- General
- The rate and its derivation
- Use in quantum optics
- See also
- References
- External links
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 constant 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 number of states per unit energy in the continuum. The main assumption in Fermi's golden rule is that the initial state has not been significantly depleted by transitions into the final states.
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.[1][2] It was given this name because, on account of its importance, Fermi dubbed it "Golden Rule No. 2." [3]
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. [4]
The rate and its derivation
Fermi's golden rule describes a system which begins in an eigenstate, 
, of an unperturbed Hamiltonian, {{mvar| H}}0 and considers the effect of a perturbing Hamiltonian, {{mvar| H'}} applied the system. If {{mvar| H'}} is time-independent, the system goes only into those states in the continuum that have the same energy as the initial state. If {{mvar| H'}} is oscillating sinusoidally as a function of time (i.e. it is a harmonic perturbation) with an angular frequency {{mvar|ω}}, the transition is into states with energies that differ by {{math| 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 {{mvar| 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.
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 |
|---|
| 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 {{mvar|H'}} of the Hamiltonian, where {{math|En}} and {{math|{{ket|n}}}} are the stationary eigenvalues and eigenfunctions of {{mvar|H}}0 . Rewrite this equation as that of the time evolution of the coefficients This equation is exact but normally cannot be solved in practice. For a weak constant perturbation {{mvar|H'}} which turns on at {{mvar|t}}=0, we can use perturbation theory. Namely, if it is evident that For states weak perturbation. Hence, one can plug in the zeroth order form correction for the amplitudes which integrates to for
The transition rate is then a sinc function peaking sharply for small {{mvar|ω}}. At By dramatic contrast, for states of energy {{mvar|E}} embedded in a continuum, they must be all accounted for collectively. For a density of states per unit energy interval {{math|ρ(E)}}, they must be integrated over their energies, and whence the corresponding {{mvar|ω}}s, For large {{mvar|t}}, the sinc function is sharply peaked at {{mvar|ω}} ≈ 0, and negligible outside {{math|[−{{pi}}/t, {{pi}}/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, {{mvar|π}}. The time dependence has vanished, and the constant decay rate of the golden rule follows.[7] As a constant, it underlies the exponential particle decay laws of radioactivity. (For excessively long times, however, the secular growth of the {{math|ak(t)}}s invalidates lowest-order perturbation theory, which requires {{math|ak ≪ ai}} .) |
,
, which simply says that the system stays in the initial state 
.
, 
becomes non-zero due to 
and these are assumed to be small due to the
, for a state with {{math|ai(0)}} =1, {{math|ak(0)}}=0, transitioning to a state with {{math|ak(t)}} (again,
, 
, so the transition rate varies linearly with {{mvar|t}} for an isolated state 
!
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.[8]
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. [9]
See also
- {{annotated link|Exponential decay}}
- {{annotated link|List of things named after Enrico Fermi}}
- {{annotated link|Particle decay}}
- {{annotated link|Sinc function}}
- {{annotated link|Time-dependent perturbation theory}}
References
2. ^{{cite journal | last = Dirac | first = P.A.M. | authorlink = Paul Dirac | title = The Quantum Theory of Emission and Absorption of Radiation | journal = Proceedings of the Royal Society A | volume = 114 | pages = 243–265 | date=1 March 1927| issue = 767 | doi = 10.1098/rspa.1927.0039 | jstor=94746|bibcode = 1927RSPSA.114..243D }} See equations (24) and (32).
3. ^{{cite book | last = Fermi | first = E. | title = Nuclear Physics | publisher = University of Chicago Press | year = 1950 |isbn= 978-0226243658 }} formula VIII.2
4. ^{{cite book | last = Fermi | first = E. | title = Nuclear Physics | publisher = University of Chicago Press | year = 1950 |isbn= 978-0226243658 }} formula VIII.19
5. ^R Schwitters' UT Notes on Derivation
6. ^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.
7. ^{{cite book|author=Merzbacher, Eugen|year=1998|title=Quantum Mechanics|edition=3rd|publisher=Wiley, John & Sons, Inc.|isbn=978-0-471-88702-7|chapter=19.7|chapter-url=http://instrumentation.tamu.edu/~ting/other/QM_Merzbacher.pdf}}
8. ^{{cite journal|title=Coordinate Shift in Semiclassical Boltzmann Equation and Anomalous Hall Effect|author=N. A. Sinitsyn, Q. Niu and A. H. MacDonald|journal=Phys. Rev. B|volume=73|year=2006|pages=075318|arxiv=cond-mat/0511310|doi=10.1103/PhysRevB.73.075318|bibcode = 2006PhRvB..73g5318S|issue=7 }}
9. ^{{cite book |last1=Fox |first1=Mark |title=Quantum Optics: An Introduction |date=2006 |publisher=Oxford University Press |location=Oxford |isbn=9780198566731 |page=51}}
External links
- More information on Fermi's golden rule
- Derivation of Fermi’s Golden Rule
- Time-dependent perturbation theory
- [https://arxiv.org/abs/1604.06916 Fermi's golden rule: its derivation and breakdown by an ideal model]
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