Radial matrix elements with Dirac`s functions in the theory of fine and hyperfine structures of hydrogen-like systems
Journal Article
·
· International Journal of Modern Physics B
- BITM, Bryansk (Russian Federation). Dept. of Mathematics
Investigating the effects of fine and hyperfine structures of hydrogen spectrum there arises practical necessity to evaluate the different matrix elements with Dirac`s radial functions f and g; usually such matrix elements being evaluated approximately. The article deals with the study of relativistic matrix elements by a new method, i.e. with the help of Appell`s hypergeometrical functions F{sub 2}(x,y) and their new properties. Such approach allows: (1) to get exact analytical expressions for these matrix elements by means of Appell`s F{sub 2} (and Clausen`s {sub 3}F{sub 2}) functions; (2) to reveal latent symmetry of diagonal matrix elements the above symmetry is connected with the property of Appell`s function F{sub 2}(1,1) mirror-like symmetry; (3) to prove that the diagonal and off-diagonal matrix elements in a pole-domain of Euler`s gamma-function are given exactly analytically by means of nonorientable series of type {dot F}{sub 2}(x,y). The examples for the evaluation of diagonal matrix elements for the K- and L-shells and also of dipole matrix elements for Lyman`s series are obtained.
- Sponsoring Organization:
- USDOE
- OSTI ID:
- 398291
- Journal Information:
- International Journal of Modern Physics B, Journal Name: International Journal of Modern Physics B Journal Issue: 20 Vol. 10; ISSN IJPBEV; ISSN 0217-9792
- Country of Publication:
- United States
- Language:
- English
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