The Solution of the Truncated Harmonic Oscillator Using Lie Groups

Transcription

1 Advanced Studies in Theoretical Physics Vol. 11, 2017, no. 7, HIKARI Ltd, The Solution of the Truncated Harmonic Oscillator Using Lie Groups D. Millan Universidad Nacional de Colombia Departamento de Física Bogotá-Colombia F. Fonseca Universidad Nacional de Colombia Grupo de Ciencia de Materiales y Superficies Departamento de Física Bogotá-Colombia Copyright c 2017 D. Millan and F. Fonseca. This article is distributed under the Creative Commons Attribution License, which permits unrestrikd use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper we solve the problem of the harmonic truncated oscillator by using the symmetry Lie group method. We get good agreement with previous analytical results. Keywords: Truncated harmonic oscillator, Lie groups 1 Introduction The symmetry concept is one of the most stimulating and profound ideas in Mathematics and Physics. This concept has been used from the solid state crystallography groups, till the relativistic invariance of quantum field theories. Among the mathematical tools that used symmetry, as the key point, we can find Lie group theory. Basically, the method looks for solutions of differential equations treating them as on a hypersurface. Basically, it is used

2 328 D. Millan and F. Fonseca the infinitesimal coefficients and symmetry generators of the equation. Of all possible particular solutions, the symmetry criterion allows to find only those, who are infinitesimal invariant under symmetry transformations, [1]-[3]. 2 Truncated Harmonic Oscillator We start with the Fokker-Planck equation for an harmonic truncated oscillator, P (x, t) t (f(x)p (x, t)) = + D 2 P (x, t) (1) x x 2 The oscillator is truncated under the next force condition: f(x) = { kx d x d 0 d > x > d } (2) For each region, eq. (1), the Fokker-Planck is: P (x, t) t = D 2 (P (x, t)) x 2 d > x > d (3) P (x, t) t (f(x)p (x, t)) = + D 2 P (x, t) d x d (4) x x 2 Where D is known as the diffusion constant [4], [6], k is the Hooke constant, and d is the position where the oscillator jumps. 3 The solution to P t + DP xx = 0, for d > x > d The space for this equation is six dimensional, each for every variable and derivative. We build up the infinitesimal coefficients according to each dependent or independent variable in the differential equation: ξ(x, t, P ) related to x, η(x, t, P ) to t y φ(x, t, P ) associated to P. Then, we construct the general expression for the symmetry generator, [1], [7]: V = ξ(x, t, P ) x + η(x, t, P ) + φ((x, t, P )) t P (5) The second prolongation of the symmetry generator, given in eq. (5), is:

3 Solution of the truncated harmonic oscillator 329 P r 2 V = ξ(x, t, P ) x + η(x, t, P ) + φ((x, t, P )) t P + φx (6) P x +φ t + φ xx + φ tt + φ xt P t P xx P tt P xt The quantities φ α,j are given by p φ α,j = D J Q α + ξ i (x, u)u α J,i (7) i=1 Q α (x, u 1 ) = φ α (x, u) p i=1 ξ i (x, u) uα x i (8) Where α represents each dependent variable, J the independent variables and as a subindex represents the derivative respect those variables. Q is known as the characteristic equation of the symmetry generator and D J is the derivative respect to J. Now, we apply the symmetry condition who establishes that the prolongation of the symmetry generator applied to the differential equation must be zero, [1], [7]: (ξ(x, t, P ) x + η(x, t, P ) + φ((x, t, P )) t P + φx (9) P x +φ t + φ xx + φ tt + φ xt )(D 2 (P (x, t)) P (x, t) ) = 0 P t P xx P tt P xt x 2 t The result is: Dφ xx φ t = 0 (10) Equations (7) and (8) let us to know the expressions φ xx and φ t, which are: φ t = D t Q + ξp tx + ηp tt (11) φ xx = D xx Q + ξp xxx + ηp xxt (12) Q = φ ξp x ηp t (13)

4 330 D. Millan and F. Fonseca Monomial Coefficient P t 2ξ x Dη xx + η t = 0 Px 2 φ P P 2ξ xp = 0 P x 2Dφ xp Dξ xx + ξ t = 0 P x P t 3ξ P 2Dη xp + ξ P = 0 Px 3 ξ P P = 0 P xt P x η P = 0 P xt η x = 0 Pt 2 0 Px 2 P t η P P = 0 1 φ t = 0 Table 1: Table of Infinitesimal coefficients, [7]. Replacing them in eq. (9), we get: 2P t ξ x + DP 2 x (φ P P 2ξ xp ) + DP x (2φ xp ξ xx ) 3P x P t ξ P (14) DP 3 x ξ P P 2DP xt P x η P 2DP xt η x P 2 t η P DP 2 x P t η P P 2DP x P t η xp DP t η xx φ t + P x P t ξ P + P x ξ t + P 2 t η P + P η t = 0 Organizing this equation as a polynomial of P i, the coefficients become a set of coupled system of differential equations for the infinitesimal coefficients. Then, we get: η = 2Da 1 t 2 + 2a 2 t + a 4 (15) ξ = 2Da 1 xt 2Da 3 t + a 2 x + a 5 (16) φ = (a 1 P )/2x 2 + a 3 P x + a 6 P + a 7 g(x) + a 8 (17) Where the constants a i are known as the Lie constants. Then, the symmetry generator, also called symmetry algebra, is: V = a 1 ( 2xt x 2Dt2 t + (P x2 )/2 P ) (18) +a 2 (x x + 2t t ) + a 3( 2Dt x + P x P ) +a 4 t + a 5 x + a 6P P + a 7g(x) P + a 8 P

5 Solution of the truncated harmonic oscillator 331 The lineal independence of the infinitesimal coefficients, let us to find particular solutions to the chosen a i. We choose the a 3 constant, then we build the characteristic equation of the particular symmetry generator and we apply the symmetry criteria, [1]-[3], [7]. Then, we get: a 3 P x + 2Da 3 t P x = 0 (19) The solution corresponds to the diffusion equation: DP xx P t = 0 (20) P (x, t) = 1 4πDt e x2 /4Dt (21) 4 The solution to DP xx + kxp x + kp P t = 0 Following the same procedure as in section (3), we find the infinitesimal coefficients corresponding to x, t and P, are: ξ = a 1 e kt (22) η = a 2 (23) φ = a 3 g(x, t) (24) We choose the constants a 1 and a 2, then, the symmetry generator is: V = a 1 e kt x + a 2 t (25) After constructing the characteristic equation for the symmetry generator, and applying the criterion of symmetry, the solution is: P (x, t) = c 1 e ( x(kx+2ae( kt) ))/2D + c 2 (π/2dk)e ( ae( kt) )/2Dk (26) e ( x(kx+2ae( kt) ))/2D Erf [ (kx + ae ( kt) )/ 2Dk ] i

6 332 D. Millan and F. Fonseca Figure 1: Probability density function for the harmonic truncated; both solutions eqs. (27) and (30), are on the same curve, [7]. Where a = a 1 /a 2. When we choose only the real part of this solution, we normalize it and express the full solution of the truncated oscillator, then, we get, [7]: P (x, t) = N 1 / 4πDte ( x2 )/4Dt, x < d N 2 (k/2πd)e ( (kx+ae( kt) ) 2 )/2Dk d x d N 3 / 4πDte ( x2 )/4Dt x > d (27) Here we have the normalization functions corresponding to the position N 1 and N 2, both of them are time dependent. These two functions are obtained from two conditions of the problem: the first one is that both parts of the solution must match in x = d, and the second one is the normalization condition, for the complete solution. Then, the two functions are, [1], [7]: N 1 = N 2 2kte ( (kd+ae kt ) 2 )/2Dk+d 2 /4Dt (28) N 2 = (k/2πd)(2e ( (kd+ae( kt) ) 2 )/2Dk+d 2 /4Dt d + e ( (kx+ae kt ) 2 )/2Dk dx) 1 d d e ( x2 )/4Dt dx (29) The two integrals in N 2 have solutions that involve error functions, so their accuracy is as great as the number of summands of the error function can be. On the other hand, the solution given in [4], for the truncated harmonic oscillator, is:

7 Solution of the truncated harmonic oscillator 333 Figure 2: The continuous line corresponds to the error generated using the Lie group method. The dashed line is the error given in [4]. The discontinuity of the curves in d = 1, 55 is due to the potential jump of the oscillator at this point, [7]. P (x, t) = N I / 4πDte ( x2)/4dt, x < d N I g(t) jm=0(1/(2 m m!) (k/2πd)) e (kx2 )/2D H m ( (k/2d)x)h m (0)e t mk d x d N I / 4πDte ( x2)/4dt, x > d (30) Where g(t) = e ( d2 /4Dt / 4πDt( j m=0 H m ( (k/2d)d)h m (0)e t mk ) 1 (1/(2 m m!) (k/2πd))e (kd2 )/2D (31) N I = (2/ 4πDt d d e ( x2)/4dt dx + g(t) (k/2πd))e (kd2 )/2D H m ( j (1/(2 m m!) (32) d m=0 (k/2d)d)h m (0)e t mk dx) The Hermite polynomials of m-order are given by H m (y). This kind of series for the Fokker-Planck is already known in a very interesting work, given in [4]- [5]. In order to compare the Lies group solution with the one given in reference [4], we use the next values d = 1, 55, D = 1 and k = 1, 4. Also, we took till the tenth summatory term of the error function, plotting for t=1. In figure (1),

8 334 D. Millan and F. Fonseca we show the probability density for m=0; we remark the total coincidence of both curves, showing the validity of the Lie group method to solve the Fokker- Planck. In figure (2), we show the error generated in both solutions when we replace the solutions, equations (27) and (30) in the oscillator differential equations (3) and (4), as appears in eq. (33), [7]. Err = { 5 Conclusions DP xx P t, d > x, x > d DP xx + kxp x + kp P t, d x d } (33) We solve the Fokker-Planck equation using Lie groups. The founded solution was contrasted with the results in reference [4]. Both solutions match quite well, fig. (1). In addition, the error because of the occurrence of error functions, during the normalization process, is lower in the case of Lie group method, fig. (2). Acknowledgements. F. Fonseca acknowledges support by Universidad Nacional de Colombia under Hermes project (32501). References [1] P. J. Olver, Applications of Lie Groups to Differential Equations, Springer- Verlag, New York, USA, [2] F. Oliveri, Lie Symmetries of Differential Equations: Classical Results and Recent Contributions, Symmetry, 2 (2010), [3] Ortiz Alvarez, H. Hernan, J. Garcia, F. Nelly, P. Agudelo, A. Enrique, Algunas Soluciones Exactas para la Ecuación Unidimensional de Fokker- Planck Usando Simetrías de Lie, Revista de Matemáticas de la Universidad de Costa Rica, 22 (2015), [4] M. T. Araujo, & E. Drigo Filho, Approximate Solution for Fokker-Planck Equation, Condensed Matter Physics, 18 (2015), no. 4, [5] M. T. Araujo, & E. Drigo Filho, A General Solution of the Fokker- Planck Equation, Journal of Statistical Physics, 146 (2012),

9 Solution of the truncated harmonic oscillator 335 [6] A. Einstein, On the Movement of Small Particles Suspended in Stationary Liquids Required by the Molecular-Kinetic Theory of Heat, Annalen der Physik, 17 (1905), [7] D. Millán De la Cruz, Solución del Problema de una Onda de Choque en un Medio Gaseoso por Métodos Estocásticos, Tesis de Maestría en Física, Universidad Nacional de Colombia, Departamento de Física, Bogotá-Colombia, Received: May 21, 2017; Published: June 6, 2017