Difference between revisions of "Fermi's Golden Rule"

Revision as of 21:14, 16 February 2017

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 Deftnition

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]H_{b}[/math]. Besides this discrete state, [math]H_{0}[/math] has a set of eigenstates {[math]\left |n \right \rangle[/math]}, with eigenenergies [math]{E_{n}}[/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 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 \cite{zjm}. 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 |[/math]

[math]V=g\sum_{n=-\infty}^{\infty} ( |b \rangle \langle n | + |n \rangle \langle b | )[/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 Schr\text\ddot{o}dinger equation is

[math]i \hbar \frac{\partial |\psi \rangle }{\partial t }(H_0 + \lambda V) |\psi \rangle[/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[/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}[/math]

[math]i \hbar \frac{\partial |\psi_1 \rangle}{\partial t }H_0 |\psi_1 \rangle+ V |\psi_0 \rangle . \label{1st}[/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}[/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 }[/math]

[math]g \frac{ 1 - e^{i (n \Delta - E_b )t/\hbar } }{n \Delta - E_b } e^{-i n \Delta t /\hbar }[/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}[/math].

We recognize this as a periodic summation of the [math]\sinc [/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[/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)[/math].