Quaternionic Elastic Scattering

Home , Delta potential

arXiv:2011.05743v1 [quant-ph] 11 Nov 2020 art ikof[] hl rs teklegltrachie later quest Stueckelberg this Ernst t fact, while this ( In [1], Would speculation? Birkhoff nonsense field. Garrett a number wh simply ask another or may by theory one However, complexes space. the Hilbert replacing complex a in functions ∗ theory a such of definition precise The mechanics. quantum a i of As space and theory. Hilbert obtained, collision real also stationary the the is and theorem quintess matrix optical scattering section quaternionic cross A a case. admits function hypot wave the contradicts quaternionic potentials p symmetric the in spherically results, of these Supporting 21]. and [20, 48, anti-hermitian [25, version, anti-hermitian this non-relativistic contradict as such results giv systems, Recent to supposed 47]. initially [6, textbooks, was theory that several territory in unexplored described and currently and established results. their of d terms a in from powerful scattering space the Hilbert and fr real [41] the the expli law [42], Lorentz novel effect quantum Furthermore, Aharonov-Bohm the the [45], [41]. u as enabled theorem such approach approach, Virial This this the [40]. of space proof Hilbert a real a using [39] als considered We be [7–27]. cannot that kind mechanics this quantum of in results several quote we anti-hermitian example, Furthermore, sho several limit. presents proposal anti-hermitian summar the is However, and operators, Hamiltonian anti-hermitian using space Ê lcrncades [email protected] address: Electronic h ahmtclfaeoko unu ehnc sbsdon based is mechanics quantum of framework mathematical The oercnl,anvlapoc to approach novel a recently, More nti ril eso htteqatmeatcsatrn c scattering elastic quantum the that show we article this In M 25,wihi osdrdeuvln to equivalent considered is which [2–5], QM) À M eutta a ee banduigteanti-hermitia the using obtained never was that result a QM, eatmnod ae´tc uaeAlcd,Universidade Aplicada, e Matemática Pura de Departamento esuyteeatcsatrn fqatmprilsbsdo based particles ( quantum mechanics of quantum scattering quaternionic elastic to the study We ASnumbers: PACS ( mechanics quantum complex ncmlxqatmmcaisrifretevldt fthe of validity h a the of result reinforce case quaternionic mechanics the new quantum for these complex theorem between in optical agreement the strong as The well as shifts, phase vnd et o¸avs90,CiaPsa 58,91501-9 15080, Postal Caixa 9500, Gon¸calves Bento Avenida Q À À Mi hsclycnitn nahge ee hnteanti-H the than level higher a in consistent physically is QM Ma hsclydfeetfrom different physically as QM AENOI LSI SCATTERING ELASTIC UATERNIONIC À À Msltosaefw novd n ifiutt nepe.B interpret. to difficult and involved, few, are solutions QM C Meiiae h nihriiiyrqieetfrteHam the for requirement anti-hermiticity the eliminated QM QM) EGOGIARDINO SERGIO I. À M n eieepeso o h aefnto,the function, wave the for expression derive and QM) INTRODUCTION À C M utrincqatmter a ae developed later was theory quantum quaternionic A QM. M uha [28–38]. as such QM, 9,nnrltvsi o-emta 4]adrelativisti and [46] non-Hermitian non-relativistic 49], eetpprw hwta h utrincscattering quaternionic the that show we paper resent ei fa smttclycomplex asymptotically an of hesis C utrincefcswr setie navreyof variety a in ascertained were effects quaternionic toig,ms motnl h l-endclassical ill-defined the importantly most rtcomings, qiaetrslswe oprdt h complex the to compared when results equivalent e M n o ipyamteaia generalization mathematical a simply not and QM, ocuigrmr,orrslsspotteve of view the support results our remark, concluding hl h utrincsatrn otisawide a contains scattering quaternionic the while e h eeomn fra unu mechanics quantum real of development the ved zdi opeesv okb tpe de [6]. Adler Stephen by book comprehensive a in ized nilydfeetfo htfudi h complex the in found that from different entially on u h xsec fqaencapplications quaternic of existence the out point o epril 4,4] h qaeptnilsolutions potential square the 44], [43, particle ee i ouin of solutions cit a aua otniyi h eerho the of research the in continuity natural a has t o a rtyakdb onvnNuanand Neumann von John by asked firstly was ion ooti eldfie lsia ii 3]and [39] limit classical well-defined a obtain to s sa xiigdrcinfrftr research. future for direction exciting an is lafnto 4] ecncndnl tt that state confidently can We [46]. function elta twudhpe oqatmmcaisafter mechanics quantum to happen would at eea oRoGad oSl(UFRGS) Sul do Grande Rio do Federal er ihrb eeaiain different a generalization, a be either heory nb bandi em ftera Hilbert real the of terms in obtained be an ∗ h ope ubrfil n ope wave complex and field number complex the À elHletsaeapproach space Hilbert real a n n h orsodn results corresponding the and s r peesatrn potential. scattering sphere ard hoy h cteigin scattering The theory. n Mgnrlzto fordinary of generalization QM 0ProAer,R,Brazil RS, Alegre, Porto 70 À Mwr undpsil using possible turned were QM ermitian À À Mbcuethe because QM M n more and QM, C Mi well is QM a of way y iltonian c 2

II. WAVE FUNCTION À Before considering the proposals to ÀQM, we notice that a comprehensive introduction to quaternions ( ) is beyond the scope of this article, and can be found elsewhere [50–52]. We simply remark that quaternions are non-commutative hyper-complex numbers that encompass three imaginary units, i, j and k, and that a general quaternion q is such that

2 2 2

q = x0 + x1i + x2j + x3k, where x0, x1, x2, x3 ∈ Ê, and i = j = k =−1. (1)

Quaternions are non-commutative because the imaginary units are anti-commutative and satisfy, by way of example, ij =−ji. In this article we adopt the symplectic notation for quaternions, where

q = z0 + z1j, so that z0 = x0 + x1i and z1 = x2 + x3i. (2)

The quaternionic non-commutativity is algebraic and global, something very different from the local and ge- ometric non-commutativity of the space-time coordinates in the formulation of Alain Connes [53]. An example of application of non-commutative geometry in physics with several references can be found in [54]. One

could naively expect that ÀQM generalized quantum mechanics by introducing an additional complex degree of freedom, and consequently increasing the range of physical phenomena that can be described. However, the anti-commutativity constrains the theory, and the net balance of the introduction of quaternions in quantum mechanics is still an open question. Let us then consider the quaternionic Schrodinger¨ equation

h¯ 2 ∂Ψ − ∇2 + V Ψ = h¯ i. (3) " 2m # ∂t where V = V(x) is a quaternionic potential and Ψ = Ψ(x, t) is a quaternionic wave function. We point out that the the imaginary unit i multiplies the right hand side of the time derivative of the wave function. Proceeding

in analogy to CQM [55], we consider a two particle scattering problem and rewrite (3) as two equations. One equation describes the motion of the center of mass with energy Ec, and another equation describes the relative motion between the two particles with energy E. If E is the total energy of the system, the wave function is

−i E t Ψ = w u e h¯ and E = Ec + E, (4) where u = u(x) and w = w(x) are quaternionic functions and a comprehensinve solution of the time-dependent equation can be found in [43, 44]. Separating the functions, the time independent Schrodinger¨ equation gives

2 2 h¯ 2 h¯ 2 − ∇ + V u = Eu, and − ∇ w = Ecw. (5) " 2µ # 2m

The center of mass behaves as a free particle of total mass m = m1+m2 and wave function w, and the scattering is described by the wave function u, where µ is the reduced mass of the system. Our aim is to determine the cross section for a cylindrically symmetric potential V in the center of mass coordinate system, where r is the distance between the two particles. Let us recall that in the complex case the general solution of (5) is

∞ ℓ

uC (r, θ)= 2ℓ + 1 Rℓ(r)(i) Pℓ(cos θ) (6) Xℓ=0 where Pℓ is a Legendre polynomial and Rℓ depends on spherical Bessel functions [55]. Our aim is to propose a quaternionic solution that generalizes (6). Using the symplectic decomposition (2), we rewrite (6) by decompos- (0) (1) (0) (1) ing Rℓ as follows Rℓ = Rℓ + Rℓ j, where Rℓ and Rℓ are complex functions. In the quantum complex case, the asymptotic behavior is expected to be

eikr ∝ ikz uC (r , θ) C0e + f(θ) , (7) r → ∞ where C0 is a complex constant, f(θ) is a complex function, k = µv/ℏ, and v is a velocity ascribed to the reduced mass. The complex exponential has to be expanded in Legendre polynomials in order to be compared with (6). 3

In the quaternionic case, we propose a series using quaternionic coefﬁcients and replace the complex function f(θ) with a quaternionic function F(θ), so that

∞ ikr 1 ℓ ℓπ e u(r , θ)= Λℓ(2ℓ + 1)(i) sin kr − Pℓ(cos θ) + F(θ) , (8) kr 2 r Xℓ=0 → ∞ where Λℓ are the unitary quaternionic coefﬁcients

iδℓ iξℓ Λℓ = cos Θℓ e + sin Θℓ e j, so that ΛℓΛℓ = 1. (9)

The complex result is recovered for Λℓ = 1 and F(θ) f(θ). The incident quaternionic particle is consequently a linear combination of quaternionic free particles that generalizes the ﬁrst term of (7). We obtain quaternionic scattering solutions from (6) using asymptotic quaternini→ c expansions for Rℓ in and imposing the equality with (8). Let us then assume the asymptotic expansion

1 ℓπ ℓπ i(kr− 2 ) −i(kr− 2 ) Rℓ ∝ Λℓ e − Λℓ e , (10) 2kr h i and consequently (6) becomes ∞ 1 ℓπ ℓπ iξℓ ℓ u(r, θ) = (2ℓ + 1)Aℓ i cos Θℓ sin kr − + δℓ + sin Θℓ cos kr − e j (i) Pℓ(cos θ), (11) kr 2 2 Xℓ=1 where also Aℓ is quaternionic and the complex case is recovered after imposing Θℓ = 0. Following [45], we interpret Θℓ a polarization angle that combines the pure complex and the pure quaternionic components of the quaternion basis elements, which are orthogonal [39, 40]. The asymptotic expressions (8) and (11) must be compatible, and the equality of both of the expressions give

Aℓ =−ΛℓiΛℓ, (12) and consequently ∞ 1 2 F(θ)= (2ℓ + 1) Λℓ i 1 − Λ Pℓ(cos θ). (13) 2k ℓ Xℓ=0 We can now study several features of this wave function. The cross-section is deﬁned by σ(θ)= |F(θ)|2 and the total cross section σ is obtained from the integral over the scattering angle, so that σ = σ(θ)dΩ. Using the orthogonality conditions for Legendre polynomials, we get R ∞ 4π 2 2 2 σ = 2ℓ + 1) sin δℓ cos Θℓ + sin Θℓ . (14) k2 Xℓ=0 In agreement with the complex case, a ﬁnite cross section requires evanescent phase and polarization angles, so that δℓ, Θℓ 0, when ℓ . Additionally, we recover the complex cross section in the Θℓ = 0 complex limit, leading to a complete agreement between the complex and quaternionic cases. Conversely, the δℓ = 0

limit generates→ a new quaternionic→ ∞ solution whose cross section reproduces the complex result. However, the C important physical interpretation is that the ÀQM predicts a larger cross section when compared to QM, and thus we expect a more intense scattering in the quaternionic case The phase shifts δℓ and the polarization angles Θℓ also ascertain the difference between the asymptotic form of the wave function and the exact solution of the Schrodinger¨ equation (5). The scalar potential V plays an important role deﬁning a radial distance r = a that determines the region of the space where V(r>a) ≈ 0 and the asymptotic approximation holds. By analogy to the complex case, we estimate the phase shifts imposing that the continuity of the radial wave function Rℓ at r = a depends on a quaternionic constant Γℓ, so that

1 dRℓ (0) (1) (0) (1)

= Γℓ where Γℓ = Γℓ + Γℓ j, and Γℓ ,Γℓ ∈ C. (15) Rℓ dr r=a

Recalling the wave function (11) as the asymptotic limit of

iξℓ Rℓ(r)= Aℓ i cos Θℓ cos δℓ jℓ(kr) − sin δℓ yℓ(kr) − sin Θℓ e yℓ(kr) j , (16) h i 4

′ ′ and using jℓ, jℓ, yℓ and yℓ for spherical Bessel functions and their derivatives at r = a, we eventually obtain ′ 2 ′ ′ yℓy tan Θℓ + y sin δℓ + j cos δℓ jℓ cos δℓ − yℓ sin δℓ (0) ℓ ℓ ℓ Γℓ = k 2 . (17) 2 2 yℓ tan Θℓ − yℓ sin δℓ − jℓ cos δℓ We remember that (17) recovers the complex result for Θℓ = 0, as expected. Furthermore, we also obtain (0) ′ 1 Γ yℓ − ky π ( ) ℓ ℓ i(ξℓ+ 2 ) Γℓ = tan Θℓ e , (18) cos δℓ jℓ − sin δℓ yℓ (0) which accordingly disappears for Θℓ = 0 as expected. We ﬁnally observe that Γℓ is a real quantity, in further (1) agreement with the complex scattering, while Γℓ is indeed complex. On the other hand, we also have that 2 (1) ′ ′ Γℓ yℓ jℓ − Γ0jℓ − kjℓ Γ0yℓ − k yℓ tan δℓ = − . (19) 2 2 (1) 2 ′ Γℓ yℓ − Γ0yℓ − k yℓ (1)

C | | The QM result is accordingly recovered for Γ ℓ = 0, as desired. This analysis permit us to be sure that the asymptotic limit can be obtained continuously, without the risk of singular points and exotic unphysical situations.

III. THE RIGID SPHERE

In this section, we examine the simplest example for the quaternionic scattering: the perfectly spherical rigid potential. In spherical coordinates, for r

The phase angle δℓ is absolutely identical to the complex case. Adopting the low energy limit, where kR ≪ 1, then (kR)2ℓ+1 tan δℓ =− . (22) 2ℓ + 1 !! 2ℓ − 1 !! Additionally, the ﬁniteness of the pure quaternionic component is obtained from 1 (kR)ℓ+1 sin Θℓ = ≈ − . (23) yℓ(kR) 2ℓ − 1 !! Consequently, both of the phase shifts rapidly go to zero with increasing ℓ. Considering the ℓ = 0 component of the total cross section (14) and using sin δ0 =− sin Θ0 = kR , we obtain 1 σ ≈ 8πR2 1 − k2R2 , (24) 2

and the result is almost twice the CQM result. In high energy limit, where kR ≫ 1, we use (21) to obtain ∞ 2 2 2 4π j (kR) + sin Θℓ y (kR) σ = 2ℓ + 1) ℓ ℓ , (25) k2 j2(kR)+ y2(kR) Xℓ=0 ℓ ℓ a result that recovers the complex result for Θℓ = 0, as desirable. 5

IV. THEOPTICALTHEOREM

In a region without sources or sinks of probability, we expect that

J · dσ = 0, (26) IS where J is the probability current density and S is a closed surface. We remember the probability current

density in real Hilbert space ÀQM [39, 40] to be deﬁned as

1 J = Ψ ΠΨ + ΠΨ Ψ , where ΠΨ =−h¯ ∇Ψ i. (27) 2m h i Deﬁning (8) to be the wave function and the sphere to be the closed surface, from (26) and σ = |F|2dΩ we obtain R 1 ∂I 1 ∂I 1 σ =− r2 I¯ i + e−ikr F¯ i − k I¯ F eikr − IFie¯ ikr + c.c. dΩ (28) 2k I ∂r r ∂r r2 where I = I(r, θ) corresponds to the incident particle element of the wave function (8), namely

∞ 1 ℓ ℓπ I(r,θ) = Λℓ(2ℓ + 1)(i) sin kr − Pℓ(cos θ). (29) kr 2 Xℓ=0 Using the orthogonality of Legendre polynomials,

π 2 Pℓ(cos θ)Pm(cos θ) sin θdθ = δℓm (30) Z0 2ℓ + 1 we obtain that ∂I ∂I¯ r2 I¯ i − i I dΩ = 0. (31) I ∂r ∂r We also observe that the term |F|2 cancels because of the complex conjugation. Furthermore, using

π Pℓ(cos θ) sin θdθ = δℓ0 (32) Z0 and integration by parts, we only keep the ℓ = 0 terms in (29) because t The trigonometric terms highly oscillate in the limit of large r and does not contribute to the result. Thus, 1 σ = iF − Fi¯ dΩ 2 I π = 2π Im iF sin θdθ. (33) Z0 The above result is similar to the complex case, where the cross section depends linearly to the imaginary component of the complex amplitude |f(θ)|. The quaternionic cross section agrees to the continuity equation for quaternionic scalar potentials [39, 40], where the imaginary unit i, associated to momentum and energy operators, is physically more relevant than the purely quaternionic imaginary units. An integrated expression for the optial theorem is an important direction for future research.

V. CONCLUSION

In this article we presented a quaternionic version for the elastic scattering, and we obtained a sound agreement with the complex results. This concordance reinforces the hypothesis for a viable quaternionic quantum mechanics in real Hilbert space. Directions for future research include the interesting phenomena of inelastic 6

scattering. From a theoretical standpoint, real Hilbert space ÀQM seems mature to be applied in the exciting area of non-hermitian complex quantum mechanics, that much developed in recent times and was never studied in quaternionic terms.

[1] J. von Neumann; G. Birkhoff. “The logic of quantum mechanics”. Ann. Math. 37, 823-843 (1936). [2] E. C. G. Stueckelberg. “Quantum theory in real Hilbert Space”. Helv. Phys. Acta 33, 752 (1960). [3] E. C. G. Stueckelberg. “Quantum theory in real Hilbert Space II: Addenda and errats”. Helv. Phys. Acta 34, 628 (1961). [4] E. C. G. Stueckelberg. “Quantum theory in real Hilbert Space III: Fields of the ﬁrst kind”. Helv. Phys. Acta 34, 698 (1961). [5] E. C. G. Stueckelberg. “Theorie´ des quanta dans l’espace de Hilbert reel.´ IV, Champs de deuxieme` espece”.` Helv. Phys. Acta 35, 695 (1962). [6] S. L. Adler. “Quaternionic Quantum Mechanics and Quantum Fields”. Oxford University Press (1995). [7] A. J. Davies; B. H. J. McKellar. “Nonrelativistic quaternionic quantum mechanics in one dimension”. Phys. Rev., A40:4209–4214, (1989). [8] A. J. Davies; B. H. J. McKellar. “Observability of quaternionic quantum mechanics”. Phys. Rev., A46:3671– 3675, (1989). [9] S. De Leo; G. Ducati. “ Quaternionic differential operators”. J. Math. Phys, 42:2236–2265, (2001). [10] S. De Leo; G. Ducati; C. Nishi. “ Quaternionic potentials in non-relativistic quantum mechanics”. J. Phys, A35:5411–5426, (2002). [11] S. De Leo; G. Ducati. “ Quaternionic bound states”. J. Phys, A35:3443–3454, (2005). [12] S. De Leo; G. Ducati; T. Madureira. “Analytic plane wave solutions for the quaternionic potential step”. J. Math. Phys, 47:082106–15, (2006). [13] S. De Leo; G. Ducati. “ Quaternionic wave packets”. J. Math. Phys, 48:052111–10, (2007). [14] A. J. Davies. “Quaternionic Dirac equation”. Phys.Rev., D41:2628–2630, (1990). [15] S. De Leo; S. Giardino. “Dirac solutions for quaternionic potentials”. J. Math. Phys., 55:022301–10, (2014) arXiv:1311.6673[math-ph]. [16] S. De Leo; G. Ducati; S. Giardino. “Quaternioninc Dirac Scattering”. J. Phys. Math., 6:1000130, (2015) arXiv:1505.01807[math-ph]. [17] S. Giardino. “Quaternionic particle in a relativistic box”. Found. Phys., 46(4):473–483, (2016) arXiv:1504.00643[quant-ph]. [18] H. Sobhani; H. Hassanabadi. “Scattering in quantum mechanics under quaternionic Dirac delta potential”. Can. J. Phys., 94:262–266, (2016). [19] L. M. Procopio; L. A. Rozema; B. Dakic;´ P. Walther. Comment on “Peres experiment using photons: No test for hypercomplex (quaternionic) quantum theories”. Phys. Rev., A96(3):036101, (2017). [20] H. Sobhani; H; Hassanabadi; W. S. Chung. “Observations of the Ramsauer-Townsend effect in quaternionic quantum mechanics”. Eur. Phys. J., C77(6):425, (2017). [21] H. Hassanabadi; H. Sobhani; A. Banerjee. “Relativistic scattering of fermions in quaternionic quantum mechanics”. Eur. Phys. J., C77(9):581, (2017). [22] H. Hassanabadi, H. Sobhani; W. S. Chung. “Scattering study of fermions due to double Dirac delta potential in quaternionic relativistic quantum mechanics”. Adv. High Energy Phys., 2018:8124073, 2018. [23] H. Sobhani; H; Hassanabadi. “New face of Ramsauer-Townsend effect by using a quaternionic double Dirac potential”. Indian J. Phys., 91(1):1–5, (2017). [24] P. A. Bolokhov. “Quaternionic wave function”. Int. J. Mod. Phys., A34(02):1950001, (2019). [25] S. De Leo; C A. A. de Souza; G. Ducati. “Quaternionic perturbation theory”. Eur. Phys. J. Plus, 134(3):113, (2019). [26] B. Muraleetharan; K. Thirulogasanthar. “Coherent state quantization of quaternions”. J. Math. Phys., 56(8):083510, (2015). [27] B. Muraleetharan; K. Thirulogasanthar; I. Sabadini. “A representation of Weyl–Heisenberg Lie algebra in the quaternionic setting”. Ann. Phys., 385:180–213, (2017). [28] A. I. Arbab. “The Quaternionic Quantum Mechanics”. Appl. Phys. Res., 3:160–170, (2011). [29] M. Kober. “Quaternionic quantization principle in general relativity and supergravity”. Int. J. Mod. Phys., A31(04n05):1650004, (2016). [30] D. C. Brody; E.-V. Graefe. “Six-dimensional space-time from quaternionic quantum mechanics”. Phys. Rev., D84:125016, (2011). [31] S. B. Tabeu; F. Fotsa-Ngaffo; A. Kenfack-Jiotsa. “Non-Hermitian Hamiltonian of two-level systems in complex quaternionic space: An introduction in electronics”. EPL, 125(2):24002, (2019). [32] B. C. Chanyal. “A relativistic quantum theory of dyons wave propagation”. Can. J. Phys., 95(12):1200–1207, (2017). [33] B. C. Chanyal. “A new development in quantum ﬁeld equations of dyons”. Can. J. Phys., 96(11):1192–1200, (2018). [34] B. C. Chanyal. “Quaternionic approach on the Dirac-Maxwell, Bernoulli and Navier-Stokes equations for dyonic 7

fluid plasma”. Int. J. Mod. Phys. A, 34(31):1950202, (2019). [35] B. C. Chanyal; S. Karnatak. “A comparative study of quaternionic rotational Dirac equation and its interpretation”. Int. J. Geom. Meth. Mod. Phys., 17(02):2050018, (2020). [36] M. Cahay; G. B. Purdy; D. Morris. “On the quaternion representation of the Pauli spinor of an electron”. Phys. Scripta, 94(8):085205, (2019). [37] M. Cahay; D. Morris. “On the quaternionic form of the Pauli-Schroedinger equation”. Phys. Scripta, 10.1088/1402- 4896/ab47d2 (2019). [38] S. Marques-Bonham; B. C. Chanyal; R. Matzner. “Yang–Mills-like field theories built on division quaternion and octonion algebras”. Eur. Phys. J. Plus, 135(7):608, (2020). [39] S. Giardino. “Non-anti-hermitian Quaternionic Quantum Mechanics”. Adv. Appl. Clifford Algebras, 28(1):19, (2018). [40] S. Giardino. “Quaternionic quantum mechanics in real Hilbert space”. J. Geom. Phys. (accept) arXiv:1803.11523[math-ph]. [41] S. Giardino. “Virial theorem and generalized momentum in quaternionic quantum mechanics”. Eur. Phys. J. Plus, 135(1):114, (2020). [42] S. Giardino. “Quaternionic Aharonov-Bohm effect”. Adv. Appl. Cliff. Alg. 27 2445–2456 (2017) arXiv:1606.08486 [quant-ph]. [43] S. Giardino. “Quaternionic quantum particles”. Adv. Appl. Clifford Algebras, 29(4):83, (2019). [44] S. Giardino. “Quaternionic quantum particles: new solutions”. Can. J. Phys. (accept) arXiv:1706.08370[quant-ph]. [45] S. Giardino. Square-well potential in quaternic quantum mechanics. EPL (accept) arXiv[quant-ph]:2009.08237 (2020). [46] M. Hasan; B. P. Mandal. “New scattering features of quaternionic point interaction in non-Hermitian quantum mechanics”. J. Math. Phys., 61(3):032104, (2020). [47] S. L. Adler. “Scattering and Decay Theory for Quaternionic Quantum Mechanics and the Structure of Induced T Nonconservation”. Phys. Rev., D37:3654, (1988). [48] L. P. Horwitz. “A Soluble model for scattering and decay in quaternionic quantum mechanics. 1. Decay”. J. Math. Phys., 35:2743–2760, (1994). [49] L. P. Horwitz. “A Soluble model for scattering and decay in quaternionic quantum mechanics. 2. Scattering”. J. Math. Phys., 35:2761–2771, (1994). [50] J. P. Ward. “Quaternions and Cayley Numbers”. Springer Dordrecht (1997). [51] J. Vaz; R. da Rocha . “An Introduction to Clifford Algebras and Spinors”. Oxford University Press (2016). [52] J. P. Morais; S. Georgiev; W. Sproßig¨ . “Real Quaternionic Calculus Handbook”. Birkhäuser (2014). [53] A. Connes. “Noncommutative geometry: Year 2000”. In: N. Alon; J. Bourgain; A. Connes; M. Gromov; V. Milman (eds) “Visions in Mathematics” Modern Birkhäuser Classics (2000). [54] M. Zahiri Abyaneh, M. Farhoudi. “Zitterbewegung in noncommutative geometry”. Int. J. Mod. Phys. A, 34(09):1950045, (2019). [55] L. I. Schiff. “Quantum Mechanics”. McGraw-Hill (1968).