Abstract
Traditional first order JWKB method ( = : ( J W K B ) 1 ) is a conventional semiclassical approximation method mainly used in quantum mechanical systems for accurate solutions. ( J W K B ) 1 general solution of the Time Independent Schrodinger’s Equation (TISE) involves application of the conventional asymptotic matching rules to give the accurate wavefunction in the Classically Inaccessible Region (CIR) of the related quantum mechanical system. In this work, Bessel Differential Equation of the first order ( = : ( B D E ) 1 ) is chosen as a mathematical model and its ( J W K B ) 1 solution is obtained by first transforming into the normal form via the change of independent variable. The ( J W K B ) 1 general solution for appropriately chosen initial values in both normal and standard form representations is analyzed via the generalized ( J W K B ) 1 asymptotic matching rules regarding the S ˜ i j matrix elements given in the literature. Instead of applying the common ( J W K B ) 1 asymptotic matching rules relying on the physical nature of the quantum mechanical system, i.e., a physically acceptable (normalizable) wavefunction, a pure semiclassical analysis is studied via the ( B D E ) 1 model mathematically. Finally, an application to a specific case of the exponential potential decorated quantum mechanical bound state problem is presented.
Highlights
The (JWKB)1 method (we refer to the “n-th order JWKB” by a simple abbreviation: (JWKB)n) is conventionally known to be a strong and effective semiclassical approximation method enabling accurate analytical solutions in quantum mechanical systems, i.e., [1,2,3,4,5,6,7,8,9,10]
In the (JWKB)1 calculations, the entire domain can be considered as a unification of two neighboring regions (CAR-Classically Inaccessible Region (CIR)), and if we start with the Classically Accessible Region (CAR) located at the left-hand side of the turning point and connect it to the CIR by using the conventional (JWKB)1 connection formulas given in [3,4,5] in the reverse direction, we find the same formulas for yL(c, ρ) and yR(c, ρ) as in ([4]) (but in (c, ρ) rather than (c, x)) to give: y(c, ρ) =
The (BDE)1 has been chosen as a mathematical model for a pure semiclassical analysis since there exist subdomains where the (JWKB)1 applicability criterion both holds and fails
Summary
The (JWKB) method (we refer to the “n-th order JWKB (or WKB)” by a simple abbreviation: (JWKB)n) is conventionally known to be a strong and effective semiclassical approximation method enabling accurate analytical solutions in quantum mechanical systems, i.e., [1,2,3,4,5,6,7,8,9,10]. Our interest here can be summarized as follows: (i) to find the (JWKB) general solution of the (BDE) whose structure is given in (2b) by using some appropriate change of variable to transform into a normal form (which is not unique); (ii) to check its accuracy in the (sub)domains of the CAR and the CIR where the (JWKB) applicability criterion in (6) holds; and (iii) to find ways to do the correct asymptotic matching in the necessary (sub)domains by semiclassical analyses mathematically only. Since our asymptotic matching rule gives a criterion for a semiclassically (not physically) acceptable (JWKB) solution, we have the following semiclassical outcomes: (i) why (JWKB) solutions are accurate in some domains in the CAR (SD1 ∈ CAR) and why they give inaccurate results in some other domains in the CIR (SD1 ∈ CIR); (ii) which complementary function in the general solution in (2b) (and in the corresponding transformed representation) should be canceled in order to give an accurate general solution as desired by a successful asymptotic matching. Preliminary work regarding the part “Calculations in The Normal Form” of Section 3 was discussed in the 19th International Conference on Applied Mathematics (AMATH’14)-Istanbul where some 2D analyses are available in [15]. (Note that Equations: (40) and (41a)–(41b) here are in the corrected form when compared with the misprints in [15])
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