Fermi's golden rule

Although the rule is named after Enrico Fermi, the first to obtain the formula was Paul Dirac,{{cite book |last1=Bransden |first1=B. H. |url=https://archive.org/details/bransden-joachain-quantum-mechanics/page/443 |title=Quantum Mechanics |last2=Joachain |first2=C. J. |publisher=Prentice Hall |year=1999 |isbn=978-0582356917 |edition=2nd |page=443}} as he had twenty years earlier formulated a virtually identical equation, including the three components of a constant, the matrix element of the perturbation and an energy difference.{{cite journal | last = Dirac | first = P. A. M. | author-link = 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 | doi-access = free }} See equations (24) and (32). It was given this name because, on account of its importance, Fermi called it "golden rule No. 2".{{cite book | last = Fermi | first = E. | title = Nuclear Physics | publisher = University of Chicago Press | year = 1950 |isbn= 978-0226243658 }} formula VIII.2

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.{{cite book | last = Fermi | first = E. | title = Nuclear Physics | publisher = University of Chicago Press | year = 1950 |isbn= 978-0226243658 }} formula VIII.19

Fermi's golden rule describes a system that begins in an eigenstate $|i\backslash rangle$ of an unperturbed Hamiltonian {{math|H₀}} and considers the effect of a perturbing Hamiltonian {{mvar|H'}} applied to 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 $|i\backslash rangle$ to a set of final states $|f\backslash rangle$ is essentially constant. It is given, to first-order approximation, by

$$\backslash Gamma\_\{i\; \backslash to\; f\}\; =\; \backslash frac\{2\; \backslash pi\}\{\backslash hbar\}\; \backslash left|\; \backslash langle\; f|H\text{'}|i\; \backslash rangle\; \backslash right|^2\; \backslash rho(E\_f),$$

where $\backslash langle\; f|H\text{'}|i\; \backslash rangle$ is the matrix element (in bra–ket notation) of the perturbation {{mvar|H'}} between the final and initial states, and $\backslash rho(E\_f)$ 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 $|i\backslash rangle$ is proportional to $e^\{-\backslash Gamma\_\{i\; \backslash to\; f\}\; t\}$.

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.[http://www.ph.utexas.edu/~schwitte/PHY362L/QMnote.pdf 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, see Wolfgang Pauli, Wave Mechanics: Volume 5 of Pauli Lectures on Physics (Dover Books on Physics, 2000) {{ISBN|0486414620}}, pp. 150–151.

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! Derivation in time-dependent perturbation theory

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| {{main|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. The total Hamiltonian is the sum of an “original” Hamiltonian {{math|H₀}} and a perturbation: $H\; =\; H\_0\; +\; H\text{'}(t)$. In the interaction picture, we can expand an arbitrary quantum state’s time evolution in terms of energy eigenstates of the unperturbed system $|n\backslash rang$, with $H\_0\; |n\backslash rang\; =\; E\_n\; |n\backslash rang$.

We first consider the case where the final states are discrete. The expansion of a state in the perturbed system at a time {{mvar|t}} is $|\backslash psi(t)\backslash rang\; =\; \backslash sum\_n\; a\_n(t)\; e^\{-\; i\; E\_n\; t\; /\; \backslash hbar\}\; |n\backslash rang$. The coefficients {{math|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:

$$H\; |\backslash psi(t)\backslash rang\; =\; i\backslash hbar\; \backslash frac\{\backslash partial\}\{\backslash partial\; t\}\; |\backslash psi(t)\backslash rang.$$

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 {{math|E_n}} and {{math|{{ket|n}}}} are the stationary eigenvalues and eigenfunctions of {{math|H₀}}.

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 {{mvar|H'}} that turns on at {{math|1=t = 0}}, we can use perturbation theory. Namely, if $H\text{'}\; =\; 0$, it is evident that $a\_n(t)\; =\; \backslash delta\_\{n,i\}$, which simply says that the system stays in the initial state $i$.

For states $k\; \backslash ne\; i$, $a\_k(t)$ becomes non-zero due to $H\text{'}\; \backslash ne\; 0$, 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 $H\text{'}$. Hence, one can plug in the zeroth-order form $a\_n(t)\; =\; \backslash delta\_\{n,i\}$ into the above equation to get the first correction for the amplitudes $a\_k(t)$:

\mathrm{i}\hbar \frac{da_k(t)}{dt} = \langle k|H'|i\rangle e^{\mathrm{i}t(E_k - E_i)/\hbar},

whose integral can be expressed as

\mathrm{i}\hbar a_k(t) = \int_0^t\langle k|H'(t')|i\rangle e^{\mathrm{i}\omega_{ki} t'} dt'

with $\backslash omega\_\{ki\}\; \backslash equiv\; (E\_k\; -\; E\_i)/\backslash hbar$, for a state with {{math|1=a_i(0) = 1}}, {{math|1=a_k(0) = 0}}, transitioning to a state with {{math|a_k(t)}}.

The probability of transition from the initial state (ith) to the final state (fth) is given by

$$w\_\{fi\}\; =\; |a\_f(t)|^2=\backslash frac\{1\}\{\backslash hbar^2\}\; \backslash left|\backslash int\_0^t\; \backslash langle\; f|H\text{'}(t\text{'})|i\backslash rangle\; e^\{\backslash mathrm\{i\}\backslash omega\_\{fi\}\; t\text{'}\}\; dt\text{'}\backslash right|^2$$

It is important to study a periodic perturbation with a given frequency $\backslash omega$ since arbitrary perturbations can be constructed from periodic perturbations of different frequencies. Since $H\text{'}(t)$ must be Hermitian, we must assume $H\text{'}(t)\; =\; Fe^\{-\backslash mathrm\{i\}\backslash omega\; t\}\; +\; F^\backslash dagger\; e^\{\backslash mathrm\{i\}\backslash omega\; t\}$, where $F$ is a time independent operator. The solution for this case isLandau, L. D., & Lifshitz, E. M. (2013). Quantum mechanics: non-relativistic theory (Vol. 3). Elsevier.

$$a\_f(t)\; =\; -\; \backslash langle\; f|F|i\backslash rangle\; \backslash frac\{e^\{\backslash mathrm\{i\}(\backslash omega\_\{fi\}-\backslash omega)\; t\}\}\{\backslash hbar\; (\backslash omega\_\{fi\}-\backslash omega)\}\; -\; \backslash langle\; f|F^\backslash dagger|i\backslash rangle\; \backslash frac\{e^\{\backslash mathrm\{i\}(\backslash omega\_\{fi\}+\backslash omega)\; t\}\}\{\backslash hbar\; (\backslash omega\_\{fi\}+\backslash omega)\}.$$

This expression is valid only when the denominators in the above expression is non-zero, i.e., for a given initial state with energy $E\_i$, the final state energy must be such that $E\_f-E\_i\; \backslash neq\; \backslash pm\; \backslash hbar\; \backslash omega.$ Not only the denominators must be non-zero, but also must not be small since $a\_f$ is supposed to be small.

Consider now the case where the perturbation frequency is such that $E\_k-E\_n=\backslash hbar(\backslash omega+\backslash varepsilon)$ where $\backslash varepsilon$ is a small quantity. Unlike the previous case, not all terms in the sum over $n$ in the above exact equation for $a\_k(t)$ matters, but depends only on $a\_n(t)$ and vice versa. Thus, omitting all other terms, we can write

$$i\backslash hbar\backslash frac\{da\_k\}\{dt\}=\; \backslash langle\; k|F|n\backslash rangle\; e^\{i\backslash varepsilon\; t\}a\_n,\; \backslash quad\; i\backslash hbar\backslash frac\{da\_n\}\{dt\}=\; \backslash langle\; n|F^\backslash dagger|k\backslash rangle\; e^\{-i\backslash varepsilon\; t\}a\_k.$$

The two independent solutions are

$$a\_n\; =\; Ae^\{\; i\backslash alpha\_1\; t\},\backslash ,\; a\_k=-A\backslash hbar\; \backslash alpha\_1\; e^\{\; i\backslash alpha\_1\; t\}/\backslash langle\; n|F^\backslash dagger|k\backslash rangle$$

$$a\_n\; =\; Be^\{-i\backslash alpha\_2\; t\},\backslash ,\; a\_k=\; B\backslash hbar\; \backslash alpha\_2\; e^\{-i\backslash alpha\_2\; t\}/\backslash langle\; n|F^\backslash dagger|k\backslash rangle$$

where

$$\backslash alpha\_1=-\backslash frac\{1\}\{2\}\backslash varepsilon+\backslash Omega,\; \backslash quad\; \backslash alpha\_2\; =\; \backslash frac\{1\}\{2\}\backslash varepsilon\; +\; \backslash Omega,\; \backslash quad\; \backslash Omega\; =\; \backslash sqrt\{\backslash frac\{1\}\{4\}\backslash varepsilon^2\; +\; |\backslash eta|^2\},\; \backslash quad\; \backslash eta\; =\; \backslash frac\{1\}\{\backslash hbar\}\backslash langle\; k|F|n\backslash rangle$$

and the constants $A$ and $B$ are fixed by the normalization condition.

If the system at $t=0$ is in the $|\backslash psi\_k\backslash rang$ state, then the probability of finding the system in the $|\backslash psi\_n\backslash rang$ state is given by

$$w\_\{kn\}=\backslash frac\{|\backslash eta|^2\}\{2\backslash Omega^2\}(1-\backslash cos\; 2\backslash Omega\; t)$$

which is a periodic function with frequency $2\backslash Omega$; this function varies between $0$ and $|\backslash eta|^2/\backslash Omega^2$. At the exact resonance, i.e., $\backslash varepsilon=0$, the above formula reduces to

$$w\_\{kn\}=\backslash frac\{1\}\{2\}(1-\backslash cos\; 2|\backslash eta|\; t)$$

which varies periodically between $0$ and $1$, that is to say, the system periodically switches from one state to the other. The situation is different if the final states are in the continuous spectrum.

Since the continuous spectrum lies above the discrete spectrum, $E\_f-E\_i>0$ and it is clear from the previous section, major role is played by the energies $E\_f$ lying near the resonance energy $E\_i+\backslash hbar\backslash omega$, i.e., when $\backslash omega\_\{fi\}\; \backslash approx\; \backslash omega$. In this case, it is sufficient to keep only the first term for $a\_f(t)$. Assuming that perturbations are turned on from time $t=0$, we have then

$$a\_f(t)\; =\; -\backslash frac\{\backslash mathrm\{i\}\}\{\backslash hbar\}\backslash int\_0^t\backslash langle\; f|H\text{'}(t\text{'})|i\backslash rangle\; e^\{\backslash mathrm\{i\}\backslash omega\_\{fi\}\; t\text{'}\}\; dt\text{'}\; =\; -\; \backslash langle\; f|F|i\backslash rangle\; \backslash frac\{e^\{\backslash mathrm\{i\}(\backslash omega\_\{fi\}-\backslash omega)\; t\}-1\}\{\backslash hbar\; (\backslash omega\_\{fi\}-\backslash omega)\}$$

The squared modulus of $a\_f$ is

$$|a\_f|^2=\; 4\; |\; \backslash langle\; f|F|i\backslash rangle|^2\; \backslash frac\{\backslash sin^2((\backslash omega\_\{fi\}-\backslash omega)t/2)\}\{\backslash hbar^2(\backslash omega\_\{fi\}-\backslash omega)^2\}$$

Therefore, the transition probability per unit time, for large t, is given by

$$\backslash frac\{dw\_\{fi\}\}\{dt\}\; =\; \backslash frac\{d\}\{dt\}|a\_f|^2\; =\; \backslash frac\{2\backslash pi\}\{\backslash hbar\}|\; \backslash langle\; f|F|i\backslash rangle|^2\; \backslash delta(E\_f\; -\; E\_i\; -\; \backslash hbar\backslash omega)$$

Note that the delta function in the expression above arises due to the following argument. Defining $\backslash Delta\; =\; \backslash omega\_\{fi\}\; -\; \backslash omega$ the time derivative of $\backslash sin^2(\backslash Delta\; t/2)\; /\; \backslash Delta^2$ is $\backslash sin(\backslash Delta\; t)\; /\; (2\backslash Delta)$, which behaves like a delta function at large {{mvar|t}} (for more information, please see Sinc function#Relationship to the Dirac delta distribution).

The constant decay rate of the golden rule follows.{{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 |author-link=Eugen Merzbacher}} As a constant, it underlies the exponential particle decay laws of radioactivity. (For excessively long times, however, the secular growth of the {{math|a_k(t)}} terms invalidates lowest-order perturbation theory, which requires {{math|a_k ≪ a_i}}.)

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Only the magnitude of the matrix element $\backslash langle\; f|H\text{'}|i\; \backslash rangle$ 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.{{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 |s2cid=119476624 }}

While the Golden rule is commonly stated and derived in the terms above, the final state (continuum) wave function is often rather vaguely described, and not normalized correctly (and the normalisation is used in the derivation). The problem is that in order to produce a continuum there can be no spatial confinement (which would necessarily discretise the spectrum), and therefore the continuum wave functions must have infinite extent, and in turn this means that the normalisation $\backslash langle\; f|f\; \backslash rangle\; =\; \backslash int\; d^3\backslash mathbf\{r\}\; \backslash left|f(\backslash mathbf\{r\})\backslash right|^2$ is infinite, not unity. If the interactions depend on the energy of the continuum state, but not any other quantum numbers, it is usual to normalise continuum wave-functions with energy $\backslash varepsilon$ labelled $|\; \backslash varepsilon\backslash rangle$, by writing $\backslash langle\; \backslash varepsilon|\backslash varepsilon\; \text{'}\; \backslash rangle=\backslash delta(\backslash varepsilon-\backslash varepsilon\; \text{'})$ where $\backslash delta$ is the Dirac delta function, and effectively a factor of the square-root of the density of states is included into $|\backslash varepsilon\_i\backslash rangle$.{{Cite book| title=Quantum Mechanics Vol II Chapter XIII Complement D_{XIII}|last1=Cohen-Tannoudji|first1=Claude| last2=Diu|first2=Bernard| last3=Laloë|first3=Franck| publisher=Wiley|year=1977|isbn=978-0471164333|author-link=Claude Cohen-Tannoudji}} In this case, the continuum wave function has dimensions of $1/\backslash sqrt\{\backslash text\{[energy]\}\}$, and the Golden Rule is now

$$\backslash Gamma\_\{i\; \backslash to\; \backslash varepsilon\_i\}\; =\; \backslash frac\{2\backslash pi\}\{\backslash hbar\}\; |\backslash langle\; \backslash varepsilon\_i|H\text{'}|i\backslash rangle|^2\; .$$

where $\backslash varepsilon\_i$ refers to the continuum state with the same energy as the discrete state $i$. For example, correctly normalized continuum wave functions for the case of a free electron in the vicinity of a hydrogen atom are available in Bethe and Salpeter.{{cite book |author1-last=Bethe | author1-first = Hans | author2-last = Salpeter | author2-first = Edwin |year=1977 |title=Quantum Mechanics of One- and Two-Electron Atoms |publisher=Springer, Boston, MA |isbn=978-0-306-20022-9 |author1-link=Hans Bethe}}

{{math proof

| title = Normalized Derivation in time-dependent perturbation theory

| proof = {{main|Perturbation theory (quantum mechanics)#Time-dependent perturbation theory}}

The following paraphrases the treatment of Cohen-Tannoudji. As before, the total Hamiltonian is the sum of an “original” Hamiltonian {{math|H₀}} and a perturbation: $H\; =\; H\_0\; +\; H\text{'}$. We can still expand an arbitrary quantum state’s time evolution in terms of energy eigenstates of the unperturbed system, but these now consist of discrete states and continuum states. We assume that the interactions depend on the energy of the continuum state, but not any other quantum numbers. The expansion in the relevant states in the Dirac picture is

$$|\backslash psi(t)\backslash rang\; =\; a\_i\; e^\{-\backslash mathrm\{i\}\backslash omega\_i\; t\}|i\backslash rang\; +\; \backslash int\_C\; d\backslash varepsilon\; a\_\backslash varepsilon\; e^\{-\backslash mathrm\{i\}\backslash omega\; t\}\; |\backslash varepsilon\backslash rangle,$$

where $\backslash omega\_i\; =\; \backslash varepsilon\_i\; /\; \backslash hbar$, $\backslash omega\; =\; \backslash varepsilon\; /\; \backslash hbar$ and $\backslash varepsilon\_i,\backslash varepsilon$ are the energies of states $|i\backslash rangle,\; |\backslash varepsilon\backslash rangle$, respectively. The integral is over the continuum $\backslash varepsilon\; \backslash in\; C$, i.e. $|\backslash varepsilon\backslash rangle$ is in the continuum.

Substituting into the time-dependent Schrödinger equation

$$H\; |\backslash psi(t)\backslash rang\; =\; \backslash mathrm\{i\}\backslash hbar\; \backslash frac\{\backslash partial\}\{\backslash partial\; t\}\; |\backslash psi(t)\backslash rang$$

and premultiplying by $\backslash langle\; i|$ produces

$$\backslash frac\{da\_i(t)\}\{dt\}\; =\; -\backslash mathrm\{i\}\; \backslash int\_C\; d\backslash varepsilon\; \backslash Omega\_\{i\backslash varepsilon\}\; e^\{-\backslash mathrm\{i\}(\backslash omega\; -\; \backslash omega\_i)t\}a\_\backslash varepsilon(t),$$

where $\backslash Omega\_\{i\; \backslash varepsilon\; \}=\backslash langle\; i|H\text{'}|\backslash varepsilon\backslash rangle/\backslash hbar$, and premultiplying by $\backslash langle\; \backslash varepsilon\; \text{'}|$ produces

$$\backslash frac\{da\_\backslash varepsilon(t)\}\{dt\}\; =\; -\backslash mathrm\{i\}\; \backslash Omega\_\{\backslash varepsilon\; i\}\; e^\{\backslash mathrm\{i\}(\backslash omega\; -\; \backslash omega\_i)t\}\; a\_i(t).$$

We made use of the normalisation $\backslash langle\; \backslash varepsilon\text{'}\; |\backslash varepsilon\backslash rangle\; =\; \backslash delta(\backslash varepsilon\text{'}-\backslash varepsilon)$.

Integrating the latter and substituting into the former,

\frac{da_i(t)}{dt} = - \int_C d\varepsilon \Omega_{i\varepsilon}\Omega_{\varepsilon i} \int_0^t dt' e^{-\mathrm{i}(\omega - \omega_i)(t-t')} a_i(t').

It can be seen here that $da\_i/dt$ at time $t$ depends on $a\_i$ at all earlier times $t\text{'}$, i.e. it is non-Markovian. We make the Markov approximation, i.e. that it only depends on $a\_i$ at time $t$ (which is less restrictive than the approximation that $a\_i\; \backslash approx\; 1$ used above, and allows the perturbation to be strong)

\frac{da_i(t)}{dt} = \int_C d\varepsilon |\Omega_{i\varepsilon}|^2 a_i(t) \int_0^t dT e^{-\mathrm{i}\Delta T},

where $T=t-t\text{'}$ and $\backslash Delta=\backslash omega\; -\backslash omega\_i$. Integrating over $T$,

\frac{da_i(t)}{dt} = - 2\pi\hbar \int_C d\varepsilon |\Omega_{i\varepsilon}|^2 a_i(t) \frac{ e^{-\mathrm{i}\Delta t/2}\sin(\Delta t/2)}{\pi\hbar\Delta} ,

The fraction on the right is a nascent Dirac delta function, meaning it tends to $\backslash delta(\backslash varepsilon-\backslash varepsilon\_i)$ as $t\backslash to\backslash infty$ (ignoring its imaginary part which leads to a very small energy (Lamb) shift, while the real part produces decay ). Finally

$$\backslash frac\{da\_i(t)\}\{dt\}\; =\; -\; 2\backslash pi\backslash hbar\; |\backslash Omega\_\{i\backslash varepsilon\_i\}|^2\; a\_i(t),$$

which can have solutions:

$a\_i(t)\; =\; \backslash exp(-\backslash Gamma\_\{i\backslash to\; \backslash varepsilon\_i\}\; t/2)$, i.e., the decay of population in the initial discrete state is

$P\_i(t)\; =\; |a\_i(t)|^2\; =\; \backslash exp(-\backslash Gamma\_\{i\backslash to\; \backslash varepsilon\_i\}\; t)$

where

$$\backslash Gamma\_\{i\backslash to\; \backslash varepsilon\_i\}\; =\; 2\backslash pi\backslash hbar\; |\backslash Omega\_\{i\backslash varepsilon\_i\}|^2\; =\; \backslash frac\{2\backslash pi\}\{\backslash hbar\}\; |\backslash langle\; i|H\text{'}|\backslash varepsilon\backslash rangle|^2.$$

}}

The Fermi's 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.{{cite book |last1=Yu |first1=Peter Y. |last2=Cardona |first2=Manuel |date=2010 |title=Fundamentals of Semiconductors - Physics and Materials Properties |edition=4 |publisher=Springer |page=260 |isbn=978-3-642-00709-5 |doi=10.1007/978-3-642-00710-1 }}

Consider a photon of frequency $\backslash omega$ and wavevector $\backslash textbf\{q\}$, where the light dispersion relation is $\backslash omega\; =\; (c/n)\backslash left|\backslash textbf\{q\}\backslash right|$ and $n$ is the index of refraction.

Using the Coulomb gauge where $\backslash nabla\backslash cdot\; \backslash textbf\{A\}=0$ and $V=0$, the vector potential of light is given by $\backslash textbf\{A\}\; =\; A\_0\backslash boldsymbol\{\backslash varepsilon\}e^\{\backslash mathrm\{i\}(\backslash textbf\{q\}\backslash cdot\backslash textbf\{r\}-\backslash omega\; t)\}\; +C$ where the resulting electric field is

$$\backslash textbf\{E\}\; =\; -\backslash frac\{\backslash partial\backslash textbf\{A\}\}\{\backslash partial\; t\}\; =\; \backslash mathrm\{i\}\; \backslash omega\; A\_0\; \backslash boldsymbol\{\backslash varepsilon\}\; e^\{\backslash mathrm\{i\}.(\backslash textbf\{q\}\backslash cdot\backslash textbf\{r\}-\backslash omega\; t)\}.$$

For an electron in the valence band, the Hamiltonian is

$$H\; =\; \backslash frac\{(\backslash textbf\{p\}\; +e\backslash textbf\{A\})^2\}\{2m\_0\}\; +\; V(\backslash textbf\{r\}),$$

where $V(\backslash textbf\{r\})$ is the potential of the crystal, $e$ and $m\_0$ are the charge and mass of an electron, and $\backslash textbf\{p\}$ is the momentum operator. Here we consider process involving one photon and first order in $\backslash textbf\{A\}$. The resulting Hamiltonian is

$$H\; =\; H\_0\; +\; H\text{'}\; =\; \backslash left[\; \backslash frac\{\backslash textbf\{p\}^2\}\{2m\_0\}\; +\; V(\backslash textbf\{r\})\; \backslash right]\; +$$

\left[ \frac{e}{2m_0}(\textbf{p}\cdot \textbf{A} + \textbf{A}\cdot \textbf{p}) \right],

where $H\text{'}$ is the perturbation of light.

From here on we consider vertical optical dipole transition, and thus have transition probability based on time-dependent perturbation theory that

$$\backslash Gamma\_\{if\}\; =\; \backslash frac\{2\backslash pi\}\{\backslash hbar\}\; \backslash left|\backslash langle\; f|H\text{'}|i\backslash rangle\; \backslash right|^2\backslash delta\; (E\_f-E\_i\; \backslash pm\; \backslash hbar\; \backslash omega),$$

with $$H\text{'}\; \backslash approx\; \backslash frac\{eA\_0\}\{m\_0\}\backslash boldsymbol\{\backslash varepsilon\}\backslash cdot\; \backslash mathbf\{p\},$$

where $\backslash boldsymbol\{\backslash varepsilon\}$ is the light polarization vector. $|i\backslash rangle$ and $|f\backslash rangle$ are the Bloch wavefunction of the initial and final states. Here the transition probability needs to satisfy the energy

conservation given by $\backslash delta\; (E\_f-E\_i\; \backslash pm\; \backslash hbar\; \backslash omega)$. From perturbation it is evident that the heart of the calculation lies in the matrix elements shown in the bracket.

For the initial and final states in valence and conduction bands, we have $|i\backslash rangle\; =\backslash Psi\_\{v,\backslash textbf\{k\}\_i,s\_i\}(\backslash textbf\{r\})$ and $|f\backslash rangle\; =\backslash Psi\_\{c,\backslash textbf\{k\}\_f,s\_f\}(\backslash textbf\{r\})$, respectively and if the $H\text{'}$ operator does not act on the spin, the electron stays in the same spin state and hence we can write the Bloch wavefunction of the initial and final states as

$$\backslash Psi\_\{v,\backslash textbf\{k\}\_i\}(\backslash textbf\{r\})=$$

\frac{1}{\sqrt{N\Omega_0}}u_{n_v,\textbf{k}_i}(\textbf{r})e^{i\textbf{k}_i\cdot\textbf{r}},

$$\backslash Psi\_\{c,\backslash textbf\{k\}\_f\}(\backslash textbf\{r\})=$$

\frac{1}{\sqrt{N\Omega_0}}u_{n_c,\textbf{k}_f}(\textbf{r})e^{i\textbf{k}_f\cdot\textbf{r}},

where $N$ is the number of unit cells with volume $\backslash Omega\_0$. Calculating using these wavefunctions, and focusing on emission (photoluminescence) rather than absorption, we are led to the transition rate

\Gamma_{cv}=\frac{2\pi}{\hbar}\left(\frac{eA_0}{m_0}\right)^2

|\boldsymbol{\varepsilon} \cdot \boldsymbol{\mu}_{cv}(\textbf{k})|^2 \delta (E_c - E_v - \hbar \omega),

where $\backslash boldsymbol\{\backslash mu\}\_\{cv\}$ defined as the optical transition dipole moment is qualitatively the expectation value $\backslash langle\; c|\; (\backslash text\{charge\})\; \backslash times\; (\backslash text\{distance\})|v\backslash rangle$ and in this situation takes the form

$$\backslash boldsymbol\{\backslash mu\}\_\{cv\}\; =$$

-\frac{i\hbar}{\Omega_0} \int_{\Omega_0} d\textbf{r}' u^*_{n_c,\textbf{k}}(\textbf{r}')

\nabla u_{n_v,\textbf{k}}(\textbf{r}').

Finally, we want to know the total transition rate $\backslash Gamma(\backslash omega)$. Hence we need to sum over all possible initial and final states that can satisfy the energy conservation (i.e. an integral of the Brillouin zone in the k-space), and take into account spin degeneracy, which after calculation results in

\Gamma(\omega) = \frac{4\pi}{\hbar}\left( \frac{eA_0}{m_0} \right)^2

|\boldsymbol{\varepsilon}\cdot \boldsymbol{\mu}_{cv}|^2 \rho_{cv}(\omega)

where $\backslash rho\_\{cv\}(\backslash omega)$ is the joint valence-conduction density of states (i.e. the density of pair of states; one occupied valence state, one empty conduction state). In 3D, this is

$$\backslash rho\_\{cv\}(\backslash omega)\; =\; 2\backslash pi\; \backslash left(\; \backslash frac\{2m^*\}\{\backslash hbar^2\}\backslash right)^\{3/2\}\backslash sqrt\{\backslash hbar\; \backslash omega\; -\; E\_g\},$$

but the joint DOS is different for 2D, 1D, and 0D.

We note that in a general way we can express the Fermi's golden rule for semiconductors as{{cite journal| last1=Edvinsson|first1=T.|title=Optical quantum confinement and photocatalytic properties in two-, one- and zero-dimensional nanostructures|journal=Royal Society Open Science|volume=5|issue=9|year=2018|pages=180387|issn=2054-5703| doi=10.1098/rsos.180387 | pmid=30839677|pmc=6170533|bibcode=2018RSOS....580387E}}

\Gamma_{vc}=

\frac{2\pi}{\hbar}\int_\text{BZ} \frac{d\textbf{k}}{4\pi^3}|H_{vc}'|^2

\delta(E_c(\textbf{k}) - E_v(\textbf{k}) - \hbar\omega).

In the same manner, the stationary DC photocurrent with amplitude proportional to the square of the field of light is

\textbf{J}=

-\frac{2\pi e \tau}{\hbar}\sum_{i,f}\int_\text{BZ} \frac{d\textbf{k}}{(2\pi)^D} |\textbf{v}_i-\textbf{v}_f|(f_i(\textbf{k})-f_f(\textbf{k}))|H_{if}'|^2

\delta(E_f(\textbf{k}) - E_i(\textbf{k}) - \hbar\omega),

where $\backslash tau$ is the relaxation time, $\backslash textbf\{v\}\_i-\backslash textbf\{v\}\_f$ and $f\_i(\backslash textbf\{k\})-f\_f(\backslash textbf\{k\})$ are the

difference of the group velocity and Fermi-Dirac distribution between possible the initial and

final states. Here $|H\_\{if\}\text{'}|^2$ defines the optical transition dipole. Due to the commutation relation between position $\backslash textbf\{r\}$ and the Hamiltonian, we can also rewrite the transition dipole and photocurrent in terms of position operator matrix using $\backslash langle\; i|\backslash textbf\{p\}|f\backslash rangle=\; -im\_0\backslash omega\backslash langle\; i|\backslash textbf\{r\}|f\backslash rangle$. This effect can only exist in systems with broken inversion symmetry and nonzero components of the photocurrent can be obtained by symmetry arguments.

{{main|Scanning tunneling microscope#Principle of operation}}

In a scanning tunneling microscope, the Fermi's golden rule is used in deriving the tunneling current. It takes the form

$$w\; =\; \backslash frac\{2\; \backslash pi\}\{\backslash hbar\}\; |M|^2\; \backslash delta\; (E\_\{\backslash psi\}\; -\; E\_\{\backslash chi\}\; ),$$

where $M$ is the tunneling matrix element.

When considering energy level transitions between two discrete states, Fermi's golden rule is written as

$$\backslash Gamma\_\{i\; \backslash to\; f\}\; =\; \backslash frac\{2\; \backslash pi\}\{\backslash hbar\}\; \backslash left|\backslash langle\; f|\; H\text{'}\; |i\; \backslash rangle\backslash right|^2\; g(\backslash hbar\backslash omega),$$

where $g(\backslash hbar\backslash omega)$ is the density of photon states at a given energy, $\backslash hbar\backslash omega$ is the photon energy, and $\backslash omega$ 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.{{cite book |last1=Fox |first1=Mark |title=Quantum Optics: An Introduction |date=2006 |publisher=Oxford University Press |location=Oxford |isbn=9780198566731 |page=51}}

File:Drexhage experiment.gif

Fermi's golden rule predicts that the probability that an excited state will decay depends on the density of states. This can be seen experimentally by measuring the decay rate of a dipole near a mirror: as the presence of the mirror creates regions of higher and lower density of states, the measured decay rate depends on the distance between the mirror and the dipole.{{cite journal |title=Variation of the Fluorescence Decay Time of a Molecule in Front of a Mirror |author1=K. H. Drexhage | author2 = H. Kuhn | author3 = F. P. Schäfer |journal=Berichte der Bunsengesellschaft für physikalische Chemie |volume=72 |pages=329 |year=1968 |issue=2 |doi=10.1002/bbpc.19680720261 |s2cid=94677437 }}{{cite journal |title=Influence of a dielectric interface on fluorescence decay time |author=K. H. Drexhage |journal=Journal of Luminescence |volume=1 |pages=693–701 |year=1970 |doi=10.1016/0022-2313(70)90082-7|bibcode=1970JLum....1..693D }}

• {{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}}
• Sargent's rule

{{Reflist|2}}

• [http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/fermi.html More information on Fermi's golden rule]
• [http://onlinelibrary.wiley.com/doi/10.1002/9783527665709.app6/pdf Derivation of Fermi’s Golden Rule]
• [http://www.tcm.phy.cam.ac.uk/~bds10/aqp/handout_dep.pdf Time-dependent perturbation theory]
• [https://arxiv.org/abs/1604.06916 Fermi's golden rule: its derivation and breakdown by an ideal model]

{{DEFAULTSORT:Fermi's Golden Rule}}

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