PATH INTEGRAL SOLUTION FOR A PARTICLE WITH POSITION DEPENDENT MASS

Transcription

1 Vol. 9 ACTA PHYSICA POLONICA B No PATH INTEGRAL SOLUTION FOR A PARTICLE WITH POSITION DEPENDENT MASS N. Bouchemla, L. Chetouani Département de Physique, Faculté de Sciences Exactes Université Mentouri, Constantine, Algéria lyazidchetouani@gmail.com Received July, 9; revised version received August, 9 The problem of the particle with variable mass is considered by the approach of path integral. The Green s function related to this problem is reduced to that of a particle with a constant mass. As examples, simple cases are considered. PACS numbers: 3.65.Db, 3.65.Ge. Introduction We know that a simple description of the motion of a particle interacting with an external environment consists of replacing the mass by a so-called effective mass, this effective mass is in general variable and dependent on the position. From the classical point of view it is known that the Lagrangian or the Hamiltonian which can be constructed and associated to the equation of motion relating the particles with variable masses, is not unique. Among various Lagrangians or Hamiltonians, there exists a form where the kinetic energy has the standard form in mxẋ or in p /mx mass being variable. From the quantum point of view and at the level of the Hamiltonian, the replacement of the classical variables x and p by operators ˆx and ˆp poses the problem of the order. Thus, the Hamiltonian operator and the Schrödinger equation are not unique and it is not possible, in spite of the limit that all the Hamiltonian operators give the classical Hamiltonian, to remove this ambiguity of the order in Ĥ, except some physical conditions such as, for example, the hermiticity condition. The problem of variable mass can be formulated by the path integral approach where the associated propagator takes the standard form of exp i Spath S being the action. For this purpose it is necessary to path 7

2 7 N. Bouchemla, L. Chetouani start with a Hamiltonian operator where the order is fixed, and then to use the usual process: application of the Trotter formula, elimination of operators ˆx and ˆp. The order problem in Ĥ is then transposed in the path integral on the various ways of carrying out the discretization post-point or mid-point,.... For example, if Ĥ is chosen following the Weyl order, it will appear in the path integral formulation the Lagrangian of classical mechanics where the mid-point plays a central role. In addition, in some cases of potentials it is necessary to introduce regulating functions or to change the parameterization of paths by using a new time s instead of the usual time t in order to obtain a regular expression. This procedure completed with a transformation on the coordinates enabled to solve practically all the problems of standard quantum mechanics. In this paper we propose, using the path integral approach, to consider the problem of position dependent mass which is not sufficiently studied, despite the large literature which has been devoted to it 3 9. Also, our aim in this paper is to show, by using the path integral formalism, how to transform the problem of a particle having a variable mass into a problem of particle with a constant mass and to establish the effective potential V eff in which was induced. For this purpose, we adapt the procedure of Duru Kleinert related to particles having a constant mass in order to determine the corrections induced by the regularization and the transformation on the coordinate. Finally, let us note that the problem of variable mass has been also studied by the so-called supersymmetric approach and that connection with the mass constant problem was shown by using the transformations on the coordinates and on the wave functions.. Green s function First, let us consider the Green s function operator G, solution of the formal equation E Ĥ Ĝ = ii, where is the Hamiltonian operator with the kinetic term Ĥ = ˆT + ˆV, T = m α x ˆpm β x ˆpm γ x + m γ x ˆpm β x ˆpm α x, 3 α + β + γ =, and V is the potential term, α, β and γ are parameters.

3 Path Integral Solution for a Particle with Position Dependent Mass 73 In order to make the kinetic term constant, we choose two arbitrary regulating functions f l x, f r x whose product is fx and we introduce them as follows f l E Ĥf rfr G = if l, 5 then, we obtain for G, an equivalent expression i G = f r f l E Ĥf f l. 6 r In the configurations space, the Green s function becomes i G x b, x a ; E = x b f r f l E Ĥf f l x a 7 r i = f r x b f l x a x b f l E Ĥf x a r = f r x b f l x a ds x b exp i f lxĥ Ef rxs x a, where in the last line, the exponential form is introduced in order to pass to the path integral formulation. Let us subdivide the time interval S into N intervals having a length each one equal to σ = S/N, G x b, x a ; E = f r x b f l x a exp i f lxĥ Ef rxσ ds lim x b exp i N f lxĥ Ef rxσ... exp i f lxĥ Ef rxσ x a, 8 and with the completeness relation dx n x n x n =, 9 G becomes G x b, x a ; E = f r x b f l x a x n exp ds lim N N dx n i f lxĥ Ef rxσ x n.

4 7 N. Bouchemla, L. Chetouani First, let us consider the term f l xĥ Ef rx, by using the commutation relations since the mass is not constant m α x, ˆp = i αm x m α x, the kinetic term is arranged by moving ˆp f l xĥ Ef rx = f l x mα ˆpm β ˆpm γ + m γ ˆpm β ˆpm α + V x E f r x. The term x n exp i f lxĥ Ef rxσ x n is then calculated and in the exponent, it appears the following expression f l x n { m α x n ˆp m β x n m γ x n +m γ x n ˆp m β x n m α x n { + iβ m α x n ˆpm x n m β x n m γ x n } + m γ x n ˆpm x n m β x n m α x n } + V x n E f r x n, 3 in the nd step, and in order to eliminate the operators, we introduce the completeness relation dp n p n p n =, with the scalar product p n x n = π e i pnxn 5 and after having performed the integrations on the canonical variables p n, the Green s function in the general case of the problem with position dependent mass becomes N Gx b, x a ; E = f r x b f l x a ds lim dx n N e iπσf l x n f r x n / m α nm β n mγ n + mγ nm β n mα n 8 < i P : xn iβσf l xnfrx n m α 9 n m n mβ n mγ n +mγ n m n mβ n mα 3 = n ; 6 m α n mβ n mγ n +mγ n mβ n mα n 7 5 σf l xnfrx n e i P σf lx nf rx n {E V x n }. 6

5 Path Integral Solution for a Particle with Position Dependent Mass 75 Let us simplify the study of the variable mass problem by considering the simple case of the following Hamiltonian i.e. we fix the parameters as follows Ĥ = ˆp ˆp + V x, 7 mx α = γ =, β =, 8 since the regulating functions are arbitrary, we choose Thus the kinetic term can be rearranged f r x = f l x = f / x. 9 fx ˆp mx ˆp fx = fx ˆp ˆp fx mx mx fx fx = mx ˆp mx + fx 3 m x m 3 x m x m, x and following this rearrangement, it appears a potential in and G takes the following form G x b, x a ; E = fx b fx a lim N exp i x n exp i exp i ds N dx n fxn E V x n fx n σ σfx n then after elimination of the operators σ fx n mx n ˆp m x n m x n 3 fx n mx n m x n m 3 x n x n,

6 76 N. Bouchemla, L. Chetouani Gx b, x a ; E = fx b fx a lim N exp exp i i i exp and by using the integrals N ds dx n fxn E V x n fx n σ dp n i exp π σ fx n mx n σ fx n dp exp ap + bp = p n x n x n fx n mx n p n m x n m x n 3 in order to eliminate the variables p n, we obtain finally G x b, x a ; E = fx b fx a lim N exp i exp i m x n m 3, x n π a exp b, 3 a N ds iπσ fx n fx n /mx n mx n x mx n mx n σ fx n frx n σfx n {E V x n dx n m x n m x n 3 Let us make the change on the regulating function f g m } x n m 3. x n gx = fx mx. 5

7 Path Integral Solution for a Particle with Position Dependent Mass 77 With this change G becomes G x b, x a ; E = where W x = gx m x b g / x b mx a g / x a lim N exp i ds N dx n mxv x E + N gxn gxn gx n x σ gx n gx n σw x n, 6 m x mx 3 m x m. 7 x At this level, we notice that the kinetic term still has an inconvenient form containing a space dependent mass. This space dependence can be removed by a coordinate transformation x = F y. 8 Obviously, this transformation generates three corrections: the first related to the measure, the second, to the action and the third correction related to the factor in front of the Green s function. The postpoint expansion of x n reads at each n x = F y n F y n = F y y F y y + 3 F 6 y 3 y The choice of F is arbitrary, we impose the following condition F = g, 3 y thereafter, the mass being in the kinetic term is constant =. First, let us develop the exponential with the kinetic term. We have i x exp σ gx n gx n i = exp y + C act, 3 σ

8 78 N. Bouchemla, L. Chetouani where { C act = i y σ F y F y } + 3 F y 3 6 F y y +... is the first correction. The measure induce also a correction N dx n = n= d x n = where J is the Jacobian of the transformation and n= J = x y = F y + C meas Jd y n, C meas = F y F y y + 3 F y F y y +... is the nd correction. Also, the prefactor in the Green s function contribute by a correction C f which is obtained in the development of where C f = F y F y y + gxn = + Cf, gx n F y F y 3 F y y +... F y is the 3rd correction. By combining this three corrections we obtain the total correction C T defined by + C T = + C meas + C f + C act. 3 The corrections terms are evaluated perturbatively using the expectation values y n = i σ n n, 33 m and C T is replaced by the following effective potential V eff = 3 F y 3 F y 3 F y. 3 8 F y

9 Path Integral Solution for a Particle with Position Dependent Mass 79 The Green s function relating to the nonrelativistic problem with position dependent mass is finally the following G x b, x a ; E = SR i m b F b m af a / ds ẏ «ds Dyse W, 35 where W = F mxv x E + y F y + 3 F y 3 F y 3 F y m x mx 3 m x m x. 36 In order to illustrate our calculations, let us make two applications... Applications Let us consider the cases treated in st case: mx = cx and V x = A/cx + Bcx. With this choice, the Green s function relating to the problem with position dependent mass can be reduced to that of a particle of mass = and subjected to the action of the combination of a harmonic force and a centrifugal barrier. In this case, it is sufficient to choose an identical transformation x = y = F y the Green s function has the following expression i G x b, x a ; E = cx b x a ds Dxs e SR ẋ cex + A+g x +B ds = cx b x a x b x a ω i e i B ωs ω iω ds sinωs exp x b + x ωxb x a a cotωs I ν, 37 i sinωs where I ν is the Bessel function with ν = A/ 7/. In order to extract the energy spectrum and the corresponding wave functions, let us separate the variables x b, x a and S with the help of the Hill Hardy formula 3 by putting X = ω x a, Y = ω x b, Z = e iωs. 38

10 7 N. Bouchemla, L. Chetouani Then G x b, x a ; E = c x b x a 3 n= exp ω ω n! / ω υ/ x a Γ n + υ + x a L υ ω n a x exp ω ω n! / ω υ/ x b Γ n + υ + x b L υ ω n b x exp iωs + n + υ β ds, 39 ω and which is reduced as G x b, x a ; E = n= ω exp ω ω x a L υ ω n x a exp ω ω x b L υ n x b with ce = ω. Let us integrate on S c x 3 / a n! ω Γ n + υ + x a c x 3 b n! Γ n + υ + exp iωs exp iωs + n + υ β ds = ω from the poles, we obtain the energy spectrum β υ/ / ω υ/ x b + n + υ β ω iω + n + υ β ω ds,, E n = cn + υ +, with n =,,..., and using the standard form of the Green s function G x b, x a ; E = i n= Ψ nx b Ψ n x a E E n, 3 we can extract from residues, the corresponding wave functions:

11 Path Integral Solution for a Particle with Position Dependent Mass 7 Ψ n x = ωβ 3 x 3 n! n + υ + Γ n + υ + exp ω ω x L υ n x / ω x υ/ suitably normalized. nd case: m = m expcx and V = V expcx. With the transformation x y defined by the Green s function becomes G x b, x a ; E = c Dys exp i x = F y = ln y /c, m expc/ x b + x a y b y a / S ẏ ds + E m c V m c y + 8 y ds, 5 which has the same form as the Green s function relating to a particle of mass = subjected to the action of a harmonic force and a centrifugal barrier. The Green s function being known G x b, x a ; E = c m expc/ x b + x a y b y a / exp iω ds exp i/e m /c S sinωs ω i y b y a y b + y a cot ωsiυ ωyb y a i sinωs, 6 with υ = i. 7 By using the same formula of separation of variables 3 the Green s function is finally written:

12 7 N. Bouchemla, L. Chetouani G x b, x a ; E = m exp c/ x b + x a c ω ω n! / ω υ/ exp y a Γ n + υ + y a L υ ω n a y n= ω ω n! / ω υ/ exp y b Γ n + υ + y b L υ ω n y b ds exp iω + n + υ Em ωc s. 8 It is then easy to extract the energies spectrum, E n = v m c n + + υ, n =,,..., 9 as well as the corresponding wave functions which are also normalized Ψ n x = ω with ω = m v /c. n! / ω Γ n+υ+ ecx υ/ cxb exp ω ω ecx L υ n ecx, 5 3. Conclusion In this paper we showed how to treat the problem of a particle having a position dependent mass variable mass by the use of Duru and Kleinert procedure related to particles having a constant mass and to determine the corrections induced by the combination of the path-dependent time reparametrization and a coordinate transformation. We have also shown how to transform a problem of position dependent mass into a problem of constant mass and how to obtain the relation which exists between the two Green s functions variable mass and constant mass. For that, in order to regularize the kinetic energy we introduced the functions f r = f l = f / and we tacked account the terms in y order of σ and after transformation we obtained for the classical trajectory in a time interval σ, an unique action or Lagrangian. Our Green s function obtained is thus completely symmetrical in respect to the initial and final points this is not the case of propagator for example.

13 Path Integral Solution for a Particle with Position Dependent Mass 73 Finally, for the general case of variable mass depend on the position and of time the study is in progress and the results can be found elsewhere. REFERENCES J. Sa Borges, L.N. Epele, H. Fanchiotti, C.A. Garcia Canal, F.R.A. Sim ao, Phys. Rev. A38, H. Kleinert, Path Integrals in Quantum Mechanics Statistics and Polymer Physics, Second Edition, World Scientific, Singapore, New Jersey A. Ganguly, L.M. Nieto, J. Phys. A, L. Dekar, L. Chetouani, T.F. Hammann, J. Math. Phys. 39, ; Phys. Rev. A59, A.D. Alhaidari, Phys. Rev. A66, 6. 6 A. Ganguly, L.M. Nieto, Mod. Phys. Lett. A7, A. Schulze-Halberg, J. Phys. A 8, C. Tezcan, R. Sever, J. Math. Chem., K.C. Yung, J.H. Yee, Phys. Rev. A5, 99. A. de Souza Dutra, M. Hott, C.A.S. Almeida, Eur. Phys. Lett. 6, 8 3. T. Tanaka, J. Phys. A 39, 9 6. A. de Souza Dutra, C.A.S. Almeida, Phys. Lett. A75, 5. 3 I.S. Gradshteyn, I.M. Ryzhik, Table of Integrals and Products, Academic, New York 965, p. 5, Eqs 8 8.