Cusp Interaction in Minimal Length Quantum

Cusp Interaction in Minimal Length Quantum Mechanics

H. Hassanabadi, S. Zarrinkamar & E. Maghsoodi

Few-Body Systems ISSN 0177-7963 Volume 55 Number 4 Few-Body Syst (2014) 55:255-263 DOI 10.1007/s00601-014-0875-6

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Author's personal copy Few-Body Syst (2014) 55:255–263 DOI 10.1007/s00601-014-0875-6

H. Hassanabadi · S. Zarrinkamar · E. Maghsoodi

Cusp Interaction in Minimal Length Quantum Mechanics

Received: 11 August 2013 / Accepted: 16 March 2014 / Published online: 1 April 2014 © Springer-Verlag Wien 2014

Abstract The modified Schrödinger equation with a minimal length is considered under a Cusp potential which includes the exponential interaction. Next, exact analytical solutions of the problem are reported and thereby the scattering states as well as the corresponding transmission and reflection coefficients are reported.

1 Introduction Many theories such Quantum gravity and string theory imply the existence of a so-called minimal length (ML) which is of order of Planck length l P = G h¯ /c3 ≈ 10−35√ m [1]. In other words, the quantum effects can be no more ignored at high energies of order of Planck mass h¯ c/G ≈ 1019 GeV/c2 . Such a ML modifies our ordinary Heisenberg uncertainty principle (which lets a zero minimum length) into the form which we call generalized uncertainty principle (GUP) in the Jargon. A direct consequence of GUP is naturally the modification of the Schrödinger equation which on the other hand affects the physics of the problem. Various authors have improved our knowledge on the subject by investigating various related topics some of which we quote in the following lines. Konishi et al. [1] studied the ML and the GUP within the framework of string theory. Maggiore [2] proposed a GUP in quantum gravity and commented on its algebraic structure [3,4]. Garay [5] provides the society with a detailed discussion of the quantum gravity in the presence of a ML. Kempf et al. [6] reviewed the corresponding Hilbert space. The Hydrogen atom problem with a GUP was solved by Brau [7]. Scardigli [8] reported evidences from via the micro-black hole gedanken experiment. Ran et al. [9] solved the coulomb problem in one dimension via the path integral approach. Amelino–Camelia [10] proposed a possible scenario to test the incorporation of relativity with a ML. Chang et al. [11] exactly solved the harmonic oscillator problem in arbitrary dimensions. Scardigli and Casadio [12] discussed the GUP in connection with the extra-dimensions and the holograph. Magueijo and Smolin [13] studied the possible consequent non-tachyonic bosonic string. The interface of the subject with doubly special relativity was reported in Ref. [14] by Cortes and Gamboa. The famous example of One-dimensional box was solved by Nozari and Azizi [15] within the framework of modified Schrödinger equation (MScE). The corresponding Pauli–Hamiltonian was investigated in Ref. [16]. The Big-Bang singularity in the framework of a GUP was analysed by Battisti and Montani [17]. Slawny [18] calculated the the corresponding position and length operators. Fityo et al. [19] discussed the WKB approximation in deformed space with ML and minimal momentum. Bambi and Urban [20] worked on the Gedanken experiment, and namely the simultaneous measurement of position and momentum in deSitter H. Hassanabadi · E. Maghsoodi (B) Department of Basic Sciences, Shahrood Branch, Islamic Azad University, Shahrood, Iran E-mail: [email protected] S. Zarrinkamar Department of Basic Sciences, Garmsar Branch, Islamic Azad University, Garmsar, Iran

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space-time, and proved a direct connection of their GUP with a nonzero cosmological constant. Das and Vagenas [21] studied the universality of quantum gravity corrections and the phenomenological implications of the GUP [22]. They provided nice comments on the modifications of relativistic wave equations due to GUP [23]. Hossain et al. [24] provided the society with a background independent GUP. Bouaziz and Ferkous [25] solved the problem of Hydrogen atom with a ML in momentum space. Ali et al. [26] proposed a new form of GUP which was in agreement with string theory, black hole physics and doubly special relativity theories. Hassanabadi et al. [27,28] studied scattering states in minimal length quantum mechanics in the non-relativistic and relativistic states. For our case of interaction, bearing in mind the successful predictions of the exponential term particle physics and cosmology [29–33], we consider the Cusp potential. The paper is arranged as follows: in Sect. 2, we first review the Generalized Uncertainty Principle (GUP) then we introduce the methodology. In Sect. 4, we have solved the modified Schrödinger equation with a minimal length under Cusp interaction and we will obtain exact analytical solutions for the incident and transmitted wave functions and the corresponding coefficients. In Sect. 5, we obtain the bound state solutions of the Cusp potential. Conclusion is given in the last section. 2 The Generalized Uncertainty Principle An immediate consequence of the ML is the GUP [1,15] x ≥

h¯ p , + αl 2p h¯ p

(1)

where the GUP parameter α is determined form a fundamental theory [1,15]. At low energies, i.e. energies much smaller than the Planck mass, the second term in the right hand side of the latter vanishes and we recover the well-known Heisenberg uncertainty principle. The GUP of Eq. (1) corresponds to the generalized commutation relation [15] [xop , pop ] = i h¯ (1 + βp 2 ), 0 ≤ β ≤ 1 (2) where xop = x and 0 ≤ β ≤ 1. The limits β → 0 and β → 1 respectively correspond to the normal quantum mechanics and extreme quantum gravity. Equation (2) gives the minimal length in this case as √ (x)min = 2l p α. 3 A Different Methodology Let us first recall few points. The Cusp potential has the form (see Fig. 1)   V (x) = V0 θ (−x)eαx + θ (x)e−αx , The minimal length Schrödinger equation for a free particle is   2 pop + V (x) ψn (x) = E n ψn (x), 2m with pop = p[1 + β( p)2 ] (for a detailed discussion see [6,7]). From the well-known relations   p 2 = 2m E n(0) − V (x) ,  2 p 4 = 4m 2 E n(0) − V (x) ,

(3)

(4)

(5)

(0)

where E n is the eigenvalue of H0 =

p2 + V (x). 2m

(6)

We have very recently showed that the MScE has a new symmetry. In other words, we obtained a second-order differential equation which resembles our ordinary Schrödinger equation with an effective potential. Therefore,

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Fig. 1 V (x) for α = 0.4

one does not need to deal with a neither six or fourth-order differential. Thus, from Eqs. (2)–(6), we may write [27] H=

 2 p2 βp 4 βp 4 β 2 p6 + + + V (x) = H0 + = H0 + 4mβ E n(0) − V (x) , 2m m 2m m

(7)

or, in differential form [27],  h¯ 2 d 2 ψ(x)  (0) 2 + E − 4mβ(E − V (x)) − V (x) ψ(x) = 0. n 2m d x 2

(8)

We will first solve Eq. (8) with Cusp potential in one dimension. Next, as the scattering states play a significant role in many physical studies, we will calculate the reflection and transmission coefficients. Afterwards, we report the bound-state solutions as well. 4 Reflection and Transmission Coefficients As we are searching for the scattering states of the equation for a Cusp potential barrier, we study the wave functions for x < 0. From substitution of Eq. (3) into Eq. (8), we find

2  h¯ 2 d 2 ψ L (x) (0) 2 2αx (0) αx αx + E − 4mβ E − 4mβV e + 8mβ E V e − V e (9) ψ L (x) = 0, 0 0 n n 0 2m d x 2 √ 0 where h¯ = 1. Considering the variable z L = 4mV 2βeαx the latter is written as α   z 2L d 2 ψ L (z L ) 1 dψ L (z L ) 1 1 4mβ E (0) + + 2 − + √ − √ zL z L dz L 4 2 2βα 2βα dz 2L zL   2m E 8m 2 β(E (0) )2 ψ L (z L ) = 0, + − (10) α2 α2 −1/2

Putting ψ L (z L ) = z L

ϕ(z L ), Eq. (10) reduces to the Whittaker equation [34]   d 2 ϕ L (z L ) 1 1/4 − η2 λ ϕ L (z L ) = 0, + − + − 4 zL dz 2L z 2L

(11)

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where 1 4mβ E (0) − √ , λ= √ 2βα 2 2βα

8m 2 β(E (0) )2 2m E η=i − , 2 α α2

2  E > 4mβ E (0)

(12)

As the forthcoming Eq. (24) reveals, when we consider the extreme limit of the wave function, the coefficient of η should include a positive quantity so that the term under the square root becomes a real number and the whole term (in the exponent) yields an imaginary term. This means that physics of the problem is preserved and we have plane waves. Otherwise, i.e. when the term under square root in negative, we have a negative real term in the exponent and in fact we have obtained a damping wave which is not acceptable. In other words, the extra condition of Eq. (12) determines the threshold of scattering and implies the conservation of flux. The wave function for the x < 0 region is −1/2

ψ L (z L ) = Q 1 z L

−1/2

Mλ,η (z L ) + Q 2 z L

where Mλ,η (z L ) =

1/2+η e−z L /2 z L F



Mλ,−η (z L ),

 1 + η − λ; 1 + 2η; z L , 2

(13)

(14)

with n 

(μ) (−1)m n! zm , F(−n; μ; z L ) = (n − m)!m! (μ + m) L

(15)

m=0

Therefore, we can write −1/2   4mV0  4mV0  − αx αx 2 ψ L (z L ) = Q 1 2β e Mλ,η 2βe α α −1/2    αx 4mV0  4mV0  2β e− 2 Mλ,−η 2βeαx , +Q 2 α α 

(16)

If we intend to study the reflection and transmission coefficients, we consider the asymptotic behavior of the wave function when x tends to infinity [34]. As x → −∞, z L → 0 and therefore, Eq. (13) becomes η −η   4mV0  4mV0  2β eαηx + Q 2 2β e−αηx , (17) ψ L (x → −∞) ∝ Q 1 α α To obtain the solution of the Cusp potential for the region x > 0 we insert Eq. (3) into Eq. (8): h¯ 2 d 2 ψ R (x) + E − 4mβ(E n(0) )2 − 4mβV02 e−2αx + 8mβ E n(0) V0 e−αx 2m d x 2

−V0 e−αx ψ R (x) = 0, which, after a change of variable of the form z R =

4mV0 √ 2βe−αx α

(18)

appears at

  z 2R 1 1 dψ R (z R ) 1 d 2 ψ R (z R ) 4mβ E (0) − √ zR + √ + + 2 − z R dz R 4 2βα 2 2βα dz 2R zR   2m E 8m 2 β(E (0) )2 ψ R (z R ) = 0, + − α2 α2

(19)

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−1/2

The latter, via ψ R (z R ) = z R

ϕ(z R ), comes into the Whit-taker form:   1 1/4 − η2 d 2 φ R (z R ) λ φ R (z R ) = 0, + − + − 4 zR dz 2R z 2R

For x > 0, the wave function takes the form −1/2    αx 4mV0  4mV0  −αx 2 2β e Mλ,η 2βe ψ R (z R ) = Q 3 α α −1/2    αx 4mV0  4mV0  2β e 2 Mλ,−η 2βe−αx , +Q 4 α α

(20)

(21)

In order to define a plane wave travelling from left to right we have to set Q 3 = 0 in Eq. (18) and we are left with −1/2    αx 4mV0  4mV0  −αx 2 ψ R (z R ) = Q 4 2β e Mλ,−η 2βe , (22) α α For x > 0, in the limit x → ∞, z R → 0 and Eq. (22) yields −η  4mV0  ψ R (x → +∞) ∝ Q 4 2β eαηx , α Therefore, in the two sections, we may write ⎧   √ η αηx √ −η −αηx ⎪ 0 0 ⎨ Q 1 4mV 2β e + Q 2 4mV 2β e x → −∞ α α ψ(x) =   −η √ ⎪ 0 ⎩ Q 4 4mV 2β eαηx x → +∞ α

(23)

(24)

In order to give the explicit expressions for the coefficients, we use the continuity conditions of the wave function and its first derivatives at x = 0 i.e. ψ R (x = 0) = ψ L (x = 0) and ψ L (x = 0) = ψ R (x = 0), where prime denotes derivative with respect to x. The former and the latter respectively yield       4mV0  4mV0  4mV0  Q 1 Mλ,η 2β + Q 2 Mλ,−η 2β = Q 4 Mλ,−η 2β , (25) α α α and

   

     1 4mV0  4mV0  α + η + λ Mλ+1,η 2β + α 2β − + 2mV0 2β − αλ Mλ,η 2 α 2 α   

       1 α 4mV0 4mV0  2β + α 2β +Q 2 − + 2mV0 2β − αλ Mλ,−η − η + λ Mλ+1,η 2 α 2 α   

       α 1 4mV0 4mV0  = −Q 4 − + 2mV0 2β − αλ Mλ,η 2β + α 2β , − η + λ Mλ+1,η 2 α 2 α (26)

Q1

In the calculations, we used the following equation      1 1 1 ∂x Mλ,η (x) = x − λ Mλ,η (x) + + η + λ Mλ+1,η (x) . x 2 2

(27)

The wave function in Eq. (16) can be written as ψ L = ψinc + ψref i.e. as a sum of the incident and the reflected waves. Equation (22) contains the transmitted wave function, i.e. ψ R = ψtrans . The incoming, reflected and transmitted fluxes are  i  ∗

∗

inc J inc = , (28) ψinc ∇ψinc − ψinc ∇ψ 2m   i ∗

∗

ref J ref = , (29) ψref ∇ψref − ψref ∇ψ 2m

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(0)

Fig. 2 Reflection (R) and transmission (T ) coefficients versus E for m = 1, β = 0.002, E 0 = 0.01, V0 = 0.6, α = 0.4

and  i  ∗

∗

trans , ψtrans ∇ψtrans − ψtrans ∇ψ J inc = 2m

(30)

respectively. The reflection coefficient R, and the transmission coefficient T , are calculated by ∗ ∇ψ

ref − ψref ∇ψ

∗ ψref J ref |Q 2 |2 ref = ∗ = ,

inc − ψinc ∇ψ

inc∗ |Q 1 |2 ψinc ∇ψ J inc ∗

trans − ψtrans ∇ψ

trans |Q 4 |2 J trans ψ ∗ ∇ψ T = = trans∗ = .

inc − ψinc ∇ψ

∗ |Q 1 |2 ψinc ∇ψ J inc inc

R=

(31)

We have plotted the coefficients in Fig. 2 which is consistent with the balance relation R + T = 1. 5 Bound-State Solutions In this section, we investigate the bound state solutions of the Cusp potential which implies the transformation V0 → −V0 . Equation (3) for x < 0 turns into   2  h¯ 2 d 2 ψ L (x) (0) αx αx + E − 4mβ E n + V0 e + V0 e (32) ψ L (x) = 0. 2m d x 2 where h¯ = 1. Using the transformation y L =

4mV0 √ 2βeαx α

the latter is written as   y L2 1 1 dψ L (y L ) 1 4mβ E (0) d 2 ψ L (y L ) + √ yL + + 2 − + − √ y L dy L 4 2 2βα 2βα dy L2 yL   8m 2 β(E (0) )2 2m E ψ L (y L ) = 0, + − α2 α2 −1/2

Setting ψ L (y L ) = y L

f L (y L ), we obtain the Whit-taker differential equation possessing the form   d 2 f L (y L ) 1 1/4 − κ 2 ν f L (y L ) = 0, + − + − 4 yL dy L2 y L2

(33)

(34)

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where 1 4mβ E (0) + √ , ν=− √ 2βα 2 2βα

8m 2 β(E (0) )2 2m E κ =i − α2 α2

(35)

The wave function for the x < 0 region is −1/2

ψ L (y L ) = Q 5 y L

−1/2

Mν,κ (y L ) + Q 6 y L

Mν,−κ (y L ),

(36)

Therefore, we can write −1/2   4mV0  4mV0  − αx αx 2 ψ L (y L ) = Q 5 2β e Mν,κ 2βe α α −1/2    αx 4mV0  4mV0  2β e− 2 Mν,−κ 2βeαx , +Q 6 α α 

(37)

Next, we obtain the solution of the following form of Eq. (3) for x > 0   2  h¯ 2 d 2 ψ R (x) (0) −αx −αx + E − 4mβ E + V e + V e ψ R (x) = 0. 0 0 n 2m d x 2 or, with y R =

(38)

4mV0 √ 2βe−αx , α

  y 2R d 2 ψ R (y R ) 1 1 dψ R (y R ) 1 4mβ E (0) + √ yR + + 2 − + − √ y R dy R 4 2βα 2 2βα dy 2R yR   2m E 8m 2 β(E (0) )2 ψ R (y R ) = 0, + − α2 α2 −1/2

considering ψ R (y R ) = y R

(39)

f R (y R ), comes into the Whit-taker form:

  1 1/4 − κ 2 ν d 2 f R (y R ) f R (y R ) = 0, + − + − 4 yR dy 2R y 2R

(40)

For x > 0, the wave function takes the form −1/2   αx 4mV0  4mV0  2β 2βe−αx e 2 Mν,κ α α −1/2    αx 4mV0  4mV0  −αx 2β e 2 Mν,−κ 2βe +Q 8 , α α 

ψ R (y R ) = Q 7

(41)

In order to have the bound-state solutions, we use the boundary conditions at the origin and infinity which give Q 6 = Q 8 = 0 and therefore −1/2   4mV0  4mV0  − αx αx 2 e Mν,κ 2β 2βe ψ L (x) = Q 5 , α α −1/2    αx 4mV0  4mV0  2β e 2 Mν,κ 2βe−αx , ψ R (x) = Q 7 α α 

(42a) (42b)

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Applying the boundary conditions of the wave function and its derivative on Eqs. (42a) and (42b) give     4mV0  4mV0  Q 5 Mν,κ 2β = Q 7 Mν,κ 2β , (43) α α   

      1 α 4mV0  4mV0  Q 5 − + 2mV0 2β − αν Mν,κ 2β + α 2β + κ + ν Mν+1,κ 2 α 2 α   

      α 1 4mV0  4mV0  2β + α 2β , = −Q 7 − + 2mV0 2β − αν Mν,κ + κ + ν Mν+1,κ 2 α 2 α (44) Using Eqs. (43) and (44) we can simply determine the energy eigenvalues. As a typical example, for (0) m = 1, β = 0.002, E 0 = 0.01, V0 = 5, α = 0.4, bearing in mind that − |V0 | < E < 0, we obtain E 1 = −3.693268, E 2 = −1.740649, E 3 = −0.809844 and E 4 = −0.302047(fm−1 ). In Fig. 2, we have portrayed the behavior of the reflection and transmission coefficients versus E n . As we expect, the relation R + T = 1 is satisfied. 6 Conclusion The solid evidences for a minimal length on the one hand, and the motivating phenomenological predictions of the exponential Cusp potential in particle physics, motivated us to consider the term within the framework of modified Schrödinger equation. On the contrary to works which discuss the problem on the basis of a fourth-order differential equation, from Ref. [27], we transformed our problem into the ordinary Schrödinger equation with a new effective potential and thereby reported the exact analytical results. To provide a more useful study, we reported the scattering states as well as the transmission and reflection parameters which are in many cases an essential ingredient. Acknowledgments We wish to give our sincere gratitude to the referees for their technical comments on the manuscript.

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