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

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: 10.1103/PhysRevA.35.1535

2. Fulling S. (1973). Nonuniqueness of Canonical Field Quantization in Riemannian Space-Time, Physical Review D, 7 (10) 2850-2862. DOI: 10.1103/PhysRevD.7.2850

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

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: 10.1103/PhysRevA.35.1535

2. Fulling S. (1973). Nonuniqueness of Canonical Field Quantization in Riemannian Space-Time, Physical Review D, 7 (10) 2850-2862. DOI: 10.1103/PhysRevD.7.2850

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

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