Fermi's golden rule
In quantum physics, Fermi's golden rule is a formula that describes the transition rate 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 and is proportional to the strength of the coupling between the initial and final states of the system as well as the density of states. It is also applicable when the final state is discrete, i.e. it is not part of a continuum, if there is some decoherence in the process, like relaxation or collision of the atoms, or like noise in the perturbation, in which case the density of states is replaced by the reciprocal of the decoherence bandwidth.
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 called 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 that begins in an eigenstate of an unperturbed Hamiltonian 0 and considers the effect of a perturbing Hamiltonian applied to the system. If is time-independent, the system goes only into those states in the continuum that have the same energy as the initial state. If is oscillating sinusoidally as a function of time with an angular frequency, the transition is into states with energies that differ by 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 of the perturbation between the final and initial states, and is the density of states 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.
Statement of the problem
The golden rule is a straightforward consequence of the Schrödinger equation, solved to lowest order in the perturbation of the Hamiltonian. The total Hamiltonian is the sum of an “original” Hamiltonian 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.Discrete spectrum of final states
We first consider the case where the final states are discrete. The expansion of a state in the perturbed system at a time is. The coefficients 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 and are the stationary eigenvalues and eigenfunctions of 0.
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 that turns on at = 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 :
whose integral can be expressed via the identity as
with, for a state with = 1, = 0, transitioning to a state with . This is the same as the generic result for the time evolution of any two-state system in a basis where the Hamiltonian is not diagonal.
The transition rate is then
a sinc function peaking sharply for small. At,, so the transition rate varies linearly with for an isolated state !
Continuous spectrum of final states
By dramatic contrast, for states of energy embedded in a continuum, they must be all accounted for collectively. For a density of states per unit energy interval, they must be integrated over their energies, and whence the corresponding values,For large, the sinc function is sharply peaked at ≈ 0, so the density of states can be taken out of the integral. We also assume that the transition element can be approximated as a constant. The rate is then
A change of variables shows that the integral is independent of t, the definite integral being.
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.
Applications
Semiconductors
The Fermi golden rule can be used for calculating the transition probability rate for an electron that is excited by a photon from the valence band to the conduction band in a direct band-gap semiconductor, and also for when the electron recombines with the hole and emits a photon. Consider a photon of frequency and wavevector, where the light dispersion relation is and is the index of refraction.Using the Coulomb gauge where and, the vector potential of the EM wave is given by where the resulting electric field is
For a charged particle in the valence band, the Hamiltonian is
where is the potential of the crystal. If our particle is an electron and we consider process involving one photon and first order in. The resulting Hamiltonian is
where is the perturbation of the EM wave.
From here on we have transition probability based on time-dependent perturbation theory that
where is the light polarization vector. From perturbation it is evident that the heart of the calculation lies in the matrix elements shown in the braket.
For the initial and final states in valence and conduction bands respectively, we have and, and if the operator does not act on the spin, the electron stays in the same spin state and hence we can write the wavefunctions as Bloch waves so
where is the number of unit cells with volume. Using these wavefunctions and with some more mathematics, and focusing on emission rather than absorption, we are led to the transition rate
where is the transition dipole moment matrix element is qualitatively the expectation value and in this situation takes the form
Finally, we want to know the total transition rate. Hence we need to sum over all initial and final states, and take into account spin degeneracy, which through some mathematics results in
where is the joint valence-conduction density of states. In 3D, this is
but the joint DOS is different for 2D, 1D, and 0D.
Finally we note that in a general way we can express the Fermi golden rule for semiconductors as
Scanning tunneling microscopy
In a scanning tunneling microscope, the Fermi golden rule is used in deriving the tunneling current. It takes the formwhere is the tunneling matrix element.
Quantum optics
When considering energy level transitions between two discrete states, Fermi's golden rule is written aswhere 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 states, i.e. the range of allowed photon energies is continuous.