The transitional charge and current densities for the radiative transition from the m = 0, 2p state in hydrogen to the Is ground state are, in the notation of (9.1) and with the neglect of spin,

Where a₀ = 4??₀h²/m?² = 0.529 ? 10^-10 m is the Bohr radius, ?₀ = 3e²/32??₀ha₀ is the frequency difference of the levels, and v₀ = e²/4??₀h = ?? ? ?/137 is the Bohr orbit speed.

(a) Show that the effective transitional (orbital) "magnetization" is

"M"(r, ?, ?, t) = ?i?ca₀/4 tan ?(x sin ? - ? cos ?) ? ?(r, ?, ?, t)

Calculate ? ? "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 (h?₀) ? (?⁴c/a₀), where ? = ?²/4??₀h? is the fine structure constant.

(c) Interpreting the classically calculated power as the photon energy (h?₀) times the transition probability; evaluate numerically the transition probability in units of reciprocal seconds.

(d) If, instead of the semi-classical charge density used above, the electron in the 2p state was described by a circular Bohr orbit of radius 2a₀, rotating with the transitional frequency ?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.

Transcribed Image Text:

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