to other states of the same energy. 9.3.2 Radiative transition rates We now use equation (9.47) to calculate the rate at which an electromagnetic wave induces an atom to make radiative transitions between its stationary states. Our treatment is valid when the quantum uncertainty in the electro- magnetic field may be neglected, and the field treated as a classical object. This condition is satisfied, for example, in a laser, or at the focus of the antenna of a radio telescope. Whereas in our derivation of Fermi’s golden rule, the system had some density of states at the energy of the initial state and we summed the transi- tion probabilities by integrating over the energy of the final state, now we ar- gue that the electromagnetic field has non-negligible power over a continuum of frequencies, and we integrate over frequency. The quantity |( E k | V 0 | E N )| 2 that occurs in equation (9.47) is now a function of frequency ω . We argue that each single-frequency contribution to the electromagnetic field induces transitions independently contributions at other frequencies. Hence equation (9.47) is valid if we make the substitution |( E k | V 0 | E N )| 2 → d ω d d ω |( E k | V 0 | E N )| 2 , (9 . 53) 5 The golden rule was actually first given by Dirac, P.A.M., 1927, Proc. Roy. Soc. A, 114 , 243
9.3 Time-dependent perturbation theory 193 which isolates the contribution to |( E k | V 0 | E N )| 2 from a frequency interval of infinitesimal width. Once we have evaluated the transition probability P k ( t ) that is generated by this frequency interval, we set t to a large value and sum over all intervals by integrating with respect to ω . That is we evaluate integraldisplay d ωP k = 1 ¯ h 2 integraldisplay d ω sin 2 ( xt ) x 2 d d ω |( E k | V 0 | E N )| 2 , (9 . 54) where x ≡ (¯ hω + E k − E N ) / 2¯ h . We change the integration variable to x using d x = 1 2 d ω and exploit equation (9.51) to evaluate the integral. The result is integraldisplay d ωP k = 2 πt ¯ h 2 d d ω |( E k | V 0 | E N )| 2 , with ω = ( E N − E k ) / ¯ h. (9 . 55) The coefficient of t on the right is the rate at which the sinusoidally oscil- lating perturbation causes transitions from | E N ) to | E k ) . This rate vanishes unless there is a component of the perturbation that oscillates at the angular frequency ( E N − E k ) / ¯ h that we associate with the energy difference between the initial and final states. This makes perfect sense physically because we know from § 3.2.1 that when there is quantum uncertainty as to whether the system is in two states that differ in energy by Δ E , the system oscillates physically with angular frequency Δ E/ ¯ h . If the system bears electromag- netic charge, these oscillations are liable to transfer energy either into or out of the oscillations of the ambient electromagnetic field at this frequency. To proceed further we need an expression for the derivative of the matrix element in equation (9.55). In vacuo the electric field of an electromagnetic wave is divergence free, being entirely generated by Faraday’s law, ∇ × E = − ∂ B /∂t . It follows that the whole electromagnetic field of the wave can be described by the vector potential A through the equations B =
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