Monday, December 16, 2013

Jackson Wrong and Hyperbolic Bessels

I'm back in the lab this week working on some numerical models for the solenoid to be used in the H-Ray experiment.  In looking for analytic solutions for the z and rho components of a magnetic field due to a coil, I came across a very interesting reference in an article by Bergeman, Erez, and Metcalf[1].  Take a look


If you're one of the many who after poring over Jackson, thought "This can't be right", in this one case, you were right.  The proper expressions according to Bergeman et al. are


Hyperbolic Bessels
Solving for quantized fields, one is often led to Bessel eigenfunctions in cylindrical coordinates.  Based on some of my recent work, I was left with the feeling that relativistic quantizations of the field in a uniformly accelerated space might give the result in terms of hyperbolic Bessel functions.  Sure enough, the Macdonald function solution that Dr. Fulling[2] finds in his paper "Nonuniqueness of Canonical Field Quantization in Riemannian Space-Time" is also known as the hyperbolic Bessel[3].  Cool!




References
1.Bergeman T., Erez G. & Metcalf H. (1987). Magnetostatic trapping fields for neutral atoms, Physical Review A, 35 (4) 1535-1546. DOI:

2. Fulling S. (1973). Nonuniqueness of Canonical Field Quantization in Riemannian Space-Time, Physical Review D, 7 (10) 2850-2862. DOI:

3.  http://en.wikipedia.org/wiki/Bessel_function#Modified_Bessel_functions_:_I.CE.B1.2C_K.CE.B1

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