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 ${\displaystyle |i\rangle }$ of an unperturbed Hamiltonian H₀ 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 ${\displaystyle |i\rangle }$ to a set of final states ${\displaystyle |f\rangle }$ is essentially constant. It is given, to first-order approximation, by

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

where ${\displaystyle \langle f|H'|i\rangle }$ is the matrix element (in bra–ket notation) of the perturbation H' between the final and initial states, and ${\displaystyle \rho (E_{f})}$ is the density of states (number of continuum states divided by ${\displaystyle dE}$ in the infinitesimally small energy interval ${\displaystyle E}$ to ${\displaystyle E+dE}$) at the energy ${\displaystyle E_{f}}$ 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 ${\displaystyle |i\rangle }$ is proportional to ${\displaystyle e^{-\Gamma _{i\to f}t}}$.

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

Main article: Perturbation theory (quantum mechanics) § 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 H0 and a perturbation: ${\displaystyle H=H_{0}+H'(t)}$. In the interaction picture, we can expand an arbitrary quantum state’s time evolution in terms of energy eigenstates of the unperturbed system ${\displaystyle |n\rangle }$, with ${\displaystyle H_{0}|n\rangle =E_{n}|n\rangle }$. 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 an(t) are yet unknown functions of time yielding the probability amplitudes in the Dirac picture. This state obeys the time-dependent Schrödinger equation: $${\displaystyle H|\psi (t)\rangle =i\hbar {\frac {\partial }{\partial t}}|\psi (t)\rangle .}$$ Expanding the Hamiltonian and the state, we see that, to first order, $${\displaystyle \left(H_{0}+H'-\mathrm {i} \hbar {\frac {\partial }{\partial t}}\right)\sum _{n}a_{n}(t)|n\rangle e^{-\mathrm {i} tE_{n}/\hbar }=0,}$$ where En and |n⟩ are the stationary eigenvalues and eigenfunctions of H0. This equation can be rewritten as a system of differential equations specifying the time evolution of the coefficients ${\displaystyle a_{n}(t)}$: $${\displaystyle \mathrm {i} \hbar {\frac {da_{k}(t)}{dt}}=\sum _{n}\langle k|H'|n\rangle a_{n}(t)e^{\mathrm {i} t(E_{k}-E_{n})/\hbar }.}$$ 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 ${\displaystyle H'=0}$, it is evident that ${\displaystyle a_{n}(t)=\delta _{n,i}}$, which simply says that the system stays in the initial state ${\displaystyle i}$. For states ${\displaystyle k\neq i}$, ${\displaystyle a_{k}(t)}$ becomes non-zero due to ${\displaystyle H'\neq 0}$, and these are assumed to be small due to the weak perturbation. The coefficient ${\displaystyle a_{i}(t)}$ which is unity in the unperturbed state, will have a weak contribution from ${\displaystyle H'}$. Hence, one can plug in the zeroth-order form ${\displaystyle a_{n}(t)=\delta _{n,i}}$ into the above equation to get the first correction for the amplitudes ${\displaystyle a_{k}(t)}$: $${\displaystyle \mathrm {i} \hbar {\frac {da_{k}(t)}{dt}}=\langle k|H'|i\rangle e^{\mathrm {i} t(E_{k}-E_{i})/\hbar },}$$ whose integral can be expressed as $${\displaystyle \mathrm {i} \hbar a_{k}(t)=\int _{0}^{t}\langle k|H'(t')|i\rangle e^{\mathrm {i} \omega _{ki}t'}dt'}$$ with ${\displaystyle \omega _{ki}\equiv (E_{k}-E_{i})/\hbar }$, for a state with ai(0) = 1, ak(0) = 0, transitioning to a state with ak(t). The probability of transition from the initial state (ith) to the final state (fth) is given by $${\displaystyle w_{fi}=|a_{f}(t)|^{2}={\frac {1}{\hbar ^{2}}}\left|\int _{0}^{t}\langle f|H'(t')|i\rangle e^{\mathrm {i} \omega _{fi}t'}dt'\right|^{2}}$$ It is important to study a periodic perturbation with a given frequency ${\displaystyle \omega }$ since arbitrary perturbations can be constructed from periodic perturbations of different frequencies. Since ${\displaystyle H'(t)}$ must be Hermitian, we must assume ${\displaystyle H'(t)=Fe^{-\mathrm {i} \omega t}+F^{\dagger }e^{\mathrm {i} \omega t}}$, where ${\displaystyle F}$ is a time independent operator. The solution for this case is [7] $${\displaystyle a_{f}(t)=-\langle f|F|i\rangle {\frac {e^{\mathrm {i} (\omega _{fi}-\omega )t}}{\hbar (\omega _{fi}-\omega )}}-\langle f|F^{\dagger }|i\rangle {\frac {e^{\mathrm {i} (\omega _{fi}+\omega )t}}{\hbar (\omega _{fi}+\omega )}}.}$$ This expression is valid only when the denominators in the above expression is non-zero, i.e., for a given initial state with energy ${\displaystyle E_{i}}$, the final state energy must be such that ${\displaystyle E_{f}-E_{i}\neq \pm \hbar \omega .}$ Not only the denominators must be non-zero, but also must not be small since ${\displaystyle a_{f}}$ is supposed to be small. Consider now the case where the perturbation frequency is such that ${\displaystyle E_{k}-E_{n}=\hbar (\omega +\varepsilon )}$ where ${\displaystyle \varepsilon }$ is a small quantity. Unlike the previous case, not all terms in the sum over ${\displaystyle n}$ in the above exact equation for ${\displaystyle a_{k}(t)}$ matters, but depends only on ${\displaystyle a_{n}(t)}$ and vice versa. Thus, omitting all other terms, we can write $${\displaystyle i\hbar {\frac {da_{k}}{dt}}=\langle k|F|n\rangle e^{i\varepsilon t}a_{n},\quad i\hbar {\frac {da_{n}}{dt}}=\langle n|F^{\dagger }|k\rangle e^{-i\varepsilon t}a_{k}.}$$ The two independent solutions are $${\displaystyle a_{n}=Ae^{i\alpha _{1}t},\,a_{k}=-A\hbar \alpha _{1}e^{i\alpha _{1}t}/\langle n|F^{\dagger }|k\rangle }$$ $${\displaystyle a_{n}=Be^{-i\alpha _{2}t},\,a_{k}=B\hbar \alpha _{2}e^{-i\alpha _{2}t}/\langle n|F^{\dagger }|k\rangle }$$ where $${\displaystyle \alpha _{1}=-{\frac {1}{2}}\varepsilon +\Omega ,\quad \alpha _{2}={\frac {1}{2}}\varepsilon +\Omega ,\quad \Omega ={\sqrt {{\frac {1}{4}}\varepsilon ^{2}+|\eta |^{2}}},\quad \eta ={\frac {1}{\hbar }}\langle k|F|n\rangle }$$ and the constants ${\displaystyle A}$ and ${\displaystyle B}$ are fixed by the normalization condition. If the system at ${\displaystyle t=0}$ is in the ${\displaystyle |\psi _{k}\rangle }$ state, then the probability of finding the system in the ${\displaystyle |\psi _{n}\rangle }$ state is given by $${\displaystyle w_{kn}={\frac {|\eta |^{2}}{2\Omega ^{2}}}(1-\cos 2\Omega t)}$$ which is a periodic function with frequency ${\displaystyle 2\Omega }$; this function varies between ${\displaystyle 0}$ and ${\displaystyle |\eta |^{2}/\Omega ^{2}}$. At the exact resonance, i.e., ${\displaystyle \varepsilon =0}$, the above formula reduces to $${\displaystyle w_{kn}={\frac {1}{2}}(1-\cos 2|\eta |t)}$$ which varies periodically between ${\displaystyle 0}$ and ${\displaystyle 1}$, 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. Continuous spectrum of final states Since the continuous spectrum lies above the discrete spectrum, ${\displaystyle E_{f}-E_{i}>0}$ and it is clear from the previous section, major role is played by the energies ${\displaystyle E_{f}}$ lying near the resonance energy ${\displaystyle E_{i}+\hbar \omega }$, i.e., when ${\displaystyle \omega _{fi}\approx \omega }$. In this case, it is sufficient to keep only the first term for ${\displaystyle a_{f}(t)}$. Assuming that perturbations are turned on from time ${\displaystyle t=0}$, we have then $${\displaystyle a_{f}(t)=-{\frac {\mathrm {i} }{\hbar }}\int _{0}^{t}\langle f|H'(t')|i\rangle e^{\mathrm {i} \omega _{fi}t'}dt'=-\langle f|F|i\rangle {\frac {e^{\mathrm {i} (\omega _{fi}-\omega )t}-1}{\hbar (\omega _{fi}-\omega )}}}$$ The squared modulus of ${\displaystyle a_{f}}$ is $${\displaystyle |a_{f}|^{2}=4|\langle f|F|i\rangle |^{2}{\frac {\sin ^{2}((\omega _{fi}-\omega )t/2)}{\hbar ^{2}(\omega _{fi}-\omega )^{2}}}}$$ Therefore, the transition probability per unit time, for large t, is given by $${\displaystyle {\frac {dw_{fi}}{dt}}={\frac {d}{dt}}|a_{f}|^{2}={\frac {2\pi }{\hbar }}|\langle f|F|i\rangle |^{2}\delta (E_{f}-E_{i}-\hbar \omega )}$$ Note that the delta function in the expression above arises due to the following argument. Defining ${\displaystyle \Delta =\omega _{fi}-\omega }$ the time derivative of ${\displaystyle \sin ^{2}(\Delta t/2)/\Delta ^{2}}$ is ${\displaystyle \sin(\Delta t)/(2\Delta )}$, which behaves like a delta function at large t (for more information, please see Sinc function#Relationship to the Dirac delta distribution). The constant decay rate of the golden rule follows. [8] As a constant, it underlies the exponential particle decay laws of radioactivity. (For excessively long times, however, the secular growth of the ak(t) terms invalidates lowest-order perturbation theory, which requires ak ≪ ai.)

Only the magnitude of the matrix element ${\displaystyle \langle f|H'|i\rangle }$ 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. ^[9]

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 ${\textstyle \langle f|f\rangle =\int d^{3}\mathbf {r} \left|f(\mathbf {r} )\right|^{2}}$ 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 ${\displaystyle \varepsilon }$ labelled ${\displaystyle |\varepsilon \rangle }$, by writing ${\displaystyle \langle \varepsilon |\varepsilon '\rangle =\delta (\varepsilon -\varepsilon ')}$ where ${\displaystyle \delta }$ is the Dirac delta function, and effectively a factor of the square-root of the density of states is included into ${\displaystyle |\varepsilon _{i}\rangle }$. ^[10] In this case, the continuum wave function has dimensions of ${\textstyle 1/{\sqrt {\text{[energy]}}}}$, and the Golden Rule is now

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

where ${\displaystyle \varepsilon _{i}}$ refers to the continuum state with the same energy as the discrete state ${\displaystyle i}$. 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. ^[11]

Normalized Derivation in time-dependent perturbation theory

Main article: Perturbation theory (quantum mechanics) § Time-dependent perturbation theory

The following paraphrases the treatment of Cohen-Tannoudji. ^[10] As before, the total Hamiltonian is the sum of an “original” Hamiltonian H₀ and a perturbation: ${\displaystyle H=H_{0}+H'}$. 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

$${\displaystyle |\psi (t)\rangle =a_{i}e^{-\mathrm {i} \omega _{i}t}|i\rangle +\int _{C}d\varepsilon a_{\varepsilon }e^{-\mathrm {i} \omega t}|\varepsilon \rangle ,}$$

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

Substituting into the time-dependent Schrödinger equation

$${\displaystyle H|\psi (t)\rangle =\mathrm {i} \hbar {\frac {\partial }{\partial t}}|\psi (t)\rangle }$$

and premultiplying by ${\displaystyle \langle i|}$ produces

$${\displaystyle {\frac {da_{i}(t)}{dt}}=-\mathrm {i} \int _{C}d\varepsilon \Omega _{i\varepsilon }e^{-\mathrm {i} (\omega -\omega _{i})t}a_{\varepsilon }(t),}$$

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

$${\displaystyle {\frac {da_{\varepsilon }(t)}{dt}}=-\mathrm {i} \Omega _{\varepsilon i}e^{\mathrm {i} (\omega -\omega _{i})t}a_{i}(t).}$$

We made use of the normalisation ${\displaystyle \langle \varepsilon '|\varepsilon \rangle =\delta (\varepsilon '-\varepsilon )}$. Integrating the latter and substituting into the former,

$${\displaystyle {\frac {da_{i}(t)}{dt}}=-\int _{C}d\varepsilon \Omega _{i\varepsilon }\Omega _{\varepsilon i}\int _{0}^{t}dt'e^{-\mathrm {i} (\omega -\omega _{i})(t-t')}a_{i}(t').}$$

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

$${\displaystyle {\frac {da_{i}(t)}{dt}}=\int _{C}d\varepsilon |\Omega _{i\varepsilon }|^{2}a_{i}(t)\int _{0}^{t}dTe^{-\mathrm {i} \Delta T},}$$

where ${\displaystyle T=t-t'}$ and ${\displaystyle \Delta =\omega -\omega _{i}}$. Integrating over ${\displaystyle T}$,

$${\displaystyle {\frac {da_{i}(t)}{dt}}=-2\pi \hbar \int _{C}d\varepsilon |\Omega _{i\varepsilon }|^{2}a_{i}(t){\frac {e^{-\mathrm {i} \Delta t/2}\sin(\Delta t/2)}{\pi \hbar \Delta }},}$$

The fraction on the right is a nascent Dirac delta function, meaning it tends to ${\displaystyle \delta (\varepsilon -\varepsilon _{i})}$ as ${\displaystyle t\to \infty }$ (ignoring its imaginary part which leads to an unimportant energy shift, while the real part produces decay ^[10] ). Finally

$${\displaystyle {\frac {da_{i}(t)}{dt}}=-2\pi \hbar |\Omega _{i\varepsilon _{i}}|^{2}a_{i}(t),}$$

which can have solutions: ${\displaystyle a_{i}(t)=\exp(-\Gamma _{i\to \varepsilon _{i}}t/2)}$, i.e., the decay of population in the initial discrete state is ${\displaystyle P_{i}(t)=|a_{i}(t)|^{2}=\exp(-\Gamma _{i\to \varepsilon _{i}}t)}$ where

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

Applications

Semiconductors

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. ^[12] Consider a photon of frequency ${\displaystyle \omega }$ and wavevector ${\displaystyle {\textbf {q}}}$, where the light dispersion relation is ${\displaystyle \omega =(c/n)\left|{\textbf {q}}\right|}$ and ${\displaystyle n}$ is the index of refraction.

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

$${\displaystyle {\textbf {E}}=-{\frac {\partial {\textbf {A}}}{\partial t}}=\mathrm {i} \omega A_{0}{\boldsymbol {\varepsilon }}e^{\mathrm {i} .({\textbf {q}}\cdot {\textbf {r}}-\omega t)}.}$$

For an electron in the valence band, the Hamiltonian is

$${\displaystyle H={\frac {({\textbf {p}}+e{\textbf {A}})^{2}}{2m_{0}}}+V({\textbf {r}}),}$$

where ${\displaystyle V({\textbf {r}})}$ is the potential of the crystal, ${\displaystyle e}$ and ${\displaystyle m_{0}}$ are the charge and mass of an electron, and ${\displaystyle {\textbf {p}}}$ is the momentum operator. Here we consider process involving one photon and first order in ${\displaystyle {\textbf {A}}}$. The resulting Hamiltonian is

$${\displaystyle H=H_{0}+H'=\left[{\frac {{\textbf {p}}^{2}}{2m_{0}}}+V({\textbf {r}})\right]+\left[{\frac {e}{2m_{0}}}({\textbf {p}}\cdot {\textbf {A}}+{\textbf {A}}\cdot {\textbf {p}})\right],}$$

where ${\displaystyle H'}$ 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

$${\displaystyle \Gamma _{if}={\frac {2\pi }{\hbar }}\left|\langle f|H'|i\rangle \right|^{2}\delta (E_{f}-E_{i}\pm \hbar \omega ),}$$

with

$${\displaystyle H'\approx {\frac {eA_{0}}{m_{0}}}{\boldsymbol {\varepsilon }}\cdot \mathbf {p} ,}$$

where ${\displaystyle {\boldsymbol {\varepsilon }}}$ is the light polarization vector. ${\displaystyle |i\rangle }$ and ${\displaystyle |f\rangle }$ are the Bloch wavefunction of the initial and final states. Here the transition probability needs to satisfy the energy conservation given by ${\displaystyle \delta (E_{f}-E_{i}\pm \hbar \omega )}$. 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 ${\displaystyle |i\rangle =\Psi _{v,{\textbf {k}}_{i},s_{i}}({\textbf {r}})}$ and ${\displaystyle |f\rangle =\Psi _{c,{\textbf {k}}_{f},s_{f}}({\textbf {r}})}$, respectively and if the ${\displaystyle H'}$ 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

$${\displaystyle \Psi _{v,{\textbf {k}}_{i}}({\textbf {r}})={\frac {1}{\sqrt {N\Omega _{0}}}}u_{n_{v},{\textbf {k}}_{i}}({\textbf {r}})e^{i{\textbf {k}}_{i}\cdot {\textbf {r}}},}$$

$${\displaystyle \Psi _{c,{\textbf {k}}_{f}}({\textbf {r}})={\frac {1}{\sqrt {N\Omega _{0}}}}u_{n_{c},{\textbf {k}}_{f}}({\textbf {r}})e^{i{\textbf {k}}_{f}\cdot {\textbf {r}}},}$$

where ${\displaystyle N}$ is the number of unit cells with volume ${\displaystyle \Omega _{0}}$. Calculating using these wavefunctions, and focusing on emission (photoluminescence) rather than absorption, we are led to the transition rate

$${\displaystyle \Gamma _{cv}={\frac {2\pi }{\hbar }}\left({\frac {eA_{0}}{m_{0}}}\right)^{2}|{\boldsymbol {\varepsilon }}\cdot {\boldsymbol {\mu }}_{cv}({\textbf {k}})|^{2}\delta (E_{c}-E_{v}-\hbar \omega ),}$$

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

$${\displaystyle {\boldsymbol {\mu }}_{cv}=-{\frac {i\hbar }{\Omega _{0}}}\int _{\Omega _{0}}d{\textbf {r}}'u_{n_{c},{\textbf {k}}}^{*}({\textbf {r}}')\nabla u_{n_{v},{\textbf {k}}}({\textbf {r}}').}$$

Finally, we want to know the total transition rate ${\displaystyle \Gamma (\omega )}$. 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

$${\displaystyle \Gamma (\omega )={\frac {4\pi }{\hbar }}\left({\frac {eA_{0}}{m_{0}}}\right)^{2}|{\boldsymbol {\varepsilon }}\cdot {\boldsymbol {\mu }}_{cv}|^{2}\rho _{cv}(\omega )}$$

where ${\displaystyle \rho _{cv}(\omega )}$ 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

$${\displaystyle \rho _{cv}(\omega )=2\pi \left({\frac {2m^{*}}{\hbar ^{2}}}\right)^{3/2}{\sqrt {\hbar \omega -E_{g}}},}$$

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 ^[13]

$${\displaystyle \Gamma _{vc}={\frac {2\pi }{\hbar }}\int _{\text{BZ}}{\frac {d{\textbf {k}}}{4\pi ^{3}}}|H_{vc}'|^{2}\delta (E_{c}({\textbf {k}})-E_{v}({\textbf {k}})-\hbar \omega ).}$$

In the same manner, the stationary photocurrent proportional to intensity of light is

$${\displaystyle {\textbf {J}}=-{\frac {2\pi e\tau }{\hbar }}\sum _{i,f}\int _{\text{BZ}}{\frac {d{\textbf {k}}}{(2\pi )^{D}}}|{\textbf {v}}_{i}-{\textbf {v}}_{f}|(f_{i}({\textbf {k}})-f_{f}({\textbf {k}}))|H_{if}'|^{2}\delta (E_{f}({\textbf {k}})-E_{i}({\textbf {k}})-\hbar \omega ),}$$

where ${\displaystyle \tau }$ is the relaxation time, ${\displaystyle {\textbf {v}}_{i}-{\textbf {v}}_{f}}$ and ${\displaystyle f_{i}({\textbf {k}})-f_{f}({\textbf {k}})}$ are the difference of the group velocity and Fermi-Dirac distribution between possible the initial and final states. Here ${\displaystyle |H_{if}'|^{2}}$ defines the optical transition dipole. Due to the commutation relation between position ${\displaystyle {\textbf {r}}}$ and the Hamiltonian, we can also rewrite the transition dipole and photocurrent in terms of position operator matrix using ${\displaystyle \langle i|{\textbf {p}}|f\rangle =-im_{0}\omega \langle i|{\textbf {r}}|f\rangle }$.

Scanning tunneling microscopy

Main article: 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

$${\displaystyle w={\frac {2\pi }{\hbar }}|M|^{2}\delta (E_{\psi }-E_{\chi }),}$$

where ${\displaystyle M}$ is the tunneling matrix element.

Quantum optics

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

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

where ${\displaystyle g(\hbar \omega )}$ is the density of photon states at a given energy, ${\displaystyle \hbar \omega }$ is the photon energy, and ${\displaystyle \omega }$ 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. ^[14]

Drexhage experiment

Both the radiation pattern and the total emitted power (which is proportional to the decay rate) of a dipole depend on its distance from a mirror.

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. ^{[15] [16]}

See also

• Exponential decay  – Decrease in value at a rate proportional to the current value
• List of things named after Enrico Fermi
• Particle decay  – Spontaneous breakdown of an unstable subatomic particle into other particles
• Sinc function  – Special mathematical function defined as sin(x)/x
• Time-dependent perturbation theory  – Approximate modelling of a quantum systemPages displaying short descriptions of redirect targets
• Sargent's rule

References

1. Bransden, B. H.; Joachain, C. J. (1999). Quantum Mechanics (2nd ed.). Prentice Hall. p. 443. ISBN   978-0582356917.
2. 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).
3. Fermi, E. (1950). Nuclear Physics. University of Chicago Press. ISBN   978-0226243658. formula VIII.2
4. Fermi, E. (1950). Nuclear Physics. University of Chicago Press. 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, see Wolfgang Pauli, Wave Mechanics: Volume 5 of Pauli Lectures on Physics (Dover Books on Physics, 2000) ISBN   0486414620, pp. 150–151.
7. Landau, L. D., & Lifshitz, E. M. (2013). Quantum mechanics: non-relativistic theory (Vol. 3). Elsevier.
8. Merzbacher, Eugen (1998). "19.7" (PDF). Quantum Mechanics (3rd ed.). Wiley, John & Sons, Inc. ISBN   978-0-471-88702-7.
9. 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. S2CID   119476624.
10. Cohen-Tannoudji, Claude; Diu, Bernard; Laloë, Franck (1977). Quantum Mechanics Vol II Chapter XIII Complement D_{XIII}. Wiley. ISBN   978-0471164333.
11. Bethe, Hans; Salpeter, Edwin (1977). Quantum Mechanics of One- and Two-Electron Atoms. Springer, Boston, MA. ISBN   978-0-306-20022-9.
12. Yu, Peter Y.; Cardona, Manuel (2010). Fundamentals of Semiconductors - Physics and Materials Properties (4 ed.). Springer. p. 260. doi:10.1007/978-3-642-00710-1. ISBN   978-3-642-00709-5.
13. Edvinsson, T. (2018). "Optical quantum confinement and photocatalytic properties in two-, one- and zero-dimensional nanostructures". Royal Society Open Science. 5 (9): 180387. Bibcode:2018RSOS....580387E. doi:10.1098/rsos.180387. ISSN   2054-5703. PMC   6170533 . PMID   30839677.
14. Fox, Mark (2006). Quantum Optics: An Introduction. Oxford: Oxford University Press. p. 51. ISBN   9780198566731.
15. K. H. Drexhage; H. Kuhn; F. P. Schäfer (1968). "Variation of the Fluorescence Decay Time of a Molecule in Front of a Mirror". Berichte der Bunsengesellschaft für physikalische Chemie. 72 (2): 329. doi:10.1002/bbpc.19680720261. S2CID   94677437.
16. K. H. Drexhage (1970). "Influence of a dielectric interface on fluorescence decay time". Journal of Luminescence. 1: 693–701. Bibcode:1970JLum....1..693D. doi:10.1016/0022-2313(70)90082-7.

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