## Symmetries and solutions of the three-dimensional Paul trap

Optics Express, Vol. 8, Issue 2, pp. 123-130 (2001)

http://dx.doi.org/10.1364/OE.8.000123

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### Abstract

Using the symmetries of the three-dimensional Paul trap, we derive the solutions of the time-dependent Schrödinger equation for this system, in both Cartesian and cylindrical coordinates. Our symmetry calculations provide insights that are not always obvious from the conventional viewpoint.

© Optical Society of America

**OCIS Codes**

(020.7010) Atomic and molecular physics : Laser trapping

(270.5570) Quantum optics : Quantum detectors

**ToC Category:**

Focus Issue: Quantum control of photons and matter

**History**

Original Manuscript: November 15, 2000

Published: January 15, 2001

**Citation**

Michael Nieto and D. Truax, "Symmetries and solutions of the three-dimensional Paul trap," Opt. Express **8**, 123-130 (2001)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-8-2-123

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### References

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- The phase factor can also be obtained [14, 15], by solving the eigenvalue equation 3 nz = (nz � ) nz, where 3 = {3 t � (Z z � ) - i/4 3z2} Then, solving the equation Jz-0=0 will yield the extremal state function up to a factor of (pi)^-1/4.
- D. R. Truax, "Symmetry of time-dependent Schr� odinger equations. II. Exact solutions for the equation {xx 2 t - 2 2(t)x2 - 2 1(t)x-2 0(t) = 0," J. Math. Phys. 23, 43-54 (1982). [CrossRef]
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- M. M. Nieto and D. R. truax, eprint quant-ph/0011062, expands the contents of this manuscript. It contains further information on Ref. [1] and, in an appendix, on J. H. Eberly.

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