Zafar Ahmed

Experts know that bound state energy eigenvalues of a potential well are poles of its transmission amplitude, t ( E ). The textbooks do well by solving the bound states and scattering states separately, but the connection goes unheeded. Here, we present interesting and instructive illustrations, where we extract bound state eigenvalues from the known transmission coefficients ( T ( E ) = | t ( E )| ² ) for six analytically solvable potential wells.

The probability of reflection R ( E ) off a finite attractive scattering potential at zero or low energies is ordinarily supposed to be 1. However, in Ahmed Z et al (2017 Eur. J. Phys. 38 025401) we proposed that a fully attractive potential presents a paradoxical result of R (0) = 0, when an effective parameter q of the potential admits special discrete values. In an interesting comment to Ahmed et al (2017 Eur. J. Phys. 38 025401); by ⁵ van Dijk and Nogami (2017 Eur. J. Phys. 38 038002), another class of finite potentials turned out to be of the well–barrier type, which can be made to possess much less reflection at zero and low energies for a band of low values of q . Such well–barrier potentials have only two real turning points for E ∈ (min ( V ( x )), max ( V ( x ))), excepting E = 0. Here, we wish to add one more exactly solvable and two numerically solved models to confirm this phenomenon in the well–barrier systems.

In an earlier article ( Eur. J. Phys. 39 055402), potential wells with discontinuous jump have been found to have the expectation value 〈 p 6 〉 to be divergent for all bound states. Here in this addendum, we prove and demonstrate that for continuous but non-differentiable wells it diverges for even states and converges for odd states; here, p denotes momentum. We present three exactly solvable models.

Historically, most of the quantum mechanical results have originated in one dimensional model potentials. However, Von-Neumann’s Bound states in the Continuum (BIC) originated in specially constructed, three dimensional, oscillatory, central potentials. One dimensional version of BIC has long been attempted, where only quasi-exactly-solvable models have succeeded but not without instigating degeneracy in one dimension. Here, we present an exactly solvable bottomless exponential potential barrier V ( x ) = − V 0 [ exp ( 2 ∣ x ∣ / a ) − 1 ] which for E < V ₀ has a continuum of non-square-integrable, definite-parity, degenerate states. In this continuum, we show a surprising presence of discrete energy, square-integrable, definite-parity, non-degenerate states. For E > V ₀ , there is again a continuum of complex scattering solutions ψ ( x ) whose real and imaginary parts though solutions of Schrödinger equation yet their parities cannot be ascertained as Cψ ( x ) is also a solution where C is an arbitrary complex non-real number.