Fermi's golden rule

In quantum physics, Fermi's golden rule is a formula that describes the transition rate (the probability of a 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 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.

Historical background

Although the rule is named after Enrico Fermi, most of the work leading to it is due to Paul Dirac, who twenty years earlier had formulated 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 called it "golden rule No. 2".^[3]

Most uses of the term Fermi's golden rule are referring to "golden rule No. 2", but Fermi's "golden rule No. 1" is of a similar form and considers the probability of indirect transitions per unit time.^[4]

The rate and its derivation

Fermi's golden rule describes a system that begins in an eigenstate $|i\rangle$ of an unperturbed Hamiltonian $H 0$ 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 $|i\rangle$ to a set of final states $|f\rangle$ is essentially constant. It is given, to first-order approximation, by

\Gamma _{i\to f}={\frac {2\pi }{\hbar }}\left|\langle f|H'|i\rangle \right|^{2}\rho (E_{f}),

where

\langle f|H'|i\rangle

is the matrix element (in bra–ket notation) of the perturbation

H'

between the final and initial states, and

\rho (E_{f})

is the density of states (number of continuum states divided by

dE

in the infinitesimally small energy interval

E

to

E+dE

) at the energy

E_{f}

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

|i\rangle

is proportional to

e^{-\Gamma _{i\to f}t}

.

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.^[5]^[6]

Derivation in time-dependent perturbation theory

Statement of the problem

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 $H 0$ and a perturbation: $H=H_{0}+H'(t)$ . In the interaction picture, we can expand an arbitrary quantum state’s time evolution in terms of energy eigenstates of the unperturbed system $|n\rangle$ , with $H_{0}|n\rangle =E_{n}|n\rangle$ .

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 $t$ is ${\textstyle |\psi (t)\rangle =\sum _{n}a_{n}(t)e^{-iE_{n}t/\hbar }|n\rangle }$ . The coefficients $a n (t)$ are yet unknown functions of time yielding the probability amplitudes in the Dirac picture. This state obeys the time-dependent Schrödinger equation:

H|\psi (t)\rangle =i\hbar {\frac {\partial }{\partial t}}|\psi (t)\rangle .

Expanding the Hamiltonian and the state, we see that, to first order,

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

where

E n

and

| n ⟩

are the stationary eigenvalues and eigenfunctions of

H 0

.

This equation can be rewritten as a system of differential equations specifying the time evolution of the coefficients $a_{n}(t)$ :

\mathrm {i} \hbar {\frac {da_{k}(t)}{dt}}=\sum _{n}\langle k|H'|n\rangle a_{n}(t)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 $H'$ that 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. The coefficient $a_{i}(t)$ which is unity in the unperturbed state, will have a weak contribution from $H'$ . Hence, one can plug in the zeroth-order form $a_{n}(t)=\delta _{n,i}$

Navigácia: Veda >

Analytika
Antropológia
Aplikované vedy
Bibliometria
Dejiny vedy
Encyklopédie
Filozofia vedy
Forenzné vedy
Humanitné vedy
Knižničná veda
Kryogenika
Kryptológia
Kulturológia
Literárna veda
Medzidisciplinárne oblasti
Metódy kvantitatívnej analýzy
Metavedy
Metodika

Metodológia vedy
Náboženstvo a veda
Náučná literatúra
Podvody vo vede
Popularizácia vedy
Potravinárstvo
Prírodné vedy
Pseudoveda
Scientometria
Spoločenské vedy
Teórie
Teatrológia
Technické vedy
Technika
Terminológia
Umenie
Výskum

Veda
Veda a technika podľa štátu
Veda a technika podľa kontinentu
Veda a technika podľa roka
Veda v kozme
Vedci
Vedecká literatúra
Vedecké databázy
Vedecké experimenty
Vedecké konferencie
Vedecké metódy
Vedecké ocenenia
Vedecké organizácie
Vedecké parky
Vedeckí spisovatelia
Vzdelávanie
Záhady

Príbuzné výrazy:

Text je dostupný za podmienok Creative Commons Attribution/Share-Alike License 3.0 Unported; prípadne za ďalších podmienok.
Podrobnejšie informácie nájdete na stránke Podmienky použitia.

[1]

[2]

[3]

[4]

[5]

[6]