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Fractional Quantum Mechanics, Relativistic Quantum Mechanics, Fractional Schrödinger Equation, Relativistic Schrödinger Equation International Journal of Theoretical and Mathematical Physics 2015, 5(4): 58-65 DOI: 10.5923/j.ijtmp.20150504.02 The Infinite Square Well Problem in the Standard, Fractional, and Relativistic Quantum Mechanics Yuchuan Wei1,2 1International Center of Quantum Mechanics, Three Gorges University, China 2Department of Radiation Oncology, Wake Forest University, NC Abstract There has been a continuous argument on the correctness of the Laskin‟s solution for the infinite square well problem in the fractional quantum mechanics. In this paper, we prove that the Laskin‟s functions are not amathematical solution to the fractional Schrödinger equation and the equation does not have any nonzero solutions at all in the sense of mathematics. As in the standard quantum mechanics, we view the infinite square well problem as the limit of the finite well problem, and define the solution for the infinite square well problem as the limit of the solution for the finite square well problem. Using the simple property of the infinite operators, we show that Laskin‟s function can be the limit solution for the infinite square well problem. The single-sided well problem and the 3 dimensional well problem are included. This operator method works for the same problems in the relativistic quantum mechanicsas well. Keywords Fractional quantum mechanics, Relativistic quantum mechanics, Fractional Schrödinger equation, Relativistic Schrödinger equation standard Schrödinger equation either, and it is nothing but 1. Introduction the limit of the solution to the finite square problem. Without a mathematical definition of the eigenvalues and In 2000, Laskin introduced the fractional quantum eigenfunctions of a Hamiltonian operator with local infinity, mechanics [1-3]. As an example he solved the infinite square the argument will be endless. Therefore, in this paper we well problem in a piecewise fashion [3]. However, in 2010 mathematically define the infinite square well problem as a Jeng, et al [4] criticized that it was meaningless to solve a limit of the finite well. This viewpoint is also useful for other nonlocal equation in a piecewise fashion and they potentials with infinity, such as the coulomb potential in the demonstrated that it was impossible for the ground state hydrogen atom, the single-sided harmonic oscillator, etc. function to satisfy the fractional Schrödinger equation near Since it is difficult to solve the fractional Schrödinger‟s the boundary inside the well. In a series of papers [5-8], equation with a finite square well potential, we wish a direct Bayin insisted that he explicitly completed the calculation in way to find the solution of the infinite square well problem. Jeng‟s paper and the wave functions did satisfy the fractional Fortunately, we can express the fractional Hamiltonian in Schrödinger equation inside the well. Hawkins and Schwarz terms of the standard Hamiltonian. In the same way, we can [9] claimed that the Bayin‟s calculation contained serious mistakes. Luchko [10] provided some evidence that the also solve the infinite square well problem in the relativistic solution did not satisfy the equation outside the well. On the quantum mechanics, since the fractional and relativistic other hand, Dong [11] re-obtained the Laskin‟s solution by quantum mechanics are closely related. We acknowledge solving the fractional Schrödinger equation with the path that these solutions need to be verified when the solutions to integral method. It is not easy for readers to judge their the finite square well problem are reported later. mathematical argument [12, 13], but weagree with Jeng‟s We first recall the infinite square well problem in standard opinion, including that the piecewise way to solve the quantum mechanics, and then solve the problem in the equation is wrong, and that the solution does not satisfy the fractional quantum mechanics and in the relativistic quantum fractional Schrödinger equation, since we will inarguably mechanics. show that the Laskin‟s functions do not satisfy the fractional Schrödinger equation ( 1) anywhere on the x-axis. However, in fact, this solution does not satisfy the 2. Definition of the Infinite Well Problem * Corresponding author: In this section we will recall the relation between the finite [email protected] (Yuchuan Wei) Published online at http://journal.sapub.org/ijtmp and infinite square well problems in the standard quantum Copyright © 2015 Scientific & Academic Publishing. All Rights Reserved mechanics, and accordingly define the infinite square well International Journal of Theoretical and Mathematical Physics 2015, 5(4): 58-65 59 problem in the fractional quantum mechanics. 0 xa , . Vxi (8) 2.1. The Finite and Infinite Potential Wells in the xa Standard Quantum Mechanics At xa , the potential has two infinite jumps. The In the standard quantum mechanics [14], the one subscript imeans the infinite square well. dimensional time-independent Schrödinger equationis Obviously the potential Vi is not real in physics and not H x E x , (1) well-defined in mathematics, since we do not know how to determine the value of the multiplication V x() x where x is a wave function defined on the x-axis, and i outside the well, even if the wave function outside the well is E is an energy. The Hamiltonian operator zero. (Does 0 equal 0,1, or ?) In physics this H T V x (2) potential is used as a simplified model for „a very deep finite square well‟. is the summation of the kinetic energy and the Conventionally the Schrödinger equation potentialenergy of a particle. 22d The standard kinetic energy operator is V x x E x (9) 2m dx2 i pd2 2 2 , (3) T 2 is solved in a piecewise way again [15]. The wave function 22mmdx outside the well is zero because the potential is infinite and where p is the one dimensional momentum operator the wavefunction inside the well is a sine or cosine function. The revised continuity condition for this case is that the wave d functions must be continuous but their derivative should not pi . (4) dx be. The infinite square well problem has the simple and well-known solution As usual, m is the mass of the particle and ℏ is the reduced Plank constant. 1 n A finite square well [15] is defined as sin x a x a (), n x a 2a (10) 0 xa 0,xa , , Vx (5) f V x a 0 n2 2 2 (11) En 2 where V0 0 is the depth of the well and 2a is the width of 8ma the well. The subscript f means the finite square well. for n 1,2,3, . The eigenequation of the finite square well problem is However, we emphasize that this solution does not satisfy the Schrödinger Equations (9). Take the ground state as an 22d V x x E x . (6) example, we have 2m dx2 f 22d This equation can be solved separately in three regions V x x 2m dx2 11i first and then the piecewise solutions are connected by the . (12) continuity condition that the wave function and its derivative 2 must be continuous. This process generates the eigenvalues E11 x ()() x a x a 4ma a Exfn ()and eigenstates fn ()x , with n 1,2,3 . The The wave function satisfies the Schrödinger equation state with n=1 is the ground state [15]. everywhere except at x=a and x=-a. This is not a small flaw One can easily verify that the explicit solutions satisfy the for a differential equation, and we have to say that the wave eigenequation at every point on the whole x-axis functions in (10) are not the solution of the Schrödinger 22d equation at all. The extra terms in the above equation comes , (7) from the discontinuity of the derivative of the wave function, 2 fn V f x fn x E fn x 2m dx but if we keep the derivative of the wavefunction continuous, which convinces us that the continuity condition we use is we can only get a trivial solution x 0 , which is suitable. In fact this continuity condition comes from the meaningless in physics. We have to say that the Schrödinger Schrödinger equation. equation with the infinite square well potential does not have The infinite square well potential is defined as solutions in mathematics. (See the appendix if one feels 60 Yuchuan Wei: The Infinite Square Well Problem in the Standard, Fractional, and Relativistic Quantum Mechanics surprised.) where the coefficient D mc2 /() mc with χ a Since the infinite square well is a limit of the finite square well, conventionally we call the limit of the solutions of the positive real number, and c is the speed of the light. When finite square well problem 2 , taking 1/ 2 and hence Dm 1/ (2 ) , the fractional kinetic energy recovers the standard kinetic energy, limfn (x ) n ( x ), lim E fn ( x ) E n , (13) VV 2 00 i.e. T2 p/ (2 m ) T . Originally Laskin [1-3] required the solution of the infinite square well, and symbolically the fractional parameter 12 , but in this paper we write allow 02 . The fractional Schrödinger equation is 22d n V i x n x E n n x . (14) 2m dx2 HxE x (17) The piecewise way to solve the Schrödinger equation for HTV. (18) the infinite square well, together with the aforementioned revised continuity condition, is nothing but an easy and For the infinite square well problem, we have direct way to get the limit of the finite square well solution d 2 2 . (19) without solving the finite square well problem. D()()2 x Vi x x E x This definition for the infinity potential is applicable for dx several important cases in quantum mechanics.
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