Classical Electrodynamics · Problem 9.10

Radiating Systems, Multipole Fields and Radiation — John David Jackson

Physics — E&M · hydrogen-2p-to-1s-transition-radiation

Problem

The transitional charge and current densities for the radiative transition from the m=0m = 0, 2p2p state in hydrogen to the 1s1s ground state are, in the notation of (9.1) and with the neglect of spin, ρ(r, θ, ϕ, t)=2e6 a04⋅re−3r/2a0Y00Y10e−iω0tJ(r, θ, ϕ, t)=−iv02(r^2+a0z z^)ρ(r, θ, ϕ, t)\begin{aligned} \rho(r,\, \theta,\, \phi,\, t) &= \frac{2e}{\sqrt{6}\, a_0^4} \cdot r e^{-3r/2a_0} Y_{00} Y_{10} e^{-i\omega_0 t} \\ \mathbf{J}(r,\, \theta,\, \phi,\, t) &= \frac{-iv_0}{2}\left(\frac{\hat{r}}{2} + \frac{a_0}{z}\, \hat{\mathbf{z}}\right)\rho(r,\, \theta,\, \phi,\, t) \end{aligned} where a0=4πϵ0ℏ2/me2=0.529×10−10a_0 = 4\pi\epsilon_0 \hbar^2/me^2 = 0.529 \times 10^{-10} m is the Bohr radius, ω0=3e2/32πϵ0ℏa0\omega_0 = 3e^2/32\pi\epsilon_0 \hbar a_0 is the frequency difference of the levels, and v0=e2/4πϵ0ℏ=αc≈c/137v_0 = e^2/4\pi\epsilon_0 \hbar = \alpha c \approx c/137 is the Bohr orbit speed. (a) Show that the effective transitional (orbital) ``magnetization'' is ‘‘M′′(r, θ, ϕ, t)=−iαca04 tan⁡θ(x^sin⁡ϕ−y^cos⁡ϕ)⋅ρ(r, θ, ϕ, t)``\mathcal{M}''(r,\, \theta,\, \phi,\, t) = -i\frac{\alpha c a_0}{4}\, \tan\theta(\hat{\mathbf{x}}\sin\phi - \hat{\mathbf{y}}\cos\phi) \cdot \rho(r,\, \theta,\, \phi,\, t) Calculate ∇⋅‘‘M′′\boldsymbol{\nabla} \cdot ``\mathcal{M}'' and evaluate all the nonvanishing radiation multipoles in the long-wavelength limit. (b) In the electric dipole approximation calculate the total time-averaged power radiated. Express your answer in units of (ℏω0)⋅(α4c/a0)(\hbar\omega_0) \cdot (\alpha^4 c/a_0), where α=e2/4πϵ0ℏc\alpha = e^2/4\pi\epsilon_0 \hbar c is the fine structure constant. (c) Interpreting the classically calculated power as the photon energy (ℏω0)(\hbar\omega_0) times the transition probability, evaluate numerically the transition probability in units of reciprocal seconds. (d) If, instead of the semiclassical charge density used above, the electron in the 2p2p state was described by a circular Bohr orbit of radius 2a02a_0, rotating with the transitional frequency ω0\omega_0, what would the radiated power be? Express your answer in the same units as in part b and evaluate the ratio of the two powers numerically.

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