Fermi's golden rule

In quantum physics, Fermi's golden rule is a formula that describes the 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 weak 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. It was given this name because, on account of its importance, Fermi dubbed it "Golden Rule No. 2."

The rate and its derivation

Consider the system to begin in an eigenstate, $|i\rangle$, of a given Hamiltonian, H₀ . 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.

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 En and |n⟩ are the stationary eigenvalues and eigenfunctions of H0 . Rewrite this equation as that of the time evolution of the coefficients $a_{n}(t)$, $\mathrm {i} \hbar {\frac {\mathrm {d} a_{k}(t)}{\mathrm {d} t}}=\sum _{n}\langle k|H'|n\rangle a_{n}(t)\mathrm {e} ^{\mathrm {i} t(E_{k}-E_{n})/\hbar }.$ This equation is exact but normally cannot be solved in practice. For a weak constant perturbation H' which turns on at t=0, we can use perturbation theory. Namely, if $H'=0$ it is evident that $a_{n}(t)=\delta _{n,i}$, which simply says that the system stays in the initial state $i$. For states $k\neq i$, $a_{k}(t)$ becomes non-zero due to $H'\neq 0$ and these are assumed to be small due to the weak perturbation. Hence, one can plug in the zeroth order form $a_{n}(t)=\delta _{n,i}$ into the above equation to get the first correction for the amplitudes $a_{k}(t)$, $\mathrm {i} \hbar {\frac {\mathrm {d} a_{k}(t)}{\mathrm {d} t}}=\langle k|H'|i\rangle \mathrm {e} ^{\mathrm {i} t(E_{k}-E_{i})/\hbar },$ 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 ai(0) =1, ak(0)=0, transitioning to a state with ak(t) (again, $k\neq i$). 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. At $\omega =0$, $\sin(\omega t)/\omega =t$, so the transition rate varies linearly with t 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 ωs, $\Gamma _{i\rightarrow f}={\frac {2}{\hbar }}\int _{-\infty }^{\infty }\mathrm {d} \omega \rho (\omega )|\langle f|H'|i\rangle |^{2}{\frac {\sin \omega t}{\omega }}~.$ 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. As a constant, it underlies the exponential particle decay laws of radioactivity. (For excessively long times, however, the secular growth of the ak(t)s invalidates lowest-order perturbation theory, which requires ak ≪ ai .)

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.

In popular culture

Fermi's golden rule is displayed on a computer screen in the movie Prince of Darkness (film) where it is explained that it was translated from Latin found in an ancient text.

See also

• Exponential decay
• List of things named after Enrico Fermi
• Particle decay
• Sinc function
• Time-dependent perturbation theory

External links

• More information on Fermi's golden rule
• Derivation of Fermi’s Golden Rule
• Time-dependent perturbation theory
• Fermi's golden rule: its derivation and breakdown by an ideal model