Fermi's golden rule

Fermi's Golden Rule, named after the physicist Enrico Fermi, indicates a much used equation from the quantum mechanical perturbation theory. The equation, the theoretical prediction of the transition rate ( the transition probability per time), which merges with an initial state under the influence of a fault in another state. If not additionally in other states are still possible transitions, the inverse of the transition rate is the average lifetime of the initial state. Illustratively stated, this is the time that will be yet to come in the middle of the quantum leap in the new state. With the golden rule can be both spontaneous conversion treatment (e.g., radioactive decay, the emission of light quanta, the decomposition of unstable elementary particles) as well as the absorbent (for example, of photons ), but also the cross-section of any reactions between two particles.

Formula

, An initial state of a disorder exposed through which it can pass into a final state in an energy continuum, it is in the first perturbative approximation, the transfer rate therefor ( i.e., the probability of transition per unit time) by

Given. It is the reduced Planck constant, the density of the observed final states of energy, and belonging to this transition matrix element of the perturbation operator. And the states are eigenstates of an undisturbed Hamiltonian when the disorder is to be considered additionally. The transition rate, the dimension of 1/time. For spontaneous decays (example: radioactivity) is the decay constant in the exponential law. has the dimension of energy and releases the energy uncertainty or half-width of the initial state at. Are conversions in various forms possible, the total decay constant and the total half-width result from the sum of the individual partial values ​​for each type of transition. is the average life of the system in the initial state.

History

The perturbative formalism of the " Golden Rule " was developed by Paul Dirac in 1927 to treat the absorption and emission of photons for the first time in quantum mechanics. A little later it was developed again by Gregor Wentzel in a work to calculate the transition probability for the ( non-radiative ) Auger -Meitner- effect in atoms, which is also a transition from a discrete state in the atom in the continuous part of the spectrum. After Fermi this "rule " [Note 1] is named, as he in 1950 in a nuclear physics textbook as the " Golden Rule No.. 2 " aufführte. In the literature, but sometimes also find the names Wentzel - Fermi Golden Rule and Fermi - Wentzel Golden Rule.

As Golden Rule # 1, the use of the term second -order perturbation theory for such transitions is at Fermi called, which is prohibited by the first order.

Applications

Due to their general applicability variety of applications can be found, for example, in atomic physics, nuclear physics and solid state physics in the absorption and emission of photons, phonons or magnons for Fermi's Golden Rule.

Herleitungsskizze

The basic assumption is extended by a perturbation with a time-constant system exactly detachable Hamiltonian.

Fermi's Golden Rule applies to any constant or time-dependent Störoperatoren. It can represent the interaction with an outer ( constant or time- dependent ) field or an additional type of interaction between the particles of the system (eg the possibility of the generation of a photon ) was not taken into account. Here the derivation of a time-constant perturbation is shown.

For the completed Hamiltonian, the time-dependent Schrödinger equation must

Be solved. At the beginning (t = 0) the system is to be in an eigenstate. One develops the unknown function of the eigenfunctions of the unperturbed Hamiltonian (energy eigenvalues ​​) and thus has time-dependent coefficients:

The initial condition is, all the others.

After insertion of the Hamiltonian and wave function in the Schrödinger equation is obtained by comparing coefficients:

Where a shorthand notation for representing.

This equation describes how to change the coefficients in time. For the approximate solution is assumed that the coefficients vary continuously with respect to their initial values ​​, so that only the term n for small times of the sum = i is considered. (. This is the meaning of the first perturbative approximation) Since is constant, the equation can be integrated:

As a result, one obtains

It should be remembered that due to the derivation of this formula can only remain valid as long as valid. The probability at time t the system is in state f to find, is the absolute square

We consider a transition into the continuum, so that the final state f numerous neighboring states of similar structure, but has with continuously varying energy, which are also available as final states. The matrix element can therefore be assumed to be equal for all, the respective transition probability is due to the bracketed factor ( in the following graph with designated ) in the last formula but different. Depending on considered, this factor is a function with a sharp maximum altitude at. The neighboring zeros are included, the maximum between them can be well approximated by a triangle with the base line. The mean factor of the staple in this interval is, therefore, approximately half of the maximum.

The peak maximum of this function in shows that transitions preferably in states lead the same energy, but variations in a field of width are possible. With increasing time this range of variation is small. This is one of the forms of the uncertainty principle for energy and time and explained the spectral lines to be found in all natural line width. Also, as t increases, the peak higher, and the total area under the peak ( approximated by half the height times width ) grows in proportion to t. If W is the sum of the transition probabilities in all states in the region of the maximum, ie, W grows proportional to t, and is the required constant transfer rate. But it should be repeated that the whole derivation can only remain valid as long as remains.

To calculate W, the average transition probability is then simply multiplied the maximum to the number of final states in the interval. This number which results from the density of states in the energy axis

Multiplied by the width. It should be noted that not necessarily is the density of states of all possible states in the energy interval, as defined eg in solid state physics, but usually only a small fraction of that. Wearing it here with only those states which are also counted in the actual measurement of the transition probability, ie, for example, only those where particles fly in certain directions.

For the sum of all the individual transition probabilities W is thus obtained

Dividing by the time this results Fermi's Golden Rule for the transition rate: