element, however, contains separate information about the transition process. It appears in expressions that complement the Golden Rule in the semiclassical Boltzmann equation approach to electron transport.^[7]

References

↑ Bransden, B. H.; Joachain, C. J. (1999). Quantum Mechanics (2nd ed.). p. 443. ISBN 978-0582356917.
↑ Dirac, P.A.M. (1 March 1927). "The Quantum Theory of Emission and Absorption of Radiation". Proceedings of the Royal Society A. 114 (767): 243–265. Bibcode:1927RSPSA.114..243D. doi:10.1098/rspa.1927.0039. JSTOR 94746. See equations (24) and (32).
↑ Fermi, E. (1950). Nuclear Physics. University of Chicago Press. ISBN 978-0226243658. , formula VIII.2
↑ R Schwitters' UT Notes on Derivation
↑ It is remarkable in that the rate is constant and not linearly increasing in time, as might be naively expected for transitions with strict conservation of energy enforced. This comes about from interference of oscillatory contributions of transitions to numerous continuum states with only approximate unperturbed energy conservation, cf. Wolfgang Pauli, Wave Mechanics: Volume 5 of Pauli Lectures on Physics (Dover Books on Physics, 2000) ISBN 0486414620 , pp. 150-151.
↑ Merzbacher, Eugen (1998). "19.7". Quantum Mechanics (PDF) (3rd ed.). Wiley, John & Sons, Inc. ISBN 0-471-88702-1.
↑ N. A. Sinitsyn, Q. Niu and A. H. MacDonald (2006). "Coordinate Shift in Semiclassical Boltzmann Equation and Anomalous Hall Effect". Phys. Rev. B. 73 (7): 075318. arXiv:cond-mat/0511310. Bibcode:2006PhRvB..73g5318S. doi:10.1103/PhysRevB.73.075318.

External links

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Fermi's golden rule

General

The rate and its derivation

See also

References

External links