#### Abstract

In the framework of the deformed quantum mechanics with a minimal length, we consider the motion of a nonrelativistic 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 . We also show that if the slope of the potential at a turning point is too steep, the WKB connection formula is no longer valid around the turning point. The effects of the minimal length on the classical motions are investigated using the Hamilton-Jacobi method. We also use the Bohr-Sommerfeld quantization to study statistical physics in deformed spaces with the minimal length.

#### 1. Introduction

One of the predictions shared by various quantum theories of gravity is the existence of a minimal observable length. For example, this fundamental minimal length scale could arise in the framework of the string theory [1–3]. For a review of a minimal length in quantum gravity, see [4]. Some realizations of the minimal length from various scenarios have been proposed. Specifically, one of the most popular models is the generalized uncertainty principle (GUP) [5, 6], derived from the modified fundamental commutation relationwhere , is the Planck mass, is the Planck length, and is a dimensionless parameter. For a review of the GUP, see [7]. With this generalization, one can easily derive the generalized uncertainty principle (GUP)This in turn gives the minimal measurable lengthEquation (1) is the simplest model where only the minimal uncertainty in position is taken into account, while the momentum can be infinite. When incorporating the GUP into quantum field theory, one needs to generalize deformed commutation relations to include time. However, the existence of the minimal length could lead to Planck scale departures from Lorentz symmetry. Therefore, the corresponding deformed commutation relations are not Lorentz invariant and give rise to some version of the doubly special relativity [8–11].

In this paper we consider one-dimensional nonrelativistic quantum mechanics with the deformed commutation relation (1). To implement the deformed commutators (1), one defines [6, 12]where , the usual canonical operators. One can easily show that to the first-order of , (1) is guaranteed. Henceforth, terms of and higher are neglected in the remainder of the paper. For a quantum system described bythe Hamiltonians can be written aswhere and . Furthermore, one can adopt the momentum representationor the position representationThe momentum representation is very handy in the discussions of certain problems, such as the harmonic oscillator [13], the Coulomb potential [14, 15], and the gravitational well [16–18]. Recently, a wide class of problems, like scattering from a barrier or a particle in a square well [19–21], 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 analogous to that in the ordinary quantum mechanics [22]. Thus, we adopt the position representation in this paper. In the position representation, the deformed stationary Schrodinger equation iswhere we define for later convenience.

Although, the homogeneous field potential 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 [22]. In addition, the deformed Bohr-Sommerfeld quantization is used to acquire energy spectra of bound states in various potentials [15, 21–24]. 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 Hamiltonian become dominant before one reaches the WKB valid region.

This paper is organized as follows: in Section 2 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 . In Section 3, we obtain the asymptotic expansions of the physically acceptable wave function at both large positive and large negative values of . Section 4 is devoted to deriving the WKB connection formula and the related discussions and applications. In Section 5, we offer a summary and conclusion.

#### 2. Deformed Schrodinger Equation

Let us consider one-dimensional motion of a particle in a homogenous field, specifically in a field with the potential . Here we take the direction of the force along the axis of and let be the force exerted on the particle in the field. As discussed in Section 1, the deformed Schrodinger equation for this scenario isIn order to solve (10), a new dimensionless variable is introduced asEquation (10) then becomeswhere we define another dimensionless variable and the derivatives are in terms of the new variable . 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.

##### 2.1. Physically Acceptable Solution

The condition validating our effective GUP model impliesThis condition is also expected in the momentum space. Since the GUP model is only valid below the energy scale , the momentum spectrum of the state should be greatly suppressed around the scale . It also leads to condition (13). Moreover, conditions (13) and (12) giveIn other words, our GUP model, which is an effective model, is valid only when condition (14) holds. Considering that the Compton wavelength of a particle should be much larger than or in the GUP model, one can also obtain condition (14) in the classical allowed region where . In a field with the potential , the kinematics energy of a nonrelativistic particle is and its momentum is . Therefore, the fact that the Compton wavelength of the particle is much larger than yields . In the remainder of our paper except Section 4.2.1, we assume which is useful to derive WKB connection formula around a smooth tuning point. One needs to consider scenario only when it comes to the WKB connection around a sharp turning point.

We notice that for . The wave function is then exponentially damped for large positive value of . Thus, one needs to evaluate asymptotic values of at large positive value of to find physically acceptable solution to (12). Note that, only when , one can analyze asymptotic behavior of at large positive value of in the physically acceptable region where . To determine the leading behavior of at large positive value of , we make the exponential substitution and then obtain for (12)Equation (15) is as difficult to solve as (12). Here our strategy to find the asymptotic behavior of from (15) is as follows [25]:(a)We neglect all terms appearing small and approximate the exact differential equation with the asymptotic one.(b)We solve the resulting equation and check that the solution is consistent with approximations made in step (a).

It is usually true that higher derivative terms than are discarded in step (a). Therefore, we reduce (15) to the asymptotic differential equationSolving (16) gives four solutions for , two of which are discarded considering . Taking asymptotic relation (13) into account, one can further reduce the quartic equation (16) to a quadratic equationwhich has only two solutions for . The two solutions are and, therefore,where − is for the physically acceptable solution. It is easy to check that the solution satisfies the assumptionsas long as .

It is interesting to note that the two discarded solutions of (16) arewhich become when The resulting wave functions are . They are not physical states since they fail to satisfy condition (13). One can also see that these two solutions are discarded according to the low-momentum consistency condition in [26]. In summary, assuming , we find that the leading asymptotic behavior of the physically acceptable solution is for . In addition, we only analyze the solution in the region , where the GUP model is valid.

##### 2.2. Integral Representation

The differential equation (12) can be solved by Laplace’s method. Please refer to mathematical appendices of [27] for more details. Define the polynomialsand the functionIntegral representations of the solutions to (12) are then given bywhere the contour is chosen so that the integral is finite and nonzero and the functionvanishes at endpoints of since the integrand of (23) is entire on the complex plane of. Now that for large , we need to begin and end the contour in sectors for which (setting ). There are five such sectors, specificallyTherefore, any contour which originates at one of them and terminates at another yields a solution to (12). One could then find four linearly independent functions of the formThe asymptotic expression for for large values of is obtained by evaluating the integral equation (26) by the method of steepest descents.

#### 3. 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 formwhere is a contour in the complex plane and and are analytic functions. The parameter is real and we are usually interested in the behaviors of as . The key step of the method of steepest descent is applying Cauchy’s theorem to deform the contours 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 . It is easy to show that is constant along steepest descent paths. When a steepest descent contour passes through a saddle point , where , and are expanded around and Watson’s lemma is used to determine asymptotic behaviors of . Specifically, consider a contour through a saddle point . A new variable is introduced as to calculate . The saddle point divides the contour into two contours and . Generally, monotonically increases from to zero along one contour, say , and monotonically decreases from zero to along . Thus, the integral becomes

The physically acceptable solution can be represented by an integralwhere is any contour which ranges from to . In fact, as we show later in the section for positive , there exists a steepest descent contour from to , to which can be deformed. Moreover, the integral on such a steepest descent contour yields the required asymptotic behavior of at large positive value of . Here the exponent in the integrand has movable saddle points. Making the change of variables , one getswhere + for and − for and in the physical region.

##### 3.1. Large Positive

For , we haveThere are four saddle points given by atOur goal now is to find a steepest descent contour emerging from to . We will show that such a contour passes through . To find the contour we substitute and identify the real and imaginary parts of :Since , the constant-phase contours passing through must satisfyTherefore, one of the constant-phase contours passing through iswhich is a steepest descent contour. In fact, around the saddle point , one finds on the contour and hencewhere is a positive real number. Since is real and positive, the contour is indeed a steepest descent contour. Note that goes to as and as . In order to evaluate asymptotic expansion of , we break up the contour into and , where () is the contour above (below) of . Definewhere monotonically decreases from zero to as one moves away from along to and along to , respectively. Since , the expression for in terms of can be expressed as a power series of . Then, noting that , one haswhere can be obtained by substituting (39) into (38) and equating powers of on both sides of the equations. It is easy to findwhere one finds . The contour is in the second quadrant and hence + sign is chosen in (39) for . Therefore,For the contour segment , the sign of occurring in (39) has to be reversed. Moreover, the limit of integration on in the variable ranges from to . Thus,Combining (41) and (42), we easily find

##### 3.2. Large Negative

As for , the exponent in the integrand of isThus, one as well finds four saddle points given by :As before, our objective is to find steepest descent contours passing through the saddle point(s) in (45) that emerges from to . Substituting , we obtain the real and imaginary parts of :We have already shown that only one steepest descent contour passing through is sufficient to evaluate asymptotic behavior of for large and positive . However, for large and negative , 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 and , respectively, to connect two endpoints at .

First consider the steepest descent contour through . Since is a pure imaginary number, the steepest descent contour must satisfySolutions to the last equation give us a constant phase contour passing through , which emanates from and finally approaches . The contour actually is composed of three segments as where we defineand is a solution to that satisfies . It is straightforward to verify that, along , monotonically increases from to as one moves from to and then monotonically decreases from to as one moves away from to . Hence, the contour is indeed the steepest descent contour passing through . Now we calculate the contour integral on . Introducewhere is real on and varies from to zero and then to along . Then, one haswhere can be obtained by substituting (50) into (51). One easily getsSince , one has for and for in (51). Therefore,

Analogously, one can readily write down a constant phase contour passing through , which starts from and ends at . As before, consists of three segments:

It is also straightforward to verify that is a steepest descent contour as well. Settingone finds that is real on and varies from to zero and then to along . Note that is an odd function and . Taking complex conjugate of both sides of (55), one then has on Since , one has for and for in (56). Therefore,

Since the values of are different on , it is obvious that we need a third contour which joins up at , respectively. Here, we consider a constant phase contour connecting to that passes through . Since on the contour , one finds thatis a curve of steepest descent. On , definewhich is real on and varies from to zero and then to along . Then, one findsSimilarly, we break up the contour into and , where () is the contour above (below) of with for and for in (60). Thus,

Note that although paths and never join up at , the integrand tends to zero exponentially. Therefore, there is no contribution from a connecting path from and at a distance from the origin in the limit . As a result, the integral equals the sum of three contour integrals on the different steepest descent curves and . Combining (53), (57), and (62) gives the full asymptotic expansion of for large and negative :

#### 4. WKB Approximation

The authors of [22] find the WKB approximation in deformed space with minimal length. In [22], they consider the deformed commutation relationwhere is an arbitrary function of . In our paper, we set . Defining ,and an inverse function of , they find the physical optics approximation to the solution of the deformed Schrodinger equation,iswhere and in (67). It is also shown there that if (67) is valid, the conditionhas to be satisfied. However, the condition equation (68) fails near a turning point where . Thus, if we want to determine bound state energies, we need to be able to match wave functions at the turning points. Here we consider a potential with its classical turning point located at . A linear approximation to the potential near the turning point iswhere . The linearized potential (69) is discussed in the previous two sections. Our discussion shows that the parameter plays an important role in analyzing asymptotic behaviors of the solutions. When , the physically acceptable solution can exist at large argument , while condition (14) still holds. Accordingly, a turning points is called a smooth one if . Otherwise, it is called a sharp turning point.

##### 4.1. WKB Connection through a Smooth Turning Point

Now we want to match WKB wave functions at a smooth turning point in the deformed space with up to . Suppose is a smooth turning point, which means , and for all . The region to the left of the turning point is classically forbidden where the wave function must be damped and becomes zero at infinity. Thus, far from , the wave function has the formTo the right of the turning point, the wave function is given byAround the turning point, is small and . In this region, we may approximate (70) and (71) byThe criteria (68) for validity of the WKB approximation are satisfied ifwhere we neglect in derivation. On the other hand, when the potential is linearized around the turning point , the Schrodinger equation becomeswhere . To solve the approximate differential equation, we make the substitutionIn terms of , the solution to (75) which matches (72) and (73) in two different limits is actually calculated in Section 3. Specifically, the solution iswhere is a constant to be determined by asymptotic matching. It is easily shown from (74) that there exist overlap regions where both WKB approximation and (75) hold. In the overlap regions, one finds and . Therefore, we approximate by its leading asymptotic behaviors for large argument in the overlap regions. The appropriate formulas arewhere for a smooth turning point and as required by condition (13). Requiring that (78) and (79) match (72) and (73) in the overlap region, respectively, gives and up to . In summary, in the overlap region, we find WKB solutions and the asymptotic values of the solution to the Schrodinger equation with a linear approximation to the potential . Then, by making (78) and (79) match (72) and (73), respectively, the WKB connection formula with the deformed commutator is obtained up to . The connection formula around a smooth turning point is put in a way thatwhich is directional, just as in ordinary quantum mechanics [25]. 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-Sommerfeld quantization conditionwhere and are two smooth turning points for the potential . Notice that although (81) is claimed in [22], one still needs to obtain the connection formula to derive (81) rigorously, which is not presented in [22].

##### 4.2. Discussion

###### 4.2.1. Sharp Turning Point

Near a sharp turning point , not only the WKB approximation is no longer valid 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 . However, such region does not exist unless , which means that the asymptotic matching fails through a sharp turning point.

It can be shown, through (68), that WKB approximations are valid as long as in the region where the potential is approximated by a linear one. To put it another way, if there exists a region where both WKB and linear approximations are valid, one finds for such a region. When , we havefor a sharp turning point. However, 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.

###### 4.2.2. versus

When can be regarded as a small quantity, the approximate solution to the deformed Schrodinger equationis easy to find using WKB analysis. To be specific, the approximate solution is expressed in an exponential power series of the formThe authors of [22] findSince here , we have for Moreover, the leading order (in terms of ) of is just WKB correction calculated in the ordinary quantum mechanics. Therefore, we obtain [27]If one uses WKB approximations to evaluate quantum gravity induced corrections, say to energy levels or tunnelling rates, one may want to haveOtherwise, the quantum gravity correction (~) on the first-order WKB approximation (~) could be overwhelmed by the second-order WKB approximation (~). Suppose is the characteristic length of the potential , for example, the width of a square-well potential. Then we can get a rough estimate on :where is the de Broglie wavelength of a particle with momentum . As a result, condition (88) becomesIt is interesting to note that condition (90) 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 (90) on the de Broglie wavelength 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. However, there is an exception if the corresponding Schrodinger equation in the ordinary quantum mechanics can be solved exactly. In this case, corrections calculated on the WKB first-order approximation are just quantum gravity corrections to exact results up to even without having (90) required. For example, if we employ WKB analysis to calculate the energy spectrum of a bound state in the deformed space, the energy levels can be represented by a series in powers of:where can be expanded in terms of :If on the first-order WKB approximation one calculates up to ,the energy levels areIn order to have (94) making sense, one requires . On the other hand, if we know the exact result with , namely, ,equation (91) becomesSince is automatically smaller than , (96) always makes sense as long as .

To illustrate our points, we use the WKB approximation to derive the energy levels of a particle confined to the one-dimensional potential whose turning points areThe energy quantization condition (81) from first-order WKB approximation then becomeswhere is the characteristic length of the potential . From the last equation, we obtainwhere . What is ? The second-order generalization of (81) with is given in [25]:which givesWe can then estimate through (101):which can also be easily obtained by dimensional analysis. If one wants the first-order approximation (99) to make sense, the second term in (99) should be comparable to or larger than and then one getsThe de Broglie wavelength of a particle with energy isThus, inequality (103) readswhich is much milder than (90). In a practical way, and can be expressed in terms of , and . In fact, it is easily shown that

##### 4.3. Application

The dimensionless number plays an important role when implications and applications of nonzero minimal length are discussed. Normally, if the minimal length is assumed to be order of the Planck length , one has . In [12], based on the precision measurement of Lamb shift, an upper bound of was given by . The authors in [28] placed constraints on from the precession of the perihelion of the Mercury, which was . However, as pointed out in [29], the effective deformation parameter was substantially reduced by a factor for a macroscopic body which consists of quarks. Thus, an upper bound on for quarks was . In the following, we first use the Hamilton-Jacobi method to study the effects of the minimal length on the classical motions. The Bohr-Sommerfeld quantization is then used to investigate statistical physics in deformed spaces with the minimal length.

###### 4.3.1. Hamilton-Jacobi Method in Deformed Spaces

In [28, 29], the classical limit of deformed spaces with the minimal length has been studied by replacing the quantum mechanical commutator by the Poisson bracket viaAlternatively, we here use Hamilton-Jacobi method to probe the classical motion of a particle with the mass under the potential in deformed spaces.

For the deformed commutation relation (64), the deformed time dependent Schrodinger equation isSubstituting the ansatz into (108) and taking the limit , one finds that the leading order of (108) gives the classical Hamilton-Jacobi equation in deformed spaceswhere is the classical action. Since the potential does not depend explicitly on time, we can separate the variables aswhere can be identified with the total energy. Thus, the Hamilton-Jacobi equation becomesDefining as an inverse function of , we obtainEquation (112) can be integrated toso thatThe solution for comes from the transformation equationwhere the constant can be determined by the initial conditions and we useNow we focus on the case with .

First we consider the motion of a particle under the homogeneous field potential . Equation (115) leads towhere we neglect the terms higher than . Solving (117) for to giveswhich indicates that the initial conditions at are and Differentiating both sides of (117) twice with respect to gives the accelerationUsing , we find from (119) that

The equivalence principle is crucial in the foundations of general relativity. The weak equivalence principle is often referred to as the universality of free fall. A measure for the breakdown of the universality of free fall is the “ Eotvos ratio” [30]:which quantifies the normalized difference in the gravitational accelerations between two different bodies and . In the modern torsion-balance experiment [31], the “Eotvos ratio” has been found to befor the gravitational acceleration of Beryllium and Titanium towards the Earth. In the experiment of [31], they used