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Auxiliary quantization constraints on the von Roos ordering-ambiguity at zero binding energies; azimuthally symmetrized cylindrical coordinates

Using azimuthally symmetrized cylindrical coordinates, we report the consequences of zero-energy quantal states on the von Roos Hamiltonian. A position-dependent mass M({\rho},\phi,z)=bz^{j}{\rho}^{2\u{psion}+1}/2 is used. We show that the zero-energy setting not only offers an additional degree of freedom towards feasible separability for the von Roos Hamiltonian, but also manifestly yields auxiliary quantized ambiguity parametric constraints.

In the current methodical proposal, our motivation is inspired by a purely theoretical mathematical curiosity as to what consequences would emerge at zero-energy (i.e.,E = 0) quantal states for the von Roos Hamiltonian (1). Again, nevertheless, we use cylindrical coordinates in an azimuthally symmetrized setting. We propose a position-dependent mass of yet a more general and mixed dimensional-dependent form M (ρ, ϕ, z) = bz j ρ 2υ+1 /2; b, j, υ ∈ R. We show that such E = 0 setting not only offers an additional degree of freedom towards feasible separability of Hamiltonian (1), but also manifestly yields quantized ordering ambiguity parametric constraints. To the best of our knowledge, such position-dependent mass settings have not been considered elsewhere.
This work is organized as follows. In section II, we recollect the most relevant and necessary equations of [42] (strictly speaking, equations (2), (4), (5), (6), and (9) of [42] summarized in (2)-(7) below, with E = 0, of course). In so doing, we make the current work almost self-contained. In the same section, we report on the separability of (1) as a consequence of E = 0 and provide the corresponding components in the 1D-Schrödinger equation format. We show, in section III, that this choice would introduce quantization recipes on the ordering ambiguity parameters. Therein, we give illustrative examples of different interaction potentials that although they may look complicated with mixed coordinates' dependence, their exact solutions are simple and straightforward. The corresponding wave functions are classified as textbook normalizable wave functions. Our concluding remarks are given in section IV.

II. CYLINDRICAL COORDINATES AT ZERO-ENERGIES AND POWER-LAW PDM
Following closely our recent works [42] on cylindrical coordinates of the PDM-Hamiltonian (1), we again consider the position-dependent mass and the interaction potential to take the forms respectively. We have reported (see Mustafa [42] for more details on this issue) that the corresponding PDM-Schrödinger equation and would (with E = 0 in (11) of [42]) imply Where and Consequently, the zero-energy, assumption secures separability of the von Roos Hamiltonian (1) for some non-zero k (z) = z j of M (ρ, ϕ, z) = g (ρ) f (ϕ) k (z) and hence a more general position-dependent mass form is manifested in the process. That is, our position-dependent mass takes the form (with f (ϕ) = 1, g (ρ) = bρ 2υ+1 /2 and k (z) = z j ) as At this point, one should notice that choosing any other value for E (i.e., E = 0) would make (11) of [42] non-separable under our current methodical proposal settings. Next, we again choose to remain within azimuthal symmetrization settings and consider thatṼ (ϕ) = 0 and f (ϕ) = 1 in (5) to imply thatΦ where m is the magnetic quantum number. Moreover, substituting k (z) = z j in (5) yields that and where an irrational magnetic quantum numberl υ is introduced in the process as and Hereby, F (α, β, γ, j) /z 2 plays the role of a manifestly repulsive and/or attractive force field. It is then convenient to use the assumption that so that the condition F (α, β, γ, j) + 1/4 ≥ 0 serves as an auxiliary constraint on the ambiguity parameters. Moreover, our interaction potential takes the general form In the following section, we use simple illustrative examples so that the message of the current methodical proposal is made clear.

III. E = 0 AND AMBIGUITY PARAMETERS' QUANTIZATION CORRESPONDENCE
In this section, we show that when E = 0, the ordering ambiguity parametric constraints indulge quantization recipes. We use the exactly solvable harmonic oscillator and the Coulomb potentials for constructive illustrations.
A priori, let us provide exact solutions for the z-dependent equation in (9) by recollecting the exact solutions for the harmonic oscillator,Ṽ (z) =ã 2 z 2 /4, and for the Coulombic,Ṽ (z) = −2B/z, potentials. In so doing, we shall introduce an impenetrable infinite wall for all z < 0 and hence work in the upper-half of the cylindrical coordinate system at hand. Mathematically speaking, we proposẽ Under such settings and in a straightforward manner, one obtains for the harmonic oscillator, and k z = ±B (n z + |L| + 1) (17) for the Coulombic interaction, where L is defined in (13). This would immediately suggest that the corresponding wave functions are also well-known exact solutions. They are, no doubt, the standard textbook normalized wave functions of either the harmonic oscillator or the Coulomb models.
In what follows, the constituentsṼ (ρ) andṼ (z) of the interaction potential V (ρ, ϕ, z) in (14) shall be chosen to be simple and exactly solvable (Coulombic and/or harmonic oscillator) in their corresponding radial (10) and/or z-coordinate (9) 1D Schrödinger-like equations. Therefore, the general forms of the corresponding wave functions are exact and textbook normalized wave functions (given by (2) along with (3), and (4)). Four complementary illustrative cases are in order.
Such proposals in (4) and (14) would imply a position-dependent mass M (ρ, ϕ, z) = bz j ρ −1 /2 moving in an interaction potential of the form V (ρ, ϕ, z) = − 2Ã bz j +ã and the z-dependent equation (9) gives Therefore, Which is now the auxiliary quantization constraint on the ambiguity parameters. Therefore, and the auxiliary quantization constraint on the ambiguity parameters reads and an interaction potential Hence, and the auxiliary quantization constraint on the ambiguity parameters is In the four cases discussed above, we observe auxiliary quantization constraints on the ordering ambiguity parameters (documented in the appearance of n ρ , n z , and m quantum numbers in (21), (25), (29), and (33)). That is, for each set of values of n ρ , n z , and m there is a corresponding quantized ordering ambiguity parametric constraint. For example, in Case 1 and for j = 0 in (21), one obtains F (α, β, γ, j) = 0 and where ζ is given in (6). It is obvious here that for every set of values of n ρ , n z , and m the profile of the ordering ambiguity parameters would change and consequently the profile of the kinetic energy operator in (1) would change.
Hence the effective potential of (1) would change its profile too. Moreover, one observes that for n ρ = n z = |m| = 0, this quantized constraint (34) is only satisfied by the set of α = γ = −1/4 and β = −1/2 of Mustafa and Mazharimousavi [19]. Such a result does not make this parametric set as a universally acceptable one, of course. Changing the quantum numbers n ρ , n z , and m would change the profile of the acceptable parametric sets. The same trend may very well be observed in Cases 2, 3, and 4. The ordering ambiguity constraints' quantization correspondence, as an obvious manifestation of E = 0 quantal states, is therefore clear.
In the current methodical proposal, however, we have discussed the consequences of choosing zero-energy states ( i.e., states with E = 0) for our position-dependent mass Hamiltonian (1) under azimuthally symmetrized cylindrical coordinates settings. Moreover, we have used a more general position-dependent mass function, M (ρ, ϕ, z) = bz j ρ 2υ+1 /2; b, j, υ ∈ R. We have shown that the choice of E = 0 setting provides not only an additional degree of freedom towards the feasible separability of Hamiltonian (1), but also manifestly yields quantized ordering ambiguity parametric constraints (documented in (21), (25), (29), (33), and (34)). We have also shown that even with the simplistic choices of the constituent interaction potentialsṼ (ρ) andṼ (z), the overall general interaction potentials, V (ρ, ϕ, z), turned out to be complicated in the sense of indulging mixed coordinates dependence (documented in (18), (22), (26), and (30)). Yet, their exact solutions are simple and straightforward. They are the exact and textbook normalizable wave functions of either the harmonic oscillator or the Coulomb models in (9) and (10). Consequently, the overall general forms of the wave functions are exact, textbook normalizable, and given through (2), (3), and (4).