A Course in Mathematics Volume II by Frederick S. And Frederick H. Bailey Woods

By Frederick S. And Frederick H. Bailey Woods

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Furthermore, if one knows φ, ˜ the two transition zeros associated with the potential θ, ϑ and φ, barrier need not lie very far away from transition points that are not associated with the barrier. 2, the barrier is assumed to lie far away from all transition points that are not associated with the barrier. 30a,b) by considering A , B , A and B formally as constants. 1 Wave function given as a standing wave The case when the wave function is given as a standing wave on one side of the barrier requires a detailed treatment, since the resonance phenomenon may occur.

However, when the energy is close to a resonance energy, a careful discussion is required. 4 49 Quantization conditions for single-well potentials In this section we shall present quantization conditions for general single-well potentials [N. Fr¨ oman (1966c, 1978), Fr¨ oman and Fr¨ oman (1965, 1978a, 1978b, 1978c, 1996, 2002), Paulsson, Karlsson and LeRoy (1983)], valid for any conveniently chosen order of the phaseintegral approximation, in forms especially adapted to the treatment of the Stark effect in a hydrogenic atom or ion.

When the barrier is underdense. , that the distance from the barrier to the transition points that are not associated with the barrier must be much larger than |t − t |. However, when the energy is close to the top of the barrier, it is the slight deviation from parabolic shape close to the top that determines the ¯ 2n , n > 0, and one needs accurate values of values of the quantities K these quantities for obtaining accurate values of φ˜ in higher orders of the phase-integral approximation. ch04 January 30, 2008 48 B550 Stark Effect in a Hydrogenic Atom or Ion 9in x 6in Stark Effect in a Hydrogenic Atom or Ion The derivative of φ˜ with respect to the energy E will be needed in Chapter 5, and therefore we shall now give formulas for this derivative.

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