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Time-Independent Schrödinger and Riccati Equations

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Quantum Theory from a Nonlinear Perspective

Part of the book series: Fundamental Theories of Physics ((FTPH,volume 191))

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Abstract

In Sect. 2.11 it has been shown how a generalization of the creation and annihilation operators, known from the algebraic treatment of the harmonic oscillator (HO) problem.

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Notes

  1. 1.

    This nomenclature is somewhat misleading, as W has the dimension of the square-root of energy, not energy like a usual potential.

  2. 2.

    In comparison with a and \(a^+\) as defined in (2.200) and (2.201), a factor \(\sqrt{\frac{1}{\hbar \omega _0}}\) is missing because the definitions in Sect. 2.11 refer to \(\tilde{H}_{\mathrm{op}} = \frac{H_{\mathrm{op}}}{\hbar \omega _0}\). Further, \(B^-\) corresponds to a.

  3. 3.

    Here \(E_0\) is the ground state energy of the conventional solution of the problems.

  4. 4.

    This “construction of families of potentials strictly isospectral to the initial (bosonic) one” can also be interpreted as a “double Darboux general Riccati” transformation of the inverse Darboux type, going in two steps from an initial bosonic to a deformed bosonic system; for details, see [4].

  5. 5.

    Note that the notation in [3] and the one used here differs as the quantities with and without tilde are interchanged.

  6. 6.

    N.B.: The kinetic energy term divided by a is just identical to \(V_{\mathrm{qu}}\)!

  7. 7.

    A similar formulation of the TISE in terms of this equation, but within a different context and with different applications, has also been given in [9]. In another paper [10, 11] the relation between the Ermakov equation (3.39) and the TISE has been extended to also include magnetic field effects and in [11] possible connections between SUSY and Ermakov theories are considered. The NL differential equation (3.39) has also been used to obtain numerical solutions of the TISE for single and double-minimum potentials as well as for complex energy resonance states; for details see [12, 13]. Here, however, we want to concentrate on the similarities between the TD and TI situation, in particular with respect to SUSY.

  8. 8.

    For a survey of Darboux transformations , also in relation to the factorization method and SUSY quantum mechanics, see [4].

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  1. Institut für Theoretische Physik, Goethe-University Frankfurt am Main, Frankfurt am Main, Germany

    Dieter Schuch

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Schuch, D. (2018). Time-Independent Schrödinger and Riccati Equations. In: Quantum Theory from a Nonlinear Perspective . Fundamental Theories of Physics, vol 191. Springer, Cham. https://doi.org/10.1007/978-3-319-65594-9_3

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