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Shape invariance and the exactness of quantum Hamilton-Jacobi formalism
Quantum Hamilton-Jacobi Theory and supersymmetric quantum mechanics (SUSYQM)
are two parallel methods to determine the spectra of a quantum mechanical
systems without solving the Schr\"odinger equation. It was recently shown that
the shape invariance, which is an integrability condition in SUSYQM formalism,
can be utilized to develop an iterative algorithm to determine the quantum
momentum functions. In this paper, we show that shape invariance also suffices
to determine the eigenvalues in Quantum Hamilton-Jacobi Theory.Comment: Accepted for publication in Phys. Lett.
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Quasilinearization Method and Summation of the WKB Series
Solutions obtained by the quasilinearization method (QLM) are compared with
the WKB solutions. Expansion of the -th QLM iterate in powers of
reproduces the structure of the WKB series generating an infinite number of the
WKB terms with the first terms reproduced exactly. The QLM quantization
condition leads to exact energies for the P\"{o}schl-Teller, Hulthen,
Hylleraas, Morse, Eckart potentials etc. For other, more complicated potentials
the first QLM iterate, given by the closed analytic expression, is extremely
accurate. The iterates converge very fast. The sixth iterate of the energy for
the anharmonic oscillator and for the two-body Coulomb Dirac equation has an
accuracy of 20 significant figures
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