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Generalized Uncertainty Principle and the Ramsauer-Townsend Effect

The scattering cross section of electrons in noble gas atoms exhibits a minimum value at electron energies of approximately 1eV. This is the Ramsauer-Townsend effect. In this letter, we study the Ramsauer-Townsend effect in the framework of the Generalized Uncertainty Principle.


Introduction
Various approaches to quantum gravity, such as string theory and loop quantum gravity as well as black hole physics, predict a minimum measurable length of the order of the Planck length, ℓ p = Gh c 3 ∼ 10 −35 m. In the presence of this minimal observable length, the standard Heisenberg Uncertainty Principle attains an important modification leading to the so-called Generalized Uncertainty Principle (GUP). As a result, corresponding commutation relations between position and momenta are generalized too [1]. In recent years a lot of attention has been attracted to extend the fundamental problems of physics in this framework (see for instance ). Since in the GUP framework one cannot probe distances smaller than the minimum measurable length at finite time, we expect it modifies the Hamiltonian of systems too. Recently it has been shown that the GUP affects Lamb shift, Landau levels, reflection and transmission coefficients of a potential step and potential barrier [26]. In addition, they speculated on the possibility of extracting measurable predictions of GUP in the future experiments. In this work we will follow the procedure introduced in the Ref. [26], but we are going to address the effect of GUP on the Ramsauer-Townsend (RT) effect. The RT effect can be observed as long as the scattering does not become inelastic by excitation of the first excited state of the atom. This condition is best fulfilled by the closed shell noble gas atoms. Physically, the RT effect may be thought of as a diffraction of the electron around the rare-gas atom, in which the wave function inside the atom is distorted in such a way that it fits on smoothly to an undistorted wave function outside. The effect is analogous to the perfect transmission found at particular energies in one-dimensional scattering from a square well. The one-dimensional treatment of scattering from a square well and also three-dimensional treatment using the partial waves analysis can be found in [27]. We generalize the one-dimensional treatment of the scattering from a square well to the GUP framework. We also address the condition for interference in the Fabry-Perot interferometer in the framework of GUP.

A Generalized Uncertainty Principle
Quantum mechanics with modification of the usual canonical commutation relations has been investigated intensively in the last few years (see [23] and references therein). Such works which are motivated by several independent streamlines of investigations in string theory and quantum gravity, suggest the existence of a finite lower bound to the possible resolution ∆X of spacetime points. The following deformed commutation relation has attracted much attention in recent years [1] [ and it was shown that it implies the existence of a minimal resolution length ∆X = This means that there is no possibility to measure coordinate X with accuracy smaller than h √ β. Since in the context of the string theory the minimum observable distance is the string length, we conclude that √ β is proportional to this length. If we set β = 0, the usual Heisenberg algebra is recovered. The use of the deformed commutation relation (1) brings new difficulties in solving the quantum problems. A part of difficulties is related to the break down of the notion of locality and position space representation in this framework [1]. The above commutation relation results in the following uncertainty relation: where β is the GUP parameter and γ is a positive constant that depends on the expectation value of the momentum operator. In fact, we have β = β 0 /(M P l c) 2 where M P l is the Planck mass and β 0 is of the order of unity. We expect that these quantities are only relevant in the domain of the Planck energy M P l c 2 ∼ 10 19 GeV. Therefore, in the low energy regime, the parameters β and γ are irrelevant and one recovers the well-known Heisenberg uncertainty principle. These parameters, in principle, can be obtained from the underlying quantum gravity theory such as string theory. Moreover, the comparison between Eqs. (1) and (2) shows that γ = β P 2 . Now, let us define [26]    X = x, where x and p obey the canonical commutation relations [x, p] = ih. One can check that using Eq. (3), Eq. (1) is satisfied up to O(β). Also, from the above equation we can interpret p as the momentum operator at low energies (p = −ih∂/∂x) and P as the momentum operator at high energies. Now, consider the following form of the Hamiltonian: which using Eq. (3) can be written as In the quantum domain, this Hamiltonian results in the following generalized Schrödinger equation where the second term in the left side is due to the generalized commutation relation (1). This equation is a fourth-order differential equation which in principle admits four independent solutions. Therefore, solving this equation in x space and separating the physical solutions is not an easy task. With these preliminaries, in the next section we solve equation (6) for a quantum well to address the RT effect and the Fabry-Perot interferometer resonance condition in the presence of the minimal observable length.

The Ramsauer-Townsend effect with GUP
We choose the following geometry of the quantum well (see Fig. 1) where V 0 is a positive constant and we assume E > 0.
The eigenfunctions of a particle in this potential well satisfy the generalized Schrödinger equation (6). We need to find the solutions in three different regions which are indicated in Fig. (1). To proceed further, we rewrite Eq. (6) in these regions separately as for 0 < x < a, and , and ℓ p =h 2β 3 . As state before, the above equations are forth-order differential equations which in general admit four independent solutions. However, some solutions would be unphysical which should be removed upon imposing the boundary conditions. As it is shown in Ref. [22], alternatively, we can find the equivalent physical solutions by adding the following constraint: the physical solutions should also satisfy the ordinary Schrödinger equation but with different eigenenergy. In fact, for the cases of a free particle and a particle in a box, this additional condition prevents us from doing equivalent but lengthy calculations [22]. Therefore, we demand that the eigenfunctions also satisfy the following second-order differential equations: for 0 < x < a, and . The solutions of Eqs. (10) and (11) in regions I, II and III are respectively. These solutions should also satisfy Eqs. (8) and (9) which result in A, B, C, and D. For our purposes, the solution for A is as follows So the reflection coefficient is given by Because of the smallness of the Planck length, we can obtain k ′ ≃ k 1 − 1 2 ℓ 2 p k 2 and q ′ ≃ q 1 − 1 2 ℓ 2 p q 2 from Eq. (13) and write the reflection coefficient in terms of the physical wavenumbers At this point, it is worth to mention that the rectangular potential well is an idealization, and it would be desirable to evaluate the admitted deviations of a real potential from this ideal one. Indeed, in reality, the sharp edges are changed to the smoothed out edges. A proper candidate for this case is the Woods-Saxon potential which has the following functional form [28]: where α and L are real and positive, and θ(x) is the Heaviside step function. For αL ≫ 1 this potential closely resembles a rectangular well with smooth edges and size L (Fig. 3). This potential is exactly solvable in relativistic and non-relativistic cases and the solutions can be written in terms of the hypergeometric functions [28][29][30]. Since these solutions smoothly converge to the plane wave solutions for αL ≫ 1, we expect that the GUP-corrected reflection coefficient of this system also continuously tends to Eq. (16) at this limit. However, for this case the wave function cannot satisfy both the ordinary and GUP-corrected Schrödinger equations simultaneously. This is due to the fact that the potential is not constant inside the well. This problem needs further investigation and we are going to study it in a separate program.
For the particular case where sin[q 1 − 1 2 ℓ 2 p q 2 a] = 0, there is no reflection, that is R = 0 and therefore we will have maximum transmission. This is the Ramsauer-Townsend effect. In this case In ordinary quantum mechanics this effect occurs at those wavenumbers that satisfy the condition q ord = nπ/a. This feature shows that there is a shift (∆ q = q GUP − q ord ) in the wavenumber of the transmission resonance and this shift itself is wavenumber-dependent. Up to the first-order in the GUP parameter, we find We also note that in ordinary quantum mechanical description, the condition for resonance is λ ord = 2π/q = 2a/n which is the same condition as in Fabry-Perot interferometer. In the presence of the minimal observable length, this condition modifies as follows Therefore, in the presence of the minimal length, the condition for interference in Fabry-Perot interferometer will change too. Amazingly, this change is itself wavelength-dependent.
Up to this point, we have addressed the RT effect and the Fabry-Perot interferometer resonance condition in the GUP framework. To complete our treatment of this interesting quantum mechanical problem, let us consider the negative energy case −V 0 < E < 0 which results in the quantized energy spectrum. In this case, Eqs. (10) and (11) cast into the following equations: for 0 < x < a, and . So the solutions are If we choose the center of the well as the center of the coordinate system, it is straightforward to check that the energy eigenvalues are given by the roots of equations for even and odd eigenstates, respectively. These eigenvalues can also be written in terms of physical quantities k and q as So, for the negative energy case, the energy eigenvalues are the roots of Eq. (26). We note that there is no trace of the RT effect for −V 0 < E < 0 case.

Conclusions
The scattering cross section of electrons in noble gas atoms exhibits a minimum value at electron energies of approximately 1eV, an effect of which is called the Ramsauer-Townsend effect. We studied the RT effect in the presence of the minimal observable length in the framework of the generalized uncertainty principle. We have shown that in the presence of the minimal observable length there is a shift ∆ q = q GUP − q ord ≃ 1 2 ℓ 2 p nπ a 3 in the wavenumber of the transmission resonance and this shift itself is wavenumber dependent. This shift also affects the resonance in the Fabry-Perot interferometer in such a way that this change is itself wavelength dependent. If in the future experiments one finds a similar shift in Fabry-Perot interferometer resonance wavelength, it would be an explicit trace of underlying quantum gravity scenario. We note also that the RT effect is in principle a 3-dimensional effect that needs a 3-dimensional analysis. However, corresponding calculations in 3-dimensions are so lengthy and the essential ingredients and outcomes are the same as presented in this one-dimensional analysis. Finally, we note that this problem can be treated with more real potentials such as the Woods-Saxon potential to have more realistic situation. This potential is exactly solvable in relativistic and non-relativistic cases and the solutions can be written in terms of the hypergeometric functions. Since these solutions smoothly converge to the plane wave solutions in appropriate limit, we expect that the GUP-corrected reflection coefficient of this system also continuously tends to our result at this limit. We are going to treat this more real situation in a separate research program.