Abstract

The magnetic Hooke-Newton transmutation, which emerges from the transformation design of the quadratic conformal mapping for the system of charged particles moving in the uniform magnetic field, is investigated in the momentum space. It is shown that there are two ways to turn the linear interaction force of the system into the inverse square interaction. The first one, which involves simply applying the mapping to the system, has the spectrum with the Landau levels of even angular momentum quantum number. The second one considers the geometric structure of the mapping as an effective potential which leads us to the transmuted Coulomb system with the novel quantum spectrum. The wave functions of momentum for the bound and scattering states of the transmutation system are given. It is also shown that the effective potential due to the geometric structure can be generalized to a general 2D surface, and the Schrödinger equation of a particle moving on the surface while under the action of the potential can be solved by the form-invariant Schrödinger equation of the free particle. The solution of a particle moving on the hyperbolic surface under the potential is given with the conclusion. The presentation manifests the transformation design of the quantum state, depending mainly on the geometric structure of the representation space, not on the action of the specific potential field. This characteristic makes it possible for us to use the geometric structure of different representation spaces to explore some novel behaviors of quantum particles.

Highlights

  • The transformation design method (TDM) was introduced in optics [1,2], and has been extensively applied to the diverse wave phenomena (e.g., [3])

  • Our discussion is arranged as follows: In Section 2, the quadratic conformal mapping is applied to the momentum representation of the Schrödinger equation for the charge particle coupling to a uniform magnetic field, and obtains the equation of a transmuted

  • This paper investigates the magnetic Hooke-Newton transmutation in the momentum space

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Summary

Introduction

The transformation design method (TDM) was introduced in optics [1,2], and has been extensively applied to the diverse wave phenomena (e.g., [3]). We investigate the magnetic HookeNewton transmutation in the p-space through reinterpreting the geometric structure of the quadratic conformal mapping as an effective interacting potential. Our discussion is arranged as follows: In Section 2, the quadratic conformal mapping is applied to the momentum representation of the Schrödinger equation for the charge particle coupling to a uniform magnetic field, and obtains the equation of a transmuted. It is shown that the mapping turns the new equation into the transmuted Coulomb system with the novel quantum spectrum discovered in the recent investigation of the transformation design of quantum states [11]. In the Appendix A, the definition of the effective potential is generalized to a general 2D surface, and we explain how to construct the form-invariant Schrödinger equation on the surface with the potential, and solve the quantum problem of a particle interacting with the potential

The Quadratic Conformal Image of the Equation for a Charge in the Uniform
Bound State
Scattering State
Conclusions

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