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 to a group of energy eigenstates in a continuum, as a result of a weak perturbation. This transition rate is effectively independent of time (so long as the strength of the perturbation is independent of time) and is proportional to the strength of the coupling between the initial and final states of the system (described by the square of the matrix element of the perturbation) as well as the density of states. It is also applicable when the final state is not part of a continuum if there is some decoherence in the process, like relaxation of the atoms or like noise in the perturbation, in which case the density of states is replaced by the reciprocal of decoherence bandwidth.
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."
Most uses of the term Fermi's golden rule are referring to "Golden Rule No.2", however, Fermi's "Golden Rule No.1" is of a similar form and considers the probability of indirect transitions per unit time.
The rate and its derivation
Fermi's golden rule describes a system which begins in an eigenstate, , of an unperturbed Hamiltonian, H0 and considers the effect of a perturbing Hamiltonian, H' applied to the system. 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 sinusoidally as a function of time (i.e. it is a harmonic perturbation) 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 transition probability per unit of time from the initial state to a set of final states is essentially constant. It is given, to first order approximation, by
where is the matrix element (in bra–ket notation) of the perturbation H' between the final and initial states and is the density of states (number of continuum states in an infinitesimally small energy interval ) at the energy of the final states. This transition probability is also called "decay probability" and is related to the inverse of the mean lifetime. Thus, the probability of finding the system in state is proportional to .
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|
The golden rule is a straightforward consequence of the Schrödinger equation, solved to lowest order in the perturbation H' of the Hamiltonian. The total Hamiltonian is the sum of an “original” Hamiltonian H0 and a perturbation, . In the interaction picture, we can expand an arbitrary quantum state’s time evolution in terms of energy eigenstates of the unperturbed system , with .
The expansion of a state in the perturbed system at a time t is . The coefficients an(t) are yet-unknown functions of time yielding the probability amplitudes in the Dirac picture. This state obeys the time-dependent Schrödinger equation:
Expanding the Hamiltonian and the state, we see that, to first order,
where En and |n⟩ are the stationary eigenvalues and eigenfunctions of H0 .
This equation can be rewritten as a system of differential equations specifying the time evolution of the coefficients ,
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 , it is evident that , which simply says that the system stays in the initial state .
For states , becomes non-zero due to and these are assumed to be small due to the weak perturbation. Hence, one can plug in the zeroth order form into the above equation to get the first correction for the amplitudes ,
which integrates to
for , for a state with ai(0) =1, ak(0)=0, transitioning to a state with ak(t) (again, ).
The transition rate is then
a sinc function peaking sharply for small ω. At , , so the transition rate varies linearly with t for an isolated state !
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,
For large t, the sinc function is sharply peaked at ω ≈ 0, and negligible outside [−π/t, π/t] ; the density and transition element can be taken out of the integral, so that the rate
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 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.
Use in quantum optics
When considering energy level transitions between two discrete states Fermi's golden rule is written as,
where is the density of photon states at a given energy, is the photon energy and is the angular frequency. This alternative expression relies on the fact that there is a continuum of final (photon) states, i.e. the range of allowed photon energies is continuous.
The Drexhage experiment
Fermi's golden rule predicts that the probability that an excited state will decay depends on the density of states. This can be seen experimentally by measuring the decay rate of a dipole near to a mirror: as the presence of the mirror creates regions of higher and lower density of states, the measured decay rate depends on the distance between the mirror and the dipole.
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- 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.
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