Abstract
In the deformed quantum mechanics with a minimal length, one WKB connection formula through a turning point is derived. We then use it to calculate tunneling rates through potential barriers under the WKB approximation. Finally, the minimal length effects on two examples of quantum tunneling in nuclear and atomic physics are discussed.
1. Introduction
Various theories of quantum gravity, such as string theory, loop quantum gravity, and quantum geometry, predict the existence of a minimal length [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 relation:where is the Planck mass, is the Planck length, and is a dimensionless parameter. With this modified commutation relation, one can easily findwhich leads to the minimal measurable length:The GUP has been extensively studied recently; see, for example, [7–14]. For a review of the GUP, see [15].
To study 1D quantum mechanics with the deformed commutators (1), one can exploit the following representation for and :where . It can easily show that such representation fulfills the relation (1) to . Furthermore, we can adopt the position representation:Therefore for a quantum system described bythe deformed stationary Schrodinger equation in the position representation iswhere and terms of order are neglected.
If (7) with can be solved exactly, one could use the perturbation method to solve (7) by treating the term with as a small correction. However for the general , one might need other methods to solve (7). In fact, the WKB approximation in deformed space has been considered [16]. In [16], the authors considered the deformed commutation relationwhere is some function. For , one could solve the differential equation:for , and denotes the inverse function of . It is interesting to note that there might be more than one inverse function for . However, one usually finds that there is only one inverse function which vanishes at . The remaining ones are called “runaway” solutions, which are not physical and should be discarded [17, 18]. Then, they used the WKB approximation to show that the solution to the deformed Schrodinger equation:waswhere . Moreover, it also showed that condition,had to be satisfied to make the WKB approximation valid. However, condition (12) fails near a turning point, where .
For the case with , we derived one WKB connection formula through turning points and Bohr-Sommerfeld quantization rule in [19]. In this paper, we continue to consider other WKB connection formulas and calculate tunneling rates through potential barriers. The remainder of our paper is organized as follows. In Section 2, we derive one WKB connection formula and use it to find the formula for the tunneling rate through a potential barrier. Then two examples of quantum tunneling in nuclear and atomic physics are discussed in Section 3. Section 4 is devoted to our conclusions.
2. Tunneling through Potential Barriers
We now consider WKB description of tunneling through a potential barrier , which vanishes as and rises monotonically to its maximum at as approaches from either the left or the right side of . In Figure 1, we plot the potential . For a particle of energy , there are two turning points and , at which . There are two classical allowed regions, Region I with and Region III with . To describe tunneling, we need to choose appropriate boundary conditions in the classical allowed regions. We postulate an incident right-moving wave in Region I, where the WKB approximation solution to (10) includes a wave incident, the barrier and a reflected wave:In Region III, there is only a transmitted wave:In the classically forbidden Region II, there are exponentially growing and decaying solutions:whereTo calculate the tunneling rate, we need to use connection formulas to relate , and to . In [19], we derived one WKB connection formula around in the case with . If , we found thatwhich givesIn what follows, we will derive a WKB connection formula around to relate and to and then calculate the tunneling rate through the potential barrier.
To match WKB solutions, we need to solve the deformed Schrodinger equation (7) in the vicinity of the turning point . A linear approximation to the potential near the turning point iswhere . To simplify (7), a new dimensionless variable could be introduced:Thus, (7) becomeswhere . The differential equation (7) can be solved by Laplace’s method [19]. Integral representations of the solutions arewhere the contour is chosen so that the integrand vanishes at endpoints of . Specifically, define five sectors:The contour must originate at one of them and terminate at another.
The asymptotic expressions of for large values of can be obtained by evaluating the integral (22) using the method of steepest descent. To do this, we make the change of variables and findwhere with + for and − for , and in the physical region [19]. We will show below that there exists a steepest descent contour ranging from to , which is the red contour in Figure 3. Such contour could let us match the asymptotic expression of at large negative value of with the WKB solution (14) in Region III. Note that and .
The method of steepest descent is very powerful to calculate integrals of the form:where is a contour in the complex plane. We are usually interested in the behavior of as . The key step of the method of steepest descent is applying Cauchy’s theorem to deform the contours to the contours coinciding with steepest descent paths. Around a saddle point , where , there are two constant-phase (steepest) contours, on which is constant, passing through if . One of them is a steepest descent contour, along which increases as we go towards . The other is a steepest ascent contour, along which decreases as we go towards . If is integrated along the steepest descent contour, the asymptotic behavior of is dominated by the contribution from the saddle point .
In Figures 2 and 3, we plot saddle points (red points in figures) of and , respectively, and constant-phase contours passing through them. Specifically, saddle points of areand these of areThe red contours in Figures 2 and 3 are the steepest descent contours connecting to , along which the integral (24) is integrated. Note that red arrows on them denote the steepest contours’ directions. On the other hand, the black arrows on the constant-phase contours around saddle points denote the directions in which values of increase. Following the black arrows on the red contour in Figure 2, we find that and are smaller than . Thus for the case with , the asymptotic expression of is dominated by the contribution from the saddle . The method of steepest descent giveswhere is used, and terms of are neglected in the second line. For the case with , Figure 3 shows that the asymptotic expression of is dominated by the contribution from the saddle , and hence we findwhere terms of are neglected in the second line.
Around the turning point , and . In this region, we find that WKB solutions (14) and (15) becomewhere we use and terms of are neglected, and we express in terms of using (20). In the overlap regions, where and , matching WKB solutions (30) with the ’s asymptotic expressions (28) and (29) giveswhich by (16) lead to and . Since , (18) givesand the transmission probability is
3. Examples
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 [9], based on the precision measurement of STM current, an upper bound of was given by . In the following, we use (33) to study effects of GUP on decay and cold electrons emission from metal via strong external electric field.
3.1. α Decay
The decay of a nucleus into an -particle (charge ) and a daughter nucleus (charge ) can be described as the tunneling of an -particle through a barrier caused by the Coulomb potential between the daughter and the -particle (Figure 4) [20]. For an -particle of energy in the potential in Figure 4, there are two turning points, the nuclear radius and the outer turning point , which is determined byThe exponent in (33) iswhere is the mass of the -particle. At low energies (relative to the height of the Coulomb barrier at ), we have and thenThe probability of emission of an -particle is proportional to and hence the lifetime of the parent nucleus is aboutThe density of nuclear matter is relatively constant, so is proportional to the number of nucleons . Empirically, we haveTherefore, we findOn the other hand, a large collection of data shows that a good fit to the lifetime data obeys the Geiger–Nuttall law [21]where and are constants. If the effects of GUP do not make (39) differ too much from the Geiger–Nuttall law, it will put an upper bound:The GUP correction to the -decay has also been considered in [22]. We both find that the effects of the GUP would increase the tunneling probability and hence decrease the lifetime .
3.2. Electron Emission from the Surface of Cold Metals
If a metal is placed in a very strong electric field, then there exists cold emission of electrons from the surface of the metal. This emission of the electrons can be explained via quantum tunneling. In [23], the shape of a tunneling barrier was assumed to be the exact triangular barrier, which has been known as the Fowler-Nordheim tunneling. Note that work must be done to remove an electron from the surface of a metal. In “free electron gas” model, one could hence take the potential energy of the electron inside the metal to be zero and for the outside to be . At the absolute zero temperature, if the Fermi energy of these electrons is less than , therefore after reaching the surface of the metal, they are reflected back into the metal. Now if the external electric field is applied toward the surface of the metal, the potential energy becomeswhere is the magnitude of the electric field. This potential is shown in Figure 5.
We now use (33) to calculate the GUP modified transmission probability. For an electron of energy , there are two turning points:The exponent in (33) iswhich gives the transmission probability .
Next we want to calculate the electric current density in this case. As a consequence of the GUP, the number of quantum states should be changed to [24]where . Therefore, the electric current density is given bywhere . The range of variations of , , and is limited to the points inside the Fermi sphereTo calculate , we use cylindrical coordinates,and haveTo simplify the result, we change toTherefore, one hasSince decreases rapidly with increasing , therefore in , we can expand :We findwhere we extend the range of integration in (51) to , and
In [25], the Fowler-Nordheim tunneling in device grade ultra-thin 3–6 nm /-Si structures has been analyzed. Typically for this electron tunneling, we have Therefore from (53), the correction due to GUP is given byThe comparison of the calculated and experimental tunnel current was plotted in FIG. 8 of [25], which implies . Then the upper bound on follows:
4. Conclusions
In this paper, we considered quantum tunneling in the deformed quantum mechanics with a minimal length. We first found one WKB connection formula through a turning point. Then the tunneling rates through potential barriers were derived using the WKB approximation. Finally, effects of the minimal length on quantum tunneling were discussed in two examples in nuclear and atomic physics, decay and the Fowler-Nordheim tunneling. Upper bounds on were given in these two examples, in which one had and , respectively.
In [9], the effects of the minimal length on electrons tunneling through a potential barrier of formwere considered. For electrons of the mass and energy , the GUP deformed Schrodinger equation was solved for the transmission coefficient, which was given bywhere is the standard tunneling amplitude and used in [9] is in this paper. Although the exact result for the transmission coefficient was given by (59), it is a good exercise to use (33) to calculate the transmission coefficient and compare the WKB approximated result with the exact one. In fact, there are two turning points and for potential (58). Thus, we haveSince , one findswhere is the standard WKB tunneling amplitude. It is noteworthy that the second term in the right-hand side of (59) is , while the third term is and hence is the leading order term of the WKB expansion. So as expected, the WKB result (61) agrees with the exact one (59) to . To put an upper bound on , the authors considered the following approximate values:In this case, we find from (59) thatwhich shows that when eV, the leading WKB order term dominates, and hence the WKB approximation is a good one. Note that it was assumed that in [9], and then the first term in the right-hand side of (63) dominates. However, if the leading WKB order term dominates, one can follow the argument in [9] and haswhich might put a stronger bound on in some other case.
In the framework of the doubly special relativity, it has been shown that there is an upper bound on the momentum for a test particle [26]. To incorporate the existence of the minimal measurable length and the maximal measurable momentum, a new form of the GUP has been proposed [11, 27]:where is the GUP parameter and can be related to through dimensional analysis with . Note that the relation implies that since . The Hilbert space representation of quantum mechanics with the deformed fundamental commutation relation (65) was investigated in [28]. Comparing to the quantum mechanics with only a minimal measurable length [6], the authors of [28] found that there were several novel and interesting features in this deformed quantum mechanics. For example, it showed that the expectation value of energy in the formal position eigenvectors and the energy at the short wavelength of a test particle were no longer divergent, while both of them were divergent in the quantum mechanics with only a minimal measurable length. Moreover, the authors also found that the position operator was symmetric but not self-adjoint, while the momentum operator was self-adjoint due to absence of the minimal uncertainty in momentum.
On the phenomenological side, the upper bounds on have been studied [29, 30]. For example, the effects of the minimal length and the maximal momentum on the transition rate of ultra-cold neutrons bouncing above a mirror in the Earth’s gravitational field were investigated in [29], and the accuracy of Nesvizhevsky experiments put an upper bound on : which implies that . Moreover, some tighter bounds came from studying the Planck scale effects on current experiments in Superconductivity and muon experiments [30], which gave and , respectively.
Competing Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors are grateful to Houwen Wu and Zheng Sun for useful discussions. This work is supported in part by NSFC (Grants nos. 11005016, 11175039, and 11375121).