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 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.
Contents
1 General
2 The rate and its derivation
3 Use in quantum optics
4 See also
5 References
6 External links
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, |i⟩irangle 
In both cases, the transition probability per unit of time from the initial state |i⟩irangle 
- Γi→f=2πℏ|⟨f|H′|i⟩|2ρ(Ef)^2rho (E_f)
- Γi→f=2πℏ|⟨f|H′|i⟩|2ρ(Ef)^2rho (E_f)
where ⟨f|H′|i⟩displaystyle langle f
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 | |
|---|---|
The golden rule is a straightforward consequence of the Schrödinger equation, solved to lowest order in the perturbation H' of the Hamiltonian,
where En and |n⟩ are the stationary eigenvalues and eigenfunctions of H0 . Rewrite this equation as that of the time evolution of the coefficients an(t)displaystyle a_n(t)
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 H′=0displaystyle H'=0 For states k≠idisplaystyle kneq i
which integrates to
for ω≡(Ek−Ei)/ℏdisplaystyle omega equiv (E_k-E_i)/hbar The transition rate is then
a sinc function peaking sharply for small ω. At ω=0displaystyle omega =0 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.[7] 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 ⟨f|H′|i⟩displaystyle langle f
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,
- Γi→f=2πℏ|⟨f|H′|i⟩|2g(ℏω)displaystyle Gamma _irightarrow f=frac 2pi hbar left
- Γi→f=2πℏ|⟨f|H′|i⟩|2g(ℏω)displaystyle Gamma _irightarrow f=frac 2pi hbar left
where g(E)displaystyle g(E)
See also
- Exponential decay
- List of things named after Enrico Fermi
- Particle decay
Sinc function – Special mathematical function defined as sin(x)/x- Time-dependent perturbation theory
References
^ Bransden, B. H.; Joachain, C. J. (1999). Quantum Mechanics (2nd ed.). p. 443. ISBN 978-0582356917..mw-parser-output cite.citationfont-style:inherit.mw-parser-output .citation qquotes:"""""""'""'".mw-parser-output .citation .cs1-lock-free abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .citation .cs1-lock-subscription abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registrationcolor:#555.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration spanborder-bottom:1px dotted;cursor:help.mw-parser-output .cs1-ws-icon abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png")no-repeat;background-position:right .1em center.mw-parser-output code.cs1-codecolor:inherit;background:inherit;border:inherit;padding:inherit.mw-parser-output .cs1-hidden-errordisplay:none;font-size:100%.mw-parser-output .cs1-visible-errorfont-size:100%.mw-parser-output .cs1-maintdisplay:none;color:#33aa33;margin-left:0.3em.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-formatfont-size:95%.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-leftpadding-left:0.2em.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-rightpadding-right:0.2em
^ Dirac, P.A.M. (1 March 1927). "The Quantum Theory of Emission and Absorption of Radiation". Proceedings of the Royal Society A. 114 (767): 243–265. Bibcode:1927RSPSA.114..243D. doi:10.1098/rspa.1927.0039. JSTOR 94746. See equations (24) and (32).
^ Fermi, E. (1950). Nuclear Physics. University of Chicago Press. ISBN 978-0226243658. formula VIII.2
^ Fermi, E. (1950). Nuclear Physics. University of Chicago Press. ISBN 978-0226243658. formula VIII.19
^ R Schwitters' UT Notes on Derivation
^ 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.
^ Merzbacher, Eugen (1998). "19.7" (PDF). Quantum Mechanics (3rd ed.). Wiley, John & Sons, Inc. ISBN 978-0-471-88702-7.
^ N. A. Sinitsyn, Q. Niu and A. H. MacDonald (2006). "Coordinate Shift in Semiclassical Boltzmann Equation and Anomalous Hall Effect". Phys. Rev. B. 73 (7): 075318. arXiv:cond-mat/0511310. Bibcode:2006PhRvB..73g5318S. doi:10.1103/PhysRevB.73.075318.
^ Fox, Mark (2006). Quantum Optics: An Introduction. Oxford: Oxford University Press. p. 51. ISBN 9780198566731.
External links
- More information on Fermi's golden rule
- Derivation of Fermi’s Golden Rule
- Time-dependent perturbation theory
- Fermi's golden rule: its derivation and breakdown by an ideal model
