# The quantum-mechanical description of an electron motion in a magnetic monopole field

The quantum-mechanical motion problem of an electron in electric field of the nucleus is well known. The quantum-mechanical description of electron motion in a magnetic field is also not difficult, since it needs to solve the Schrödinger equation of the form: $$\frac{(\hat p + eA)^2} {2m} \psi = E \psi$$ But if we want to consider the motion of an electron in a magnetic monopole field, the difficulty arises because the definition of the vector potential in the whole space. See, for example. Was this problem solved? What interesting consequences derived from this task? (for energy levels, angular momentum etc.)

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This gives you coherent state representation for angular momentum. – Slaviks Mar 16 '12 at 21:54
The difficulty is not in the vector potential--- you can just use a Dirac string. The difficulty is getting the actual solution. – Ron Maimon Apr 15 '12 at 7:42

## 1 Answer

The classical version of this problem was solved by Henri Poincaré way back in 1896. This is also problem 5.43 in Electrodynamics by Griffiths. The classical trajectories are geodesics on the surface of a cone. A recent treatment of the classical version of this problem is here.

The quantum mechanical version was also solved long back by Igor Tamm in 1931. This is discussed in section 2.3 of the book Magnetic monopoles by Y M Shnir, who follows the treatment in Charge quantization and nonintegrable Lie algebras by Hurst.

The quantum mechanical version of the problem turns out to be separable in spherical polar coordinates. The angular part has the generalized spherical harmonics as its eigenvalues, while the radial solution is the same as the radial wave function of the standard Schroedinger equation. The centrifugal potential in the Schroedinger equation turns out to be always repulsive which implies that there are no bound states for this system of an electron in a magnetic monopole field. However a dyon field does have bound state solutions.

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