A new method of calculating the mean value of 1/ r 3 for keplerian systems in quantum mechanics
John Hasbrouck Van Vleck · Proceedings of the Royal Society of London Series A Containing Papers of a Mathematical and Physical Character · 1934
Abstract The usual method of calculating the diagonal matrix elements of an integral power of the radius r in an inverse square quantum system is that due to Waller. His procedure is based on the Schrödinger form of the theory, and utilizes in particular the generating series for the Langueree polynomials. For negative powers, Dirac's very elegant theory of "q-numbers," developed in these 'proceedings' during the early days of quantum mechanics, furnishes an interesting alternative method which appears to have been overlooked, and which we believe is easier. From his theory the following rule can be derived: Suppose that we desire the mean value (diagonal element) of 1/rs, where s is an integer greater than unity. We write down the experssion with A(αl + βle-ix + γleix)s-2 A = 16π4m2z2e4/(l + ½)n3h4, αl = 4π2mze2/h2l(l+1), βl = 2π2mZe2/h2(l+½)(l+1)[1-(l+1)2/n2]½, γl = 2π2mze2/h2l(l+½)[1-l2/n2]½.