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Exact solution of the classical mechanical quadratic Zeeman effect

DSpace at IIT Bombay

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Title Exact solution of the classical mechanical quadratic Zeeman effect
 
Creator DATTA, SN
PANDEY, A
 
Subject magnetic-field
hydrogen-atom
systems
zeeman effect
 
Description We address the curious problem of quadratic Zeeman effect at the classical mechanical level. The problem has been very well understood for decades, but an analytical solution of the equations of motion is still to be found. This state of a. airs persists because the simultaneous presence of the Coulombic and quadratic terms lowers the dynamical symmetry. Energy and orbital angular momentum are still constants of motion. We find the exact solutions by introducing the concept of an image ellipse. The quadratic effect leads to a dilation of space-time, and a one-to-one correspondence is observed for pairs of physical quantities like energy and angular momentum, and the maximum and minimum distances from the Coulomb center for the Zeeman orbit and the corresponding pairs for the image ellipse. Thus, instead of finding additional conserved quantities, we find constants of motion for an additional dynamics, namely, the image problem. The trajectory is open, in agreement with Bertrand's theorem, but necessarily bound. A stable unbound trajectory does not exist for real values of energy and angular momentum. The radial distance, the angle covered in the plane of the orbit, and the time are uniquely determined by introducing further the concept of an image circle. While the radial distance is defined in a closed form as a transcendental function of the image-circular angle, the corresponding orbit angle and time variables are found in the form of two convergent series expansions. The latter two variables are especially contracted, thereby leading to a precession of the open cycles around the Coulomb center. It is expected that the space time dilation effect observed here would somehow influence the solution of the quantum mechanical problem at the non-relativistic level.
 
Publisher INDIAN ACAD SCIENCES
 
Date 2011-08-01T22:06:17Z
2011-12-26T12:53:31Z
2011-12-27T05:39:47Z
2011-08-01T22:06:17Z
2011-12-26T12:53:31Z
2011-12-27T05:39:47Z
2007
 
Type Article
 
Identifier PRAMANA-JOURNAL OF PHYSICS, 68(6), 967-982
0304-4289
http://dx.doi.org/10.1007/s12043-007-0096-3
http://dspace.library.iitb.ac.in/xmlui/handle/10054/8559
http://hdl.handle.net/10054/8559
 
Language en