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Chapter 1
56
squared wavefunction at the origin inside of the nucleus.) In addition to satisfaction of the Haus’ condition given, the electron currents given by Eq. (1.12) are shown to be nonradiative with the same condition as that of Eq. (1.16) applied to the vector potential based on the electromagnetic fields and the Poynting power vector as shown in Appendix I: Nonradiation Condition. From Eq. (1.12), the charge and intrinsic magnetization terms are zero. Also, the current J x, t is in the ˆ direction; thus, the aE , m coefficient given by Eq. (1.7) is zero since r J 0 . Substitution of Eq. (1.12) into Eq. (1.8) gives the magnetic multipole coefficient aM , m : a M , m
n Nj krn sin ks c 1 2 ek 2
(1.17)
For the electron source current given by Eq. (1.12), each comprising a multipole of order , m with a time dependence eit , the far-field solutions to Maxwell’s equations given by Eq. (1.3) are i B aM , m g kr X ,m k E aM , m g kr X ,m and the time-averaged power radiated per solid angle
(1.18) dP , m is d
dP , m 2 2 c a M , m X , m 2 d 8 k
(1.19)
where aM , m is given by Eq. (1.17). In the case that k is the lightlike k 0 , then k n / c regarding an emitted photon, in Eq. (1.17), and Eqs. (1.18-1.19) vanishes for s vTn R rn n
(1.20)
There is no radiation. There is no radiation due to the azimuthal charge density wave even in an excited state. However, for excited states there exists a radial dipole that is unstable to radiation as shown in the Instability of Excited States section, and this instability gives rise to a radial electric dipole current. In a nonradiative state, there is no emission or absorption of radiation corresponding to the absence of radial motion wherein Eq. (1.7) is zero since r J 0 . Conversely, there is motion in the radial direction only when the energy of the system is changing, and the radiation emitted or absorbed during electron transitions is the multipole radiation given by Eq. (1.2) as given in the Excited States of the One-Electron Atom (Quantization) section and the Equation of the Photon section wherein Eqs. (4.18-4.23) give a macro-spherical wave in the far-field. Thus, radial motion corresponds to the emission or absorption of photons. The form of the radial solution during a transition is then the corresponding electron source current comprising a time-dependent radial Dirac delta function that connects the initial and final states as boundary conditions. The photon carries fields and corresponding angular momentum. The physical characteristics of the photon and the electron are the basis of physically solving for excited states according to Maxwell’s equations. The discontinuous harmonic radial current in Eq. (1.7) that connects the initial and final states of the transition is rJ
er 1 t ' sin u t ' u t ' 4 r 2
(1.21)
where is the lifetime of the transition given by Eq. (2.107) and t ' is time during the transition as given in the Excited States of the One-Electron Atom (Quantization) section. The vector potential of the current that connects the initial and final states of a transition, each having currents of the form given by Eq. (1.12) is A r
0 e 1 e ik r iz 2 me rn rn 4 r r
i
(1.22)
f
The magnetic and electric fields are derived from the vector potential and are used in the Poynting power vector to give the power. The transition probability or Einstein coefficient Aki for initial state ni and final state n f of atomic hydrogen given by the power divided by the energy of the transition is 1
1 e me c 2 24 me a02
2
1
n f ni
2
2.678 X 109
1
n f ni
2
s 1
(1.23)
which matches the NIST values for all transitions extremely well as shown in Excited States of the One-Electron Atom (Quantization) section.