Fermi's Golden Rule
J. M. Zhang
Fujian Provincial Key Laboratory of Quantum Manipulation and New Energy Materials, College of Physics and Energy, Fujian Normal University, Fuzhou 350007, China and Fujian Provincial Collaborative Innovation Center for Optoelectronic Semiconductors and Efficient Devices, Xiamen, 361005, China.
1 Definition
Fermi's golden rule gives an approximate prediction of the decay rate of a discrete level imbedded in and coupled to a continuum. As a nontrivial and practical result within the first order time- dependent perturbation theory, it finds widespread use in all branches of physics. Sometimes, it appears under the guise of other names. For example, the so-called Born approximation in scattering theory is a Fermi’s golden rule result in essence.
2 History
The rule was first derived by Dirac [1, 2]. However, it was Fermi who conferred the title golden on the rule [3], possibly because of its instrumental role in his most cherished theory of Beta decay [4, 5].
3 Statement
The following scenario is quite common in quantum dynamics. Initially the system is in some discrete state [math]\left |b \right \rangle[/math], which is an eigenstate of the unperturbed Hamiltonian [math]H_{0}[/math] with eigenenergy [math]E_{b}[/math]. Besides this discrete state, [math]H_{0}[/math] has a set of eigenstates {[math]\left |n \right \rangle[/math]}, with eigenenergies [math]\left \{ E_{n} \right \}[/math] which form a continuum. The discrete state [math]\left |b \right \rangle[/math] is coupled to the continuum states {[math]\left |n \right \rangle[/math]} by a perturbation [math]V[/math], which induces the transition from the former to the latter. The coupling strengthes are denoted as [math]g_n = \langle n | V | b \rangle[/math]. In many cases, both [math]E_n[/math] and [math]g_n[/math] are slowly and smoothly varying functions of [math]n[/math]. Hence, it is legitimate to introduce the coupling function [math]g(E)[/math] such that [math]g(E_n) = g_n [/math], and the density of states [math]\rho(E)[/math] such that [math]\rho(E) d E [/math] is the number of continuum states in the interval [math][E, E + d E ][/math].
Fermi's golden rule then states that up to the first order perturbation theory, the probability [math]P[/math] of finding the system in the continuum states is a linear function of time. This behavior allows the definition of a transition rate with the explicit expression of
[math]w = \frac{d P}{d t} = \frac{2\pi}{\hbar } |g (E_b )|^2 \rho(E_b).\tag {1}[/math]
The characteristic feature is that [math]w[/math] is proportional to the density of state of the continuum and the squared modulus of the coupling strength both at [math]E= E_b[/math].
4 Derivation by an ideal model
The rule is derived in every textbook on quantum mechanics. Here we will give a heuristic derivation based on an ideal model [6]. Such a model is motivated by the expression above. The transition rate [math]w[/math] is proportional to the local values of [math]\rho (E)[/math] and [math]|g(E)|^2[/math] at [math]E = E_b [/math]. Hence, for the sake of simplicity, we simply take both functions as constant. That is, [math]E_{n+1} - E_n = \Delta [/math] and [math]g_n = g[/math], where [math]\Delta [/math] and [math]g[/math] are two constants. Moreover, we assume that the continuum band extends from [math]-\infty [/math] to [math]+ \infty [/math]. The unperturbed Hamiltonian and the perturbation are then
[math]H_{0}=E_{b} |b \rangle \langle b | + \sum_{n=-\infty}^{\infty} n \Delta | n\rangle \langle n |,\tag {2} [/math]
[math]V=g\sum_{n=-\infty}^{\infty} ( |b \rangle \langle n | + |n \rangle \langle b | ). \tag{3}[/math]
To develop the time-dependent perturbation theory, we introduce a control parameter [math]\lambda [/math] whose value will be set to unity in the end. The time-dependent Schrödinger equation is
[math]i \hbar \frac{\partial |\psi \rangle }{\partial t }=(H_0 + \lambda V) |\psi \rangle,\tag {4}[/math]
with the initial value [math]|\psi(t=0 ;\lambda ) \rangle=|b\rangle[/math]. Expand the wave function [math]| \psi(t;\lambda ) \rangle[/math] into a power series of [math]\lambda[/math],
[math]|\psi(t;\lambda) \rangle=\sum_{s=0}^\infty \lambda^s |\psi_{s} (t) \rangle .\tag {5}[/math]
By the initial condition of [math]| \psi \rangle[/math], we have [math]| \psi_{0} (0) \rangle =|b\rangle [/math] and [math] | \psi_{s}(0) \rangle= 0[/math] for [math]s\geq 1 [/math]. Now substitute the wave function into the Schr\"odinger equation and compare the coefficients of the powers of [math]\lambda[/math]. To the lowest nontrivial order, we get
[math]i\hbar \frac{\partial | \psi_{0} \rangle }{\partial t }=H_0 | \psi_0 \rangle , \label{zeroth}\tag {6}[/math]
[math]i \hbar \frac{\partial |\psi_1 \rangle}{\partial t }=H_0 |\psi_1 \rangle+ V |\psi_0 \rangle . \label{1st}\tag {7}[/math]
By the first equation we solve [math]|\psi_0(t) \rangle= e^{-i E_b t/\hbar } |b\rangle [/math]. Substituting this result into the second equation, and projecting both sides of (\ref{1st}) onto [math]|n\rangle [/math], we get
[math]i\hbar \frac{\partial }{\partial t } \langle n | \psi_1 \rangle =n \Delta \langle n | \psi_1 \rangle + g e^{-i E_b t /\hbar}. \tag {8}[/math]
This first order ordinary differential equation can be easily solved, yielding
[math]\langle n | \psi_1(t) \rangle=\frac{1}{i \hbar}\int_0^t d \tau g e^{-i E_b \tau /\hbar } e^{-i n \Delta (t- \tau)/\hbar }=g \frac{ 1 - e^{i (n \Delta - E_b )t/\hbar } }{n \Delta - E_b } e^{-i n \Delta t /\hbar } .\tag {9}[/math]
Therefore, to the first order approximation, the transition probability is
[math]P=\sum_n | \langle n | \psi_1(t) \rangle |^2 = g^2 \sum_n \frac{4 \sin^2 ((n \Delta - E_b )t / 2 \hbar)}{(n \Delta - E_b )^2}. \tag {10}[/math]
We recognize this as a periodic summation of the [math]\sin c[/math] function squared. For clarity, introduce the dimensionless time [math]T \equiv \Delta t /2 \hbar [/math] and the offset parameter (here [math]\lfloor \cdot \rfloor[/math] is the floor function)
[math]\alpha = E_b/\Delta - \lfloor {E_b /\Delta } \rfloor ,\tag {11}[/math]
which characterizes the location of [math]E_b[/math] relative to the continuum spectrum (e.g., if [math]\alpha = 0.5[/math], [math]E_b[/math] is in the middle of two [math]E_n[/math]'s). The transition probability can be then written as
[math]P = \left( \frac{4g^2}{\Delta^2 } \right ) W_\alpha (T) . \tag {12} [/math]
Here the function [math]W_\alpha[/math] is defined as
[math]W_\alpha (T) \equiv T^2 \sum_{m =-\infty}^{\infty } \sin c^2 [(m-\alpha )T], \tag {13}[/math]
where the infinite summation is a very regular one---it samples the [math]\sin c^2 x [/math] function uniformly with the period given by [math]T[/math] and the offset determined by [math]\alpha[/math]. It is apparent that [math]W_\alpha = W_{-\alpha} = W_{1-\alpha}[/math], i.e., [math]W_\alpha[/math] is an even and periodic function of [math]\alpha[/math]. Note that in eq. 12, the time dependence is only through the function [math]W_\alpha[/math] and the coupling strength [math]g[/math] appears only in the prefactor.
The function [math]W_\alpha[/math] can be calculated by using the Poisson summation formula [9]. The idea is that, if we need to perform the summation
[math]I = \sum_{n=-\infty}^{+\infty } f(a+ n T) , \tag {14}[/math]
we can do it in a roundabout way. We can first take the Fourier transform of the function [math]f[/math], i.e., calculate
[math]F(q)= \int_{-\infty}^{+\infty } dx f(x) e^{-iq x} . \tag {15}[/math]
Then the problem is converted to a summation with respect to [math]F[/math]. That is,
[math]I = \frac{1}{T} \sum_{n=-\infty}^{+\infty } F\left(\frac{2\pi n }{T}\right) \exp\left( \frac{i 2\pi n a }{T} \right) . \tag {16}[/math]
The problem gets simplified if the new summation is simpler than the original one. This is exactly the case for us. Anyone familiar with the Fraunhofer diffraction of a single slit knows that the Fourier transform [math]F_1 [/math] of the [math]\sin c[/math] function is the window function. That is,
[math]F_{1}(q) = \begin{cases} \pi, &|q|\leq 1,\\ 0, &|q|\gt 1. \end{cases}\tag {17}[/math]
The Fourier transform of [math]\sin c^2 x[/math] can then be calculated by convolution. It is
[math]F_2(q ) = \frac{1}{2\pi } \int_{-\infty}^{+\infty} d p F_1(p) F_1(q-p) = \begin{cases} \frac{\pi}{2} (2-|q|) , & |q| \leq 2 ; \\ 0, & |q| \gt 2 . \end{cases} \tag {18}[/math]
It has the shape of a triangle and is nonzero only on the support [math][-2, 2][/math]. This fact means that invoking the Poisson summation formula indeed simplifies the original summation task, because in the summation involving [math]F_2[/math], only a finite number of terms are nonzero.
In particular, if [math]0\lt T \leq \pi[/math], only the [math]n = 0 [/math] term is nonzero. Consequently, in this interval, [math] W_\alpha (T) = \pi T[/math], which is independent of the value of [math]\alpha[/math]. Translated in terms of the real time [math]t[/math], the result is that, for [math]0 \lt t \lt t_H \equiv 2\pi \hbar/ \Delta [/math], the probability of transition into the continuum is
[math]P = \frac{2\pi}{\hbar} \left( \frac{g^2}{\Delta } \right) t . \tag {19}[/math]
This is exactly Fermi's golden rule.
However, for later times, if [math]m \pi \lt T \leq (m+1) \pi [/math], there are [math]2m +1[/math] nonzero terms in the summation. After some straightforward calculation, we get ([math]\theta \equiv 2\pi \alpha [/math])
\begin{eqnarray}\label{wfull} W_\alpha (T) = \pi \frac{\sin\frac{(2m+1)\theta}{2}}{\sin \frac{\theta}{2} } (T-m \pi ) + \pi^2 \frac{\sin^2 \frac{m \theta}{2}}{\sin^2 \frac{\theta}{2}} , \tag {20} \end{eqnarray}
which is still a linear function of [math]T[/math] but now its slope is both [math]\alpha[/math]-dependent and [math]m[/math]-dependent. In other words, [math]W_\alpha [/math] is a piecewise linear function of [math]T[/math] and it shows cusps whenever [math]T[/math] is an integral multiplier of [math]\pi[/math]. In terms of the real time, the transition probability [math]P[/math] shows kinks when [math]t[/math] is an integral multiplier of [math]t_H[/math], the so-called Heisenberg time. This means that beyond the Heisenberg time, the golden rule breaks down generally.
5 Comments
The heuristic derivation of Fermi’s golden rule with the ideal model as the setting is advantageous in several aspects. First, it is more lucid than the usual approach in most textbooks, which deals with the general case. The simplicity of the model enables us to circumvent the obscure arguments inevitable in the generic case, thus highlighting the key characteristics of the rule. Second, the model allows us to address the problem of the breakdown of the rule. It is well noticed that the linearly increasing behavior of the transition probability cannot last forever, because for the first order approximation to be valid we need the nondepletion condition [math]P\ll 1[/math]. This gives an upper bound of the validity domain of the rule as [math]t\ll 1/w[/math]. However, here the model exemplifies the importance of quantum recurrence on this issue, as we see that the rule can break down even if the first order approximation itself is still valid and there is another time scale, namely, the Heisenberg time. Third, although here we have only worked out the first order approximation, the model is actually exactly solvable. It was first shown by Stey and Gibberd that in the first interval, i.e., [math]0\lt t \lt t_{H}[/math], [math]P[/math] decays exponentially rigorously [8]. Afterwards, [math]P[/math] shows cusps each time when [math] t/t_{H} [/math] is an integer. Thus, we can also see that the golden rule is just an approximation and how it can deviate from the exact dynamics.
To apply Fermi’s golden rule, one first has to identify the initial state and the continuum to which it is coupled and into which it will decay. For example, in the process of spontaneous emission of an atom, the initial state is the atom in some excited state and all the electromagnetic modes in the vacuum state (note that the system is the atom plus the electromagnetic modes, not just the atom), while the continuum consists of states in which the atom is in the ground state and one of the electromagnetic modes is excited to the first excited state. It is a continuum because the frequency of the electromagnetic modes forms a continuum. The fact that the initial state is imbedded in the continuum is also essential. Consider the photoelectric effect of the hydrogen atom. A laser beam of frequency [math]\omega[/math] drives a hydrogen atom in the ground state (with energy [math]E_{g}\lt0[/math]). If [math]\omega+E_{g}\gt0[/math], the ground state (plus [math]\omega[/math]) is degenerate with some continuum states, hence Fermi’s golden rule is applicable if the laser amplitude is weak enough. However, if [math]\omega+E_{g}\lt0[/math], the ground state (plus [math]\omega[/math]) is not degenerate with any continuum state, and the rule is not applicable. In this case, the picture of the dynamics is qualitatively different. For instance, the atom might perform Rabi oscillation between the ground state and some excited bound state. In the Forster resonance energy transfer process, the continuum comes from the rotational-vibrational energy levels of the acceptor molecule, which are dense enough to be viewed as a quasi-continuum. For the imbedding condition to be fulfilled, the fluorescence emission spectrum of the donor molecule must overlap the absorption or excitation spectrum of the acceptor molecule.
Finally, we have to remind the readers the caveat in invoking the golden rule. The rule is easy to interpreter and easy to use and is indeed routinely used in the literature—one just needs to find out the density of state and the coupling strength. However, one should also keep in mind its perturbative nature—it is based on the lowest order perturbation approximation. It thus breaks down in many cases, such as when the coupling is too strong. As an example, the Einstein explanation of the photoelectric effect of metals predicts a threshold frequency of the light. But this is just a golden rule prediction. It holds only when the light is weak enough. If the light field is strong enough, as in a femtosecond laser pulse, electrons can be kicked out regardless of the lightfrequency.
6 Further Reading
The fact that the transition rate is proportional to the density of state leads to the possibility of suppressing or enhancing the spontaneous emission rate of an atom by modifying its environment. That is the so-called Purcell effect. For an experimental demonstration, see D. Kleppner (1981). “Inhibited Spontaneous Emission”. Physical Review Letters. 47, 233.
7 External Sources
MIT lecture video: https://www.youtube.com/watch?v=zC78xmSbhAY
8 References
[1] P. A. M. Dirac, “The quantum theory of the emission and absorption of radiation,” Proc. R. Soc. Lond. A 114 (1927), 243-265.
[2] P. A. M. Dirac, The Principles of Quantum Mechanics (Oxford: Oxford University Press, 1958).
[3] E. Fermi, Nuclear Physics (University of Chicago Press, Chicago, 1950), page 142.
[4] E. Fermi, “Versuch einer Theorie der beta-Strahlen. I,” Z. Physik 88 (1934): 161-177.
[5] F. L. Wilson, “Fermi’s theory of Beta decay,” Am. J. Phys. 36 (1968): 1150-1160.
[6] J. M. Zhang and Y. Liu, “Fermi’s golden rule: its derivation and breakdown by an ideal model,” Eur. J. Phys. 37 (2016): 065406.
[7] G. Mussardo, Statistical Field Theory (Oxford University Press, Oxford, 2010).
[8] G. C. Stey and R. W. Gibberd, “Decay of quantum states in some exactly soluble models,” Physica 60 (1972): 1-26.
