1. For the classical, non-relativistic hydrogen atom, the principal action is I
n
= I
r
+ I
θ
+ I
ϕ
; the total angular momentum is I
t
= I
θ
+ I
ϕ
; and the component of the angular momentum onto the polar axis is I
m
= I
ϕ
, with 2πI
i = ∮ p
i
dr
i
for i = r, θ, and ϕ (and no sum convention); see Refs. [22,111]. The quantization conditions become I
n
= nh, I
l
= kh, and I
m
= mh. In the old quantum theory n = 1, 2, ... is the principal quantum number, and k is the subsidiary (or azimuthal) quantum number [22,124]. This is not k = 1, 2, ..., n, however, because modern quantization methods (Einstein-Brillouin-Keller) require k = l 1/2 with l = 0, 1, 2,. ..., n, in which the 1/2 refers not to the spin but to the Maslov (or Morse) index in semiclassical quantization. This index counts the number of classical turning points a encountered by a closed trajectory in the classical phase space; for a more general description in terms of caustics, see [12]. At each turning point a phase loss of π/2, which is equivalent to one-quarter of a wave, has to be taken into account. Continuity of the phase of the wavefunction then leads to I = (n+α/4)h. As one example, which is a case is usually associated with Wentzel-Kramers-Brillouin, it is well known that the one-dimensional harmonic oscillator, which has two turning points per period, has a ‘zero-point’ energy, i.e., a non-zero energy when v=0 in the quantized energy expression E
v
= (v+1/2)h, where v = 0, 1, .... For further reading and to see how this may be understood from conditions to be satisfied under a coordinate transformation of a path integral from Cartesian into spherical coordinates, requiring a new term h
2/4 in the classical Hamiltonian to be added to the angular momentum part |L|2 = l(l + 1)h
2, giving (l + 1/2)2
h
2, see p. 203 and p. 212 etc., of Ref. [59].
2. Secondary resonances are defined in a better way as the result of the perturbation Hamiltonian, and they give rise to secondary island chains in Poincaré sections of the phase space. An important difference between these resonances is that the strength of primary resonances depends only weakly on ε, namely as ε1/2, whereas for secondary resonances, it decreases much faster than this for small ε, where e is the (linear) interaction coupling parameter. See Chap. 2.4b of Ref. [97].
3. Andrei, E.Y. Yücel, S., and Menna, L. Phys. Rev. Lett. (1991) 67, 3704–7.
4. Arndt, M., Buchleitner, A., Mantegna, R.N., and Walther, H. (1991) Phys. Rev. Lett. 67, 2435–8.
5. Banks, D. and Leopold, J.G. (1978) J. Phys. B 11, 37–46 and 2833-43.