Homogeneous Field and WKB Approximation In Deformed Quantum Mechanics with Minimal Length

In the framework of the deformed quantum mechanics with minimal length, we consider the motion of a non-relativistic particle in a homogeneous external field. We find the integral representation for the physically acceptable wave function in the position representation. Using the method of steepest descent, we obtain the asymptotic expansions of the wave function at large positive and negative arguments. We then employ the leading asymptotic expressions to derive the WKB connection formula, which proceeds from classically forbidden region to classically allowed one through a turning point. By the WKB connection formula, we prove the Bohr-Sommerfeld quantization rule up to $\mathcal{O}(\beta)$. We also show that, if the slope of the potential at a turning point is too steep, the WKB connection formula fall apart around the turning point.

(1) is the simplest model where only the minimal uncertainty in position is taken into account while the momentum can be infinite.
In this paper we consider one dimensional quantum mechanics with the deformed commutation relation (1). To implement the deformed commutators (1), one defines [12,15] X = X 0 , P = P 0 1 + β 3 where [X 0 , P 0 ] = i , the usual canonical operators. One can easily show that to the first order of β, eqn. (1) is guaranteed. Henceforth, terms of O (β 2 ) and higher are neglected in the remainder of the paper. For a quantum system described by the Hamiltonians can be written as where H 0 = P 2 0 2m + V (X 0 ) and H 1 = 2β 3 P 2 0 . Furthermore, one can adopt the momentum representation or the position representation The momentum representation is very handy in the discussions of certain problems, such as the harmonic oscillator [16], the Coulomb potential [17,18] and the gravitational well [19,20]. Recently, a wide class of problems, like scattering from a barrier or a particle in a square well [21][22][23] are discussed in position representation. Moreover, in the position representation, it is much easier to derive and discuss WKB approximation in the deformed quantum mechanics analogously to in the ordinary quantum mechanics [24]. Thus, we adopt the position representation in this paper. In the position representation, the deformed stationary Schrodinger equation is where we define ℓ 2 β = 2 3 2 β 2 for later convenience.
Although, the homogeneous field potential V (X) = F X is not studied so intensively as the quantum well, it has an important application in theoretical physics. In the ordinary quantum mechanics, the solutions to the Schrodinger equation with the linear potential are Airy functions, which are essential to derive the WKB connection formulas through a turning point. This motivates us to study the linear potential in the deformed quantum mechanics.
In the deformed quantum mechanics with minimal length, the WKB approximation formulas are obtained in [24]. In addition, the deformed Bohr-Sommerfeld quantization is used to acquire energy spectra of bound states in various potentials [18,[23][24][25][26]. Therefore, it is interesting to derive the WKB connection formulas through a turning point and rigorously verify the Bohr-Sommerfeld quantization rule claimed before, which are presented in our paper. Besides, we find that, if the slope of the potential is too steep at a turning point, the WKB connection algorithm fails around the turning point. This is not unexpected because, if one makes linear approximation to the potential around such a turning point for asymptotic matching, the corrections to the wave functions due to the Hamiltonian H 1 become dominant before one reaches the WKB valid region.
This paper is organized as follows: In section II we give the integral representation of the physically acceptable wave function of the homogeneous field and its leading asymptotic behavior at large positive value of ρ. In section III, we obtain the asymptotic expansions of the physically acceptable wave function at both large positive and large negative values of ρ.
Section IV is devoted to deriving the WKB connection formula and the related discussions.
In section V, we offer a summary and conclusion.

II. DEFORMED SCHRODINGER EQUATION
Let us consider one-dimensional motion of a particle in a homogenous field, specifically in a field with the potential V (X) = F X. Here we take the direction of the force along the axis of −x and let F be the force exerting on the particle in the field. As discussed in the introduction, the deformed Schrodinger equation for this scenario is In order to solve eqn. (10), a new dimensionless variable ρ is introduced as Eqn. (10) then becomes where we define another dimensionless variable α 2 = ℓ 2 β (2mF/ 2 ) 2 3 and the derivatives are in terms of the new variable ρ. The linear differential equation (12) is quartic and then there are four linearly independent solutions. We will shortly show that only one of them is physically acceptable.

A. Physically Acceptable Solution
The condition βP 2 ≪ 1 validating our effective GUP model implies This condition is also expected in the momentum space. Since the GUP model is only valid below the energy scale β − 1 2 , the momentum spectrum of the state |ψ should be greatly suppressed around the scale β − 1 2 . It also leads to the condition (13). Moreover, the condition (13) and eqn. (12) give In other words, our GUP model, which is an effective model, is valid only when the condition (14) holds. Considering that the Compton wavelength of a particle should be much larger than √ β or ℓ β in the GUP model, one can also obtain the condition (14) in the classical allowed region where ρ < 0. In a field with the potential V (x), the kinematics energy of a non-relativistic particle is E − V (x) and its momentum is 2m (E − V (x)). Therefore, the fact that the Compton wavelength of the particle λ c = √ is much larger than ℓ β yields α 2 |ρ| ≪ 1. In the remainder of our paper except subsubsection IV B 1, we assume α ≪ 1 which is useful to derive WKB connection formula around a smooth tuning point.
One needs to consider α 1 scenario only when it comes to the WKB connection around a sharp turning point.
We notice that E < V for ρ > 0. The wave function ψ is then exponentially damped for large positive value of ρ. Thus, one needs to evaluate asymptotic values of ψ (ρ) at large positive value of ρ to find physically acceptable solution to eqn. (12). Note that, only when α ≪ 1, one can analyze asymptotic behavior of ψ (ρ) at large positive value of ρ in the physically acceptable region where α 2 |ρ| ≪ 1. To determine the leading behavior of ψ (ρ) at large positive value of ρ, we make the exponential substitution ψ (ρ) = e s(ρ) and then obtain for eqn. (12) Eqn. (15) is as difficult to solve as eqn. (12). Here our strategy to find the asymptotic behavior of ψ (ρ) from eqn. (15) is as follows [27]: (a) We neglect all terms appearing small and approximate the exact differential equation with the asymptotic one. It is usually true that higher derivative terms than s ′ are discarded in step (a). Therefore, we reduce eqn. (15) to the asymptotic differential equation Solving eqn. (16) gives four solutions for s ′ , two of which are discarded considering βP 2 ≪ 1.
Taking asymptotic relation (13) into account, one can further reduce the quartic equation (16) to a quadratic equation which has only two solutions for s ′ . The two solutions are s ′ ∼ ± √ ρ, and, therefore, , at large positive value of ρ, where − is for the physically acceptable solution. It is easy to check that the solution s ′ ∼ ± √ ρ satisfy the assumptions s ′′ , s ′2 s ′′ , s (3) , s ′′2 , s ′ s (3) and s (4) ≪ ρ, as long as ρ ≫ 1.
It is interesting to note that the two discarded solutions of eqn. (16) are which become s ′ ∼ ± 1 √ α when α 2 ρ ≪ 1. The resulting wave functions are ψ (ρ) ∼ exp ± ρ √ α . They are not physical states since they fail to satisfy the condition (13). One can also see that these two solutions are discarded according to the low-momentum consistency condition in [28]. In summary, assuming α ≪ 1, we find that the leading asymptotic behavior of the physically acceptable solution is exp − 2 3 ρ 3 2 for ρ ≫ 1. In addition, we only analyze the solution in the region α 2 |ρ| ≪ 1 where the GUP model is valid.

B. Integral Representation
The differential equation eqn. (12) can be solved by Laplace's method. Please refer to mathematical appendices of [29] for more details. Define the polynomials and the function Integral representations of the solutions to eqn. (12) are then given by where the contour C is chosen so that the integral is finite and non-zero and the function vanishes at endpoints of C since the integrand of eqn. (22) is entire on the complex plane for large t, we need to begin and end the contour C in sectors for which cos 5θ < 0 (setting t = |t|e iθ ). There are five such sectors, Therefore, any contour which originates at one of them and terminates at another yields a solution to eqn. (12). One could then find four linearly independent functions of the form The asymptotic expression for I i (ρ) for large values of ρ is obtained by evaluating the integral eqn. (25) by the method of steepest descents.

III. ASYMPTOTIC EXPANSION
First we briefly review the method of steepest descent to introduce some useful formulas.
This technique is very powerful to calculate integrals of the form where C is a contour in the complex plane and g (z) and f (z) are analytic functions. The parameter ρ is real and we are usually interested in the behaviors of I (ρ) as ρ → ±∞.
The key step of the method of steepest descent is applying Cauchy's theorem to deform the contours C to the contours consisting of steepest descent paths and other paths joining endpoints of two different steepest descent paths if necessary. Usually, the joining paths are chosen to make negligible contributions to I (ρ). It is easy to show that Im f (z) is constant along steepest descent paths. When a steepest descent contour passes through a saddle point and g (z) are expanded around z 0 and Watson's lemma is used to determine asymptotic behaviors of I (ρ). Specifically, consider a contour C through a saddle The saddle point z 0 divides the contour C into two contours C 1 and C 2 . Generally, τ monotonically increases from −∞ to zero along one contour, say C 1 and monotonically decreases from zero to −∞ along C 2 . Thus, the integral becomes The physically acceptable solution can be represented by an integral where C is any contour which ranges from t = exp − 3πi In fact, as we show later in the section for positive ρ, there exists a steepest descent contour from ∞, which C can be deformed to. Moreover, the integral on such a steepest descent contour yields the required asymptotic behavior of I (ρ) at large positive value of ρ. Here the exponent in the integrand has movable saddle points. Making the change of variables s = |ρ| 1 2 t, one gets where + for ρ > 0 and − for ρ < 0 and a = α 2 |ρ| ≪ 1 in the physical region.

A. Large Positive ρ
For ρ > 0, we have There are four saddle points given by f ′ + (s) = 0 at Our goal now is to find a steepest descent contour emerging from s = exp − 3πi ∞. We will show that such a contour passes through s = −λ + . To find the contour we substitute s = u + iv and identify the real and imaginary parts of f + (s) Since Im f + (−λ + ) = 0, the constant-pahse contours passing through s = −λ + must satisfy Therefore, one of the constant-phase contours passing through s = −λ + is which is a steepest descent contour. In fact, around the saddle point s = −λ + , one finds on the contour C and hence, where b is a positive real number. Since f ′′ + (−λ + ) is real and positive, the contour C is indeed a steepest descent contour. Note that C goes to s = exp − 3πi In order to evaluate asymptotic expansion of I (ρ), we break up the contour C into C 1 and C 2 , corresponding to above and below of s = −λ + . Define where τ monotonically decreases from zero to −∞ as one moves away from s = −λ + along ∞ and along C 2 to s = exp − 3πi 5 ∞, respectively. Since f ′ + (−λ + ) = 0, the expression for s in terms of τ can be expressed as a power series of √ −τ . Then, noting that −τ = ± √ −τ 2 , one has where a i can be obtained by substituting eqn. (37) into eqn. (36) and equating powers of √ −τ on both sides of the equations. It is easy to find where one finds Im a 1 > 0. The contour C 1 is in the second quadrant and hence, + sign is chosen in eqn. (37) for C 1 . Therefore, For the contour segment C 2 , the sign of √ τ occurring in eqn. (37) has to be reversed.
Moreover, the limit of integration on C 2 in the variable τ ranges from −∞ to 0. Thus, Combining eqn. (39) and eqn. (40), we easily find B. Large Negative ρ As for ρ < 0, the exponent in the integrand of I (ρ) is Thus, one as well finds four saddle points given by f ′ As before, our objective is to find steepest descent contours passing through the saddle We have already shown that only one steepest descent contour passing through s = −λ + is sufficient to evaluate asymptotic behavior of I (ρ) for large and positive ρ. However for large and negative ρ, things are a little bit more complicated. Instead of one steepest descent contour, it turns out that we need three steepest descent contours passing through ±λ − and η − , respectively, to connect two endpoints at s = exp ± 3πi 5 ∞.
First consider the steepest descent contour through s = −λ − . Since f + (−λ − ) is a pure imaginary number, the steepest descent contour must satisfy Solutions to the last equation give us a constant phase contour C −λ − passing through s = −λ − , which emanates from s = exp − 3πi 5 ∞ and finally approaches s = exp − πi 5 ∞. The contour C −λ − actually is composed of three segments as where we define and v 0 is a solution to F +λ − (v) = 0 that satisfies 0 < v 0 ≪ 1. It is straightforward to verify that, along C −λ − , Re f − (s) monotonically increases from −∞ to 0 as one moves from s = exp − 3πi 5 ∞ to s = −λ − and then monotonically decreases from 0 to −∞ as one moves away from s = −λ − to s = exp − πi 5 ∞. Hence, the contour C −λ − is indeed a the steepest descent contour passing through s = −λ − . Now we calculate the contour integral on C −λ − . Introduce which τ is real on C −λ − and varies from −∞ to zero and then to −∞ along C −λ − . Then, one has where b i can be obtained by substituting eqn. (46) into eqn. (47). One easily gets Since Re exp π 4 i > 0, one has − √ −τ for C −λ − ,1 and √ −τ for C −λ − ,2 + C −λ − ,3 in eqn. (47). Therefore, Analogously, one can readily write down a constant phase contour C +λ − passing through s = −λ − , which starts from s = exp πi 5 ∞ and ends at s = exp 3πi 5 ∞. As before, C +λ − consists of three segments It is also straightforward to verify that C +λ − is a steepest descent contour as well. Setting one finds τ is real on C +λ − and varies from −∞ to zero and then to −∞ along C +λ − . Note that f + (s) is an odd function and λ * − = −λ − . Taking complex conjugate of both sides of eqn. (50), one then has on C +λ − Since Re b * 1 > 0, one has √ −τ for C +λ − ,1 + C +λ − ,2 and − √ −τ for C +λ − ,3 in eqn. (51). Therefore, Since the values of Im f − (s) are different on C ±λ − , it is obvious that we need a third contour which joins C ±λ − up at s = exp ± πi 5 ∞, respectively. Here, we consider a constant phase contour C η − connecting s = exp − πi 5 ∞ to exp πi 5 ∞ that passes through η − . Since Im f (s) = Im f − (η − ) = 0 on the contour C η − , one finds is a curve of steepest descent. On C η − , define which is real on C η − and varies from −∞ to zero and then to −∞ along C η − . Then, one and Similarly, we break up the contour C η − into C η − ,1 and C η − ,2 , corresponding to above and below of s = η − with √ −τ for C η − ,1 and − √ −τ for C η − ,2 in eqn. (54). Thus, Note that although paths C η − and C ±λ − never join up at s = exp ± πi 5 ∞, the integrand exp [f − (s)] ∼ exp a|ρ| 3 2 5 s 5 tends to zero exponentially. Therefore, there is no contribution from a connecting path from C η − and C ±λ − at a distance R from the origin in the limit R → ∞. As a result, the integral I (ρ) equals to the sum of three contour integrals on the different steepest descent curves C η − and C ±λ − . Combining eqn. (49), eqn. (52) and eqn.
(56) gives the full asymptotic expansion of I (ρ) for large and negative ρ (57)

IV. WKB APPROXIMATION
The authors of [24] find the WKB approximation in deformed space with minimal length.
In [24], they consider the deformed commutation relation where f (P ) is an arbitrary function of P . In our paper, we set f (P ) = 1 + βP 2 . Defining and p (P ) an inverse function of P (p) , they find the physical-optics approximation to the solution of the deformed Schrodinger equation is where P = 2m (E − V (x)) in eqn. (61). It is also shown there that, if eqn. (61) is valid, the condition has to be satisfied. However, the condition eqn. (62) fails near a turning point where P = 0. Thus, if we want to determine bound state energies, we need to be able to match wave functions at the turning points. Here we considers a potential V (x) with its classical turning point located at x = 0. A linear approximation to the potential V (x) near the where F = V ′ (0). The linearized potential (63) is discussed in the previous two sections.
Our discussion shows that the parameter α = ℓ β (2m |F | / 2 ) To the right of the turning point, the wave function is given by Around the turning point, x is small and P ∼ √ 2mF √ −x. In this region, we may approximate eqn. (64) and eqn. (65) by The criteria (62) for validity of the WKB approximation is satisfied if where we neglect βP 2 in derivation. On the other hand, when the potential is linearized around the turning point x = 0, the Schrodinger equation becomes where β = 3ℓ 2 β 2 2 . To solve the approximate differential equation, we make the substitution In terms of ρ, the solution to eqn. (69) which matches eqn. (66) and eqn. (67) in two different limits is actually I (ρ) calculated in the section III. Specifically, the solution is where D is a constant to be determined by asymptotic matching. It is easily shown from (68) that there exists overlap regions where both WKB approximation and eqn. (69) hold.
In the overlap regions, one finds |ρ| ≫ 1 and |x| ≪ 1. Therefore, we approximate I (ρ) by its leading asymptotic behaviors for large argument in the the overlap regions. The appropriate formulas are where α ≪ 1 for a smooth turning point and a = ℓ 2 β (2mF/ 2 ) which is directional, just as in ordinary quantum mechanics [27]. The analysis always proceeds from classically forbidden region to classically allowed one. For bound states, the uniqueness of the wave function in the classically allowed region leads to the Bohr- where a and b are two smooth turning points for the potential V (x). Notice that although eqn. (75) is claimed in [24], one still needs to obtain the connection formula to derive eqn.
B. Discussion

Sharp Turning Point
Near a sharp turning point x = 0, not only the WKB approximation falls apart but also matching the two WKB solutions across the turning point stops making sense. In fact, from the previous subsection, one finds that the asymptotic matching is valid as long as there exists an overlap region where 1 ≪ |ρ| ≪ α −2 . However, such region doesn't exist unless α ≪ 1, which means that the asymptotic matching fails through a sharp turning point.
It can be shown, through (62) for a sharp turning point. However, |βP 2 | ≪ 1 is required by the GUP model. This means that, as moving away from the sharp turning point, one is far beyond the region where the linear approximation to the potential is good before reaching the WKB valid region. One might resort to a higher order approximation to the potential and asymptotic matching in the overlap region to find WKB connection formula through a sharp turning point.

O (β) vs. O ( )
When can be regarded as a small quantity, the approximate solution to the deformed Schrodinger equation is easy to find using WKB analysis. To be specific, the approximate solution is expressed in an exponential power series of the form The authors of [24] finds Since here f (P ) = 1 + βP 2 , we have for S 1 Moreover, the leading order (in terms of β) of the S 2 is just the WKB O ( 2 ) correction calculated in the ordinary quantum mechanics. Therefore, we obtain [29] If one uses WKB approximations to evaluate quantum gravity induced corrections, say to energy levels or tunnelling rates, one may want to have Otherwise, the quantum gravity correction (∼ O (β)) on the first order WKB approximation (∼ O ( 0 )) could be overwhelmed by the second order WKB approximation (∼ O ( )).
Suppose a is the characteristic length of the potential V (x), for example the width of a square-well potential. Then we can get a rough estimate on S 2 where λ is the de Broglie wavelength of a particle with momentum P . As a result, the condition (82) becomes It is interesting to note that the condition (84) is a rough estimate and a more accurate estimate could be obtained once the form of the potential is given.
Taking into account the constraints (84) on the de Broglie wavelength λ of a particle, one may conclude that the WKB approximation is not a powerful tool to calculate quantum gravity corrections unless the energy of the particle considered is high enough.
where E n,j (β) can be expanded in terms of β If on the first order WKB approximation, one calculates E n,0 (β) up to O (β) the energy levels are eqn. (85) becomes Since O ( ) O (β) is automatically smaller than βE 1 0 , eqn. (90) always makes sense as long as O ( ) ≪ 1.
To illustrate our points, we use the WKB approximation to derive the energy levels of a particle confined to the one-dimensional potential V (x) = F |x| whose turning points are The energy quantization condition (75) from first order WKB approximation then becomes where ℓ F = ( 2 /2mF ) 1 3 is the characteristic length of the potential V (x) = F |x|. From the last equation, we obtain where ℓ n = ℓ F with β = 0 is given in [27] ℓ − 3 which gives E We can then estimate O ( ) through (95) which can also be easily obtained by dimensional analysis. If one wants the first order approximation (93) to make sense, the second term in (93) should be comparable to or larger than O ( ) and then one gets The de Broglie wavelength of a particle with energy E n ∼ F ℓ n is Thus, the inequality (97) reads which is much milder than (84). In a practical way, and β can be expressed in terms of ℓ β , l F and l n . In fact, it is easily shown that V. CONCLUSIONS In this paper, we considered a homogeneous field in the deformed quantum mechanics with minimal length. The physical motivation for this is to obtain the WKB connection formula and prove the Bohr-Sommerfeld quantization rule rigorously in the deformed quantum mechanics. By studying the leading asymptotic behavior of the physically acceptable wave function in the physical region, we found the contour for its integral representation.
Through the integral representation, the asymptotic expansions of the physically acceptable wave function at both large positive and large negative values of ρ were given.
Then, we used the obtained asymptotic expansions to get the WKB connection formula, which proceeds from classically forbidden region to classically allowed one through a smooth turning point, and had the Bohr-Sommerfeld quantization rule proved rigorously up to O (β). A new interesting feature appearing in the presence of deformation is that our WKB connection formula do not work for a sharp turning point. The connection through such a point might need a higher order approximation to the potential near it.
Finally, we discussed the competition between the quantum gravity correction on the first order WKB approximation and the second order WKB approximation. If the former is not overwhelmed by the latter, the energy of the particle considered should be high enough according to (84). We also showed that, if the energy levels E n of a bound state are given in the ordinary quantum mechanics, the deformed energy levels are where βE 1 n,0 is the O (β) quantum gravity correction on the first order WKB approximation.