Non-Central Potentials and Spherical Harmonics Using Supersymmetry and Shape Invariance

It is shown that the operator methods of supersymmetric quantum mechanics and the concept of shape invariance can profitably be used to derive properties of spherical harmonics in a simple way. The same operator techniques can also be applied to several problems with non-central vector and scalar potentials. As examples, we analyze the bound state spectra of an electron in a Coulomb plus an Aharonov-Bohm field and/or in the magnetic field of a Dirac monopole.Comment: Latex, 12 pages. To appear in American Journal of Physic

Semiclassical Approach to Quantum-mechanical Problems with Broken Supersymmetry

The semiclassical WKB approximation method is reexamined in the context of nonrelativistic quantum-mechanical bound-state problems with broken supersymmetry (SUSY). This gives rise to an alternative quantization condition (denoted by BSWKB) which is different from the standard WKB formula and also different from the previously studied supersymmetric (SWKB) formula for unbroken SUSY. It is shown that to leading order in ħ, the BSWKB condition yields exact energy eigenvalues for shape-invariant potentials with broken SUSY (harmonic oscillator, Pöschl-Teller I and II) which are known to be analytically solvable. Further, we show explicitly that the higher-order corrections to these energy eigenvalues, up to sixth order in ħ, vanish identically. We also consider a number of examples of potentials with broken supersymmetry that are not analytically solvable. In particular, for the broken SUSY superpotential W=Ax2d [A\u3e0, d=(integer)], we evaluate contributions up to the sixth order and show that these results are in excellent agreement with numerical solutions of the Schrödinger equation. While the numerical BSWKB results in lowest order are not always better than the corresponding WKB results, they are still a considerable improvement because they guarantee equality of the corresponding energy eigenvalues for the supersymmetric partner potentials V+ and V-. This is of special importance in those situations where these partner potentials are not related by parity

Exactness of supersymmetric WKB spectra for shape-invariant potentials

Potentials which have the property of "shape-invariance" are known to be exactly solvable. For all these potentials, it is proved that the supersymmetric WKB (SWKB) quantization condition (to leading order in h) also has the nice property of reproducing the exact bound-state spectra. These results are not modified even if the next higher-order correction in h is taken into account

Supersymmetry-inspired WKB approximation in quantum mechanics

The supersymmetry-inspired WKB approximation (SWKB) in quantum mechanics is discussed in detail. The SWKB method can be easily applied to any potential whose ground-state wave function is known. It yields eigenvalues that are exact for large quantum numbers n (as any WKB approximation should in the classical limit). Furthermore, for the important special case of "shape-invariant" potentials, the SWKB approach gives the exact analytic expressions for the entire bound-state spectra. A study of some nonshape-invariant, but solvable, potentials suggests that shape invariance is not only sufficient but perhaps even necessary for the SWKB approximation to be exact. A comparison of the WKB and SWKB predictions for the bound-state spectra of a number of potentials reveals that in many cases the SWKB approach does better than the usual WKB approximation

Supersymmetry, shape invariance, and exactly solvable potentials

It is well known that the harmonic oscillator potential can be solved by using raising and lowering operators. This operator method can be generalized with the help of supersymmetry and the concept of "shape-invariant" potentials. This generalization allows one to calculate the energy eigenvalues and eigenfunctions of essentially all known exactly solvable potentials in a simple and elegant manner

Thermodynamics of a free q-fermion gas

We study the thermodynamics of a q-fermion gas for complex values of q on the unit circle. Special emphasis is given to the study of the virial coefficients and the specific heat of this gas. In particular, it is shown that if any state can accommodate up to p q-fermions, then the first p virial coefficients of such a gas are the same as that of a gas of free bosons. Explicit expressions for the deviation of higher virial coefficients from the corresponding values for a Bose gas are obtained. Further, as for ordinary fermions, it is shown that the specific heat of a q-fermion gas at low temperature is proportional to T. Numerical computations show that the derivative of the specific heat as a function of T has no discontinuity