The use of complex variables and conformal mapping have long been useful tools in many branches of physics. However, these methods have not been widely applied in mechanics and elementary quantum mechanics. In this paper, it is shown that there are many useful applications of complex variables and conformal mapping in these subjects. It will be proven that central force problems in the plane have conformal duals. The inverse fifth power law force is self-dual in all dimensions. These results are extended to noncentral forces. Many of these results also apply to the quantum mechanics of a particle. For central force problems and certain noncentral force problems in the plane the Schrödinger equation preserves its form under conformal mapping and the inverse fifth power law force is again self-dual in all dimensions. The methods may also be applied in the presence of magnetic fields.The use of complex variables and conformal mapping have long been useful tools in many branches of physics. However, these methods have not been widely applied in mechanics and elementary quantum mechanics. In this paper, it is shown that there are many useful applications of complex variables and conformal mapping in these subjects. It will be proven that central force problems in the plane have conformal duals. The inverse fifth power law force is self-dual in all dimensions. These results are extended to noncentral forces. Many of these results also apply to the quantum mechanics of a particle. For central force problems and certain noncentral force problems in the plane the Schrödinger equation preserves its form under conformal mapping and the inverse fifth power law force is again self-dual in all dimensions. The methods may also be applied in the presence of magnetic fields.

American Journal of PhysicsMar 1, 199220 citations

The Inverse-Cube Central Force Field in Quantum Mechanics

George H. Shortley

The problem of the motion of a particle in an inverse-cube central force field is fully treated by quantum mechanics and the results compared with the classical theory. Taking the effective radial potential energy as $\frac{S}{{r}^{2}}$ , although the solutions for negative energy for $0\ensuremath{\geqq}S\ensuremath{\geqq}\frac{\ensuremath{-}{h}^{2}}{32{\ensuremath{\pi}}^{2}\ensuremath{\mu}}$ satisfy the usual boundary conditions, they can not be admitted because the Hamiltonian is not Hermitian in these solutions. This corresponds to taking ${(l+\frac{1}{2})}^{2}$ in place of $l(l+1)$ as the analogue of the square of the classical angular momentum. If we do this, we get a complete analogy between the classical and quantum mechanically allowed solutions, with no quantization. The solutions involve Bessel functions of both real and imaginary orders with both real and imaginary arguments.The problem of the motion of a particle in an inverse-cube central force field is fully treated by quantum mechanics and the results compared with the classical theory. Taking the effective radial potential energy as $\frac{S}{{r}^{2}}$, although the solutions for negative energy for $0\ensuremath{\geqq}S\ensuremath{\geqq}\frac{\ensuremath{-}{h}^{2}}{32{\ensuremath{\pi}}^{2}\ensuremath{\mu}}$ satisfy the usual boundary conditions, they can not be admitted because the Hamiltonian is not Hermitian in these solutions. This corresponds to taking ${(l+\frac{1}{2})}^{2}$ in place of $l(l+1)$ as the analogue of the square of the classical angular momentum. If we do this, we get a complete analogy between the classical and quantum mechanically allowed solutions, with no quantization. The solutions involve Bessel functions of both real and imaginary orders with both real and imaginary arguments.

Physical ReviewJul 1, 193153 citations

Green's functions in quantum mechanics courses

William J. Herrera, Herbert Vinck-Posada, Shirley Gómez Páez

The use of Green's functions is valuable when solving problems in electrodynamics, solid-state physics, and many-body physics. However, its role in quantum mechanics is often limited to the context of scattering by a central force. This work shows how Green's functions can be used in other examples in quantum mechanics courses. In particular, we introduce time-independent Green's functions and the Dyson equation to solve problems with an external potential. We calculate the reflection and transmission coefficients of scattering by a Dirac delta barrier and the energy levels and local density of states of the infinite square well potential.The use of Green's functions is valuable when solving problems in electrodynamics, solid-state physics, and many-body physics. However, its role in quantum mechanics is often limited to the context of scattering by a central force. This work shows how Green's functions can be used in other examples in quantum mechanics courses. In particular, we introduce time-independent Green's functions and the Dyson equation to solve problems with an external potential. We calculate the reflection and transmission coefficients of scattering by a Dirac delta barrier and the energy levels and local density of states of the infinite square well potential.

arXiv:2107.14104American Journal of PhysicsSep 22, 20226 citations

\frac{S}{{r}^{2}}

, although the solutions for negative energy for $0\ensuremath{\geqq}S\ensuremath{\geqq}\frac{\ensuremath{-}{h}^{2}}{32{\ensuremath{\pi}}^{2}\ensuremath{\mu}}$ satisfy the usual boundary conditions, they can not be admitted because the Hamiltonian is not Hermitian in these solutions. This corresponds to taking

{(l+\frac{1}{2})}^{2}

in place of

l(l+1)

as the analogue of the square of the classical angular momentum. If we do this, we get a complete analogy between the classical and quantum mechanically allowed solutions, with no quantization. The solutions involve Bessel functions of both real and imaginary orders with both real and imaginary arguments.