Quantization as an eigenvalue problem

Mon, 01 Jan 0001 00:00:00 +0000

E < 0.

In this case the condition (15) is not eo ipso excluded, but for the moment we hold onto its exclusion, as arranged. Then, according to (14′′) and (17), U1 grows unlimited for r = ∞, whereas U2 vanishes exponentially. Our entire transcendent U (and the same holds true for χ) will then remain finite if and only if U is identical to U2, a part from a numerical factor. However, this is not the case. One realises this so: choosing in (12) for the integration contour L a closed circuit about both the points c1 and c2 (due to the integerness of the sum α1 + α2 such contour is then really closed on the Riemann surface of the integrand), upon fulfillment of the very condition (13), one can easily show that the integral (12) represents then our entire transcendent U . Namely, it can be developed in a series of positive powers of r, which always converges for sufficiently small values of r, hence it satisfies the differential equation (7′), and thus the power series must coincide with that for U . So: U is represented by (12) when L is a closed contour about both the points c1 and c2. However, this closed contour can be distorted so that it appears as the combination of the two previously considered integration paths, those related to U1 and U2, and indeed without vanishing factors, say 1 and e2πiα1 . QED17

Quantization as an eigenvalue problem on Superphysics

Quantization as an eigenvalue problem