Minimal Length Schrödinger Equation Via Factorisation Approach

Pramana – J. Phys. (2019) 92:53 © Indian Academy of Sciences https://doi.org/10.1007/s12043-019-1717-3

Minimal length Schrödinger equation via factorisation approach

S A KHORRAM-HOSSEINI1, S ZARRINKAMAR2 ,∗ and H PANAHI1

1Physics Department, University of Guilan, Rasht, Iran 2Department of Basic Sciences, Garmsar Branch, Islamic Azad University, Garmsar, Iran ∗Corresponding author. E-mail: [email protected]

MS received 8 June 2018; revised 17 August 2018; accepted 4 September 2018; published online 15 February 2019

Abstract. The fourth-order modiﬁed Schrödinger equation due to the generalised uncertainty principle is considered in one dimension with a box problem. The factorisation of fourth-order self-adjoint differential equations is then discussed and thereby the wave functions and energy spectra of the modiﬁed Schrödinger equation are derived.

Keywords. Minimal length; generalised uncertainty principle; Planck scale; Schrödinger equation; factorisation; particle in box.

PACS Nos 04.60.−m; 03.65.Ge

1. Introduction Caruntu for the factorisation of fourth- and sixth-order ordinary differential equations [19,20]. Next, we apply The goal of unification of quantum mechanics and their novel idea to the fourth-order minimal length gravity has been investigated by many theories includ- Schrödinger equation with box example and obtain the ing loop quantum gravity, double special relativity and wave function and the energy spectra. string theory [1Ð5]. Although such scenarios are based on different assumptions and approaches, all imply the generalisation of the Heisenberg uncertainty principle. 2. Factorisation of self-adjoint differential The so-called generalised uncertainty principle (GUP) equations corresponds to the existence of a minimal length of 3 other Planck length lp = hG¯ /c [3]. The generali- Let us consider a self-adjoint ordinary differential oper- ator of the form [19,20] sation modifies the wave function’s form and therefore the basic characteristics of the equation, i.e. the wave n n 1 d n d function and the energy eigenvalue are altered. It goes L(2n) = ρβ , (1) ρ dxn dxn without saying that the order of energy at such theories is much higher than the order of our present experiments where the scalar functions ρ(x), β(x) and α(x) satisfy and we have to work on theoretical bases. The existing 1 dρ α = , (2) literature indicates that all basic equations of quan- ρ dx β tum mechanics including non-relativistic Schrödinger, d2α d3β relativistic KleinÐGordon, Dirac and DuffinÐKemmerÐ + = 0. (3) Petiau equations have been analysed within this mod- dx2 dx3 ified framework with various interactions ([6Ð18]and For n = 2, eq. (1) can be factorised as references therein). The arising equation, in its simplest L = L (L − δ ), (4) form, i.e. the one-dimensional Schrödinger equation, 4 2 2 2 appears in the sixth-order form. The latter is frequently where approximated by a fourth-order differential equation. dα k(k − 1) d2β δ = (k − ) + . This is just when the problem arises; unlike the second- k 1 2 (5) order differential equations, we know so little about dx 2 dx higher-order equations and their analytical study. In In the fourth-order case, we have the present work, we first review the elegant idea of (L2n − μ)[y]=0(6) 53 Page 2 of 3 Pramana – J. Phys. (2019) 92:53 and the equation is factorised as [19,20] δ + δ2 + μ 2 2 4 λ1 = , n 2 ( − λ )[ ]= . L2 k y 0 (7) δ − δ2 + μ 2 2 4 k=1 λ = . (14b) 2 2 The constants λk are given by ⎧ ⎪ λ + λ +···+λ = δ +···+δ , ⎪ 1 2 n 2 n 3. Generalised uncertainty principle ⎨ λ1λ2 + λ1λ3 +···+λn−1λn = δ2δ3 +···+δn−1δn, (8) We consider a GUP of the form [3,4] ⎪ ⎩⎪ ··· λ λ ···λ = (− )n−1μ. h¯ 1 2 n 1 xp ≥ (1 + β(p)2 + γ), (15) 2 The general solution of the spectral-type equation (6)is where β and γ are positive constants. Such a form given by √ corresponds to the minimal length (x)min = h¯ β, 3 n supposed to be of the Planck length lp = hG¯ /c [ , ]= y = yk, (9) order. Equation (15) implies the GUP [3,4] x p ( + β 2) k=1 ih¯ 1 p . For non-Hermitian quantum mechanics with minimal length, the interested reader may find use- where yk is obtained from ful content in [21,22]. Up to the first order in β,the modified GUP implies [3,4] L2[yk]−λk yk = 0, k = 1, 2,...,n. (10) h¯ ∂ β h¯ ∂ 2 P = 1 + . (16) The extended form of eq. (9)is op i ∂x 3 i ∂x

2 d yk dyk Let us now consider the so-called box example β(x) +[α(x) + β (x)] − λk yk = 0. (11) dx2 dx 0, 0 < x < a, V(x) = (17) In summary, we may write [19,20] ∞, elsewhere.

4 3 d d In this case, the resulting modified Schrödinger equation L = β2 + 2β(α + 2β ) +{[β(α + 2β )] 4 dx4 dx3 appears as 2 d ∂4ψ( ) ∂2ψ( ) + α(α + 2β )} , (12) x − 3 x − 3mE ψ( ) = . dx2 x 0 (18) ∂x4 2βh¯ 2 ∂x2 βh¯ 4 which gives [19,20] Equation (18) can be factorised as 2 2 2 2 2 1 d 2 d d d d d ρβ = β + (α + β ) − λ − λ ψ(x) = 0 (19a) ρ dx2 dx2 dx2 dx 2 1 2 2 dx dx 2 d d × β + (α + β ) − (α + β ) , (13) with dx2 dx √ 3 + 48βmE + 9 δ = α+β λ1 = , where 2 . Therefore, the factorisation appears 4βh¯ 2 as 12mE λ =− √ . (19b) 2 2 ¯ 2( + β + ) β d + (α + β) d − λ h 3 48 mE 9 dx2 dx 1 For the basic concepts of factorisation, we refer to 2 d d the instructive book of Dong [23]. Also, some related × β + (α + β ) − λ = 0, (14a) dx2 dx 2 aspects to supersymmetry quantum mechanics can be found in [18,24] and references therein. Now, eq. (18) where λ1 and λ2 are given by can be factorised as Pramana – J. Phys. (2019) 92:53 Page 3 of 3 53 √ 2 + β + d − 3 48 mE 9 which reduces to the well-known case of ordinary dx2 4βh¯ 2 quantum mechanics in the case of vanishing minimal length parameter. d2 12mE × + √ ψ( ) = x 0 dx2 h¯ 2(3 + 48βmE + 9) 4. Conclusions (20) We considered the one-dimensional modified and the general solution of eq. (4)isgivenby Schrödinger equation due to minimal length with the ψ(x) = ψ1(x) + ψ2(x), (21) problem of a particle in a box. Weconsidered the approx- imate case in which the arising Schrödinger equation ψ ( ) ψ ( ) where 1 x and 2 x are the general solutions of the appears in the fourth-order form. We next considered second-order differential equations obtained as the factorisation of the arising fourth-order equation and ⎛ ⎞ √ thereby reported the wave function and energy spectra. + β + ⎝ 3 48 mE 9 ⎠ The idea looks interesting when we bear in mind the very ψ1(x) = c1 sinh x 4βh¯ 2 few analytical works in the field and the deep insight the ⎛ ⎞ √ analytical approaches provide us. We hope to generalise the idea to more realistic interactions. ⎝ 3 + 48βmE + 9 ⎠ + c2 cosh x , (22) βh¯ 2 4 References 12mE [1] D Amati, M Ciafaloni and G Veneziano, Phys. Lett. ψ2(x) = c3 sin √ x B h¯ 2(3 + 48βmE + 9) 216, 41 (1989) [2] F Scardigli, Phys. Lett. B 452, 39 (1999) [3] S Hossenfelder, Liv. Rev. Gen. Rel. 16, 2 (2013) 12mE [4] A Kempf, G Mangano and R B Mann, Phys. Rev. D 52, + c4 cos √ x . h¯ 2(3 + 48βmE + 9) 1108 (1995) [5] M Maggiore, Phys. Lett. B 304, 65 (1993) (23) [6] F Brau, J. Phys. A 32, 7691 (1999) Phys. Considering the boundary condition at the origin [7] L N Chang, D Minic, N Okamura and T Takeuchi, Rev. D 65, 125028 (2002) removes the terms cosh x and cos x. On the other hand, Gen. Rel. Gravit. = [8] K Nozari and T Azizi, 38, 735 (2006) as the wave function vanishes at x a, the term sinh x [9] D Bouaziz and M Bawin, Phys. Rev. A 76, 032112 is neglected and the wave function takes the form (2007) Phys. Lett. B (x) = c sin( jx) (24a) [10] A F Ali, S Das and E C Vagenas, 678, 497 3 (2009) with [11] Y Chargui, A Trabelsi and L Chetouani, Phys. Lett. A 374, 531 (2010) 12mE [12] Z Lewis and T Takeuchi, Phys. Rev. D 84, 105029 (2011) √ = j. (24b) Phys. h¯ 2(3 + 48βmE + 9) [13] H Hassanabadi, S Zarrinkamar and E Maghsoodi, Lett. B 718, 678 (2012) The boundary condition at x = a implies sin ja = 0 [14] L Menculini, O Panella and P Roy, Phys. Rev. D 87, where j = nπ/a andn = 1, 2, 3,.... Using the nor- 065017 (2013) malisation condition a |c |2 sin2((nπx)/a) = 1, we [15] O Panella and P Roy, Phys. Rev. A 90, 042111 (2014) 0 3 Ann. simply find [16] V. Balasubramanian, S Das and E C Vagenas, Phys. 360, 1 (2015) 2 nπx [17] S Masood, M Faizal, Z Zad, A F Ali, J Reza and M B ψ(x) = sin . (25) Shah, Phys. Lett. B 763, 218 (2016) a a [18] M Alimohammadi and H Hassanabadi, Nucl. Phys. The energy condition A 957, 439 (2017) [19] D I Caruntu, Appl. Math. Comput. 219, 7622 (2013) 12mE nπ [20] D I Caruntu, Appl. Math. Comput. 224, 603 (2013) √ = (26) Phys. Lett. A h¯ 2(3 + 48βmE + 9) a [21] B Bagchi and A Fring, 373, 4307 (2009) [22] T K Jana and P Roy, SIGMA 5, 083 (2009) leads to [23] S-H Dong, Factorization methods in quantum mechanics 2 2 2 4 4 4 (Springer, Dordrecht, 2007) n π h¯ β n π h¯ Pramana – J. Phys. E = + , (27) [24] P Mandal and A Banerjee, 87:1 n 2ma2 3 ma4 (2017)