Free-particle and harmonic-oscillator propagators in two and three dimensions

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1 Free-particle harmonic-oscillator propagators in two three dimensions Ferno Chaos U. Departamento de Física, Escuela Superior de Física y Matemáticas Instituto Politécnico Nacional, México, D. F., MEXICO chaos@gina.esfm.ipn.mx L. Chaos-Cador E. Ley-Koo Instituto de Física, Universidad Nacional Autónoma de México. Apartado Postal , México, D. F. México lorea@fisica.unam.mx ; eleykoo@fisica.unam.mx (Recebido: 21 de outubro de 2002) Abstract: This contribution illustrates how to construct free-particle harmonic-oscillator quantum-mechanical propagators in two three dimensions in cartesian, in circular spherical coordinates, respectively, starting from the corresponding one-dimensional propagators in cartesian coordinates. Key words: quantum-mechanical propagator, free particle, harmonic oscillator Resumo: Esta contribuição ilustra como construir propagadores para partícula livre e oscilador harmônico em duas e três dimensões, em coordenadas cartesianas, polares e esféricas, partindo do propagador correspondente em uma dimensão em coordenada cartesiana. Palavras-chave: propagador quântico, partícula livre, oscilador harmônico 1 Introduction The study of quantum-mechanical propagators, or Green functions, in most textbooks is restricted to one-dimensional systems, including closed analytical forms for the free particle the harmonic oscillator [1]-[4]. This contribution illustrates how to construct the corresponding Green functions in two three dimensions

2 146 Revista Ciências Exatas e Naturais, Vol. 4, n o 2, Jul/Dez 2002 in cartesian, in circular spherical coordinates, respectively. The basis for such an extension is the separability of the Schrödinger equation the consequent factorability of its eigenfunctions Green functions in the different coordinates. Green functions, K(~r,~r 0 ; t), determine the time evolution of the quantum system s wave function, ψ(~r, t) Z t 0 d N r 0 K(~r,~r 0 ; t)ψ(~r, 0). (1) They satisfy the Schrödinger equation i " # t K(~r, ~r 0 ; t) v(~r) K(~r,~r 0 ; t), (2) the initial condition K(~r,~r 0 ; t 0)δ(~r ~r 0 ). (3) Correspondingly, the Green function can be constructed as the linear superposition K(~r,~r 0 ; t) ψn(~r 0 )ψ n (~r)e ient/ (4) n0 of the orthonormal eigenfunctions ψ n (~r), each one evolving harmonically in time with the frequency E n / determined by the corresponding energy eigenvalues of the quantum system. For t 0, the initial condition of Eq. (3) is obviously satisfied because of the completeness, or closure, property of the eigenfunctions. In particular, Eq. (4) for the one-dimensional free-particle harmonic-oscillator propagators takes the respective forms [1]-[6] K fp (x, x 0 ; t) 1 dke ik(x x0) e ik2 t/2 2π 1/2 e (x x0 ) 2 /(2it), (5) q q ω 1/2 K ho (x, x 0 X H n ( ω ; t) x0 )H n ( ω x) π 2 n0 n e ω(x02 +x 2 )/2 e i(n+ 1 2 )ωt n! ω 1/2 e ω(x2 cos ωt 2xx 0 +x 02 cos ωt)/(2i sin ωt), (6) where H n (z) are the Hermite polynomials. The task of constructing the corresponding propagators in two three dimensions is accomplished in the next section in two successive stages.the first one in cartesian coordinates is straightforward, the second one involves the change to circular or spherical coordinates as well as the change to the corresponding eigenfunctions energy eigenvalues. In Section 3 we discuss some points of didactic interest.

3 Ferno Chaos U., L. Chaos-Cador E. Ley-koo Construction of the quantum propagators in two three dimensions The extensions of Eqs. (5) (6) to N-dimensions in cartesian coordinates, x 1 x, x 2 y, x 3 z, are obtained by direct multiplication K fp ({x q }, {x 0 q}; t) 1 (2π) N dk s e iks(xs x0s) e ik2 s t/2 s1 N/2 Y N e (xs x0 s) 2 /(2it), (7) s1 K ho ({x q }, {x 0 q}; t) 1/2 ωs X π s1 n s ns n s! H n s r r ωs ωs x0 s H ns x s e ωs(x02 s+x 2 s)/2 e i(ns+ 1 2 )ωst ω 1/2 s e ωs(x2 s cos ω st 2x sx 0 s+x 02 s cos ω st)/(2i sin ω st). (8) 2πi sin ω s t s1 In Eq. (7) the product of the gaussian exponentials is equal to a single gaussian exponential with argument 2it (x s x 0 s )2 2it (~r ~r 0 ) 2. s1 The propagator of Eq. (8) is valid in general for anisotropic harmonic oscillators, i. e., ω x 6 ω y 6 ω z ; but, it can be simplified further for isotropic harmonic oscillators, i. e., ω x ω y ω z ω, forwhich x 2 s rs 2 s1 x s x 0 s ~r s ~r 0 s intheproductoftheexponentials. The adaptations of Eq. (4) the changes of Eq. (7) to circular spherical coordinates the corresponding eigenfunctions eigenvalues lead to the respective free-particle propagators in N 2 3 dimensions: s1 K fp (R, ϕ; R 0, ϕ 0 ; t) dk kj m (kr 0 ) )J m (kr) eim(ϕ ϕ0 e ik2 t/2 m 0 2π ω e [R2 2RR 0 cos(ϕ ϕ 0 )+R 02 ]/(2it), (9)

4 148 Revista Ciências Exatas e Naturais, Vol. 4, n o 2, Jul/Dez 2002 K fp (r, θ, ϕ; r 0, θ 0, ϕ 0 ; t) lx l0 m l Z 2 dk k 2 j l (kr 0 )j l (kr) π 0 Ylm(θ 0, ϕ 0 )Y lm (θ, ϕ)e ik2 t/2 ω 3/2 e [r2 2~r ~r 0 +r 02 ]/2it, (10) where J m (z) j l (z) are the ordinary spherical Bessel functions, respectively, Y lm (θ, ϕ) are the spherical harmonics. Likewise, the propagators for the isotropic harmonic oscillators in circular spherical coordinates are constructed via Eqs. (4)(8)withtheresults K ho (R, ϕ; R 0, ϕ 0 ; t) ω X 2(n!) (n + m)! n0 m r ω m R L n ( m ) ω ω R2 r m ω ω R0 L n ( m ) R02 e ω(r02 +R 2 )/2 eim(ϕ ϕ0 ) 2π e i(2n+ m +1)ωt e ω(r2 cos ωt 2 ~ R ~R 0 +R 02 cos ωt)/(2i sin ωt), (11) K ho (r, θ, ϕ; r 0, θ, ϕ 0 ; t) ω 3/2 X lx r 2(n!) ω l Γ(n + l +3/2) r0 L (l+ 1 2 ) ω n r02 n0 l0 m l r ω l r L (l+ 1 2 ) ω n r2 e ω(r02 +r 2 )/2 Ylm(θ 0, ϕ 0 )Y lm (θ, ϕ)e i(2n+l+3/2)ωt ω 3/2 e ω(r2 cos ωt 2~r ~r 0 +r 02 cos ωt)/(2i sin ωt). (12) in terms of the circular spherical radial Laguerre polynomials L (α) n (z) angular harmonics, respectively. This section can be concluded by recognizing that Eqs. (7) (8) are the quantum propagators in N dimensions in cartesian coordinates for the free particle harmonic oscillator, respectively. Correspondingly, the respective propagators for N 2 in circular coordinates are given by Eqs. (9) (11) for N 3in spherical coordinates by Eqs. (10) (12).

5 Ferno Chaos U., L. Chaos-Cador E. Ley-koo Discussion The general construction of quantum propagators, Eq. (4), requires the solution of the Schrödinger Eq. (2) subject to the initial condition of Eq. (3). For the free particle the harmonic oscillator in one dimension cartesian coordinates the sumations of Eqs. (5) (6) have been explicitly done [1]-[6]. The extension to higher dimensions in cartesian coordinates, Eqs. (7) (8), are immediate for both quantum systems on account of the separability of the Schrödinger equation. It is well known that the Schrödinger equation for both systems is also separable in circular spherical coordinates for two three dimensions, respectively. Consequently, the steps from Eqs. (7) to (9) (10) correspond to the changes from cartesian coordinates plane waves to circular spherical coordinates waves, respectively. Their study can be implemented after having learned about the free particle in states with well-defined angular momentum. Similar comments are valid for the steps from Eq. (8) to (11) (12) for the harmonic oscillators. Comparison of Eqs. (9)-(12) with Eq. (4) allows the identification of the orthonormal eigenfunctions ψ n (~r) eigenenergies for each system in the corresponding coordinates. The solution of Eq. (2) subject to the initial condition of Eq. (3) is the same in the different coordinate systems, guaranteeing by construction the validity of Eqs. (9)-(12). Of course, if the reader is interested in proving their validity independently of the context of this paper, the easiest way to do it is to exp the exponentials using the Rayleigh expansion for the angular dependent factor resumming. Part of the didactic value of our results is the ease with which they can be obtained, understood explained. While Eqs. (7) (8) are valid for any dimension N, the propagators of (9)-(12) have been written explicitly for N 2 3. The extension to higher dimensions using hyperspherical coordinates hyperspherical harmonics can also be implemented for both the free particle the harmonic oscillator. References [1] L. I. Schiff, Quantum Mechanics, Third Edition. New York: McGraw-Hill, [2] D. S. Saxon, Elementary Quantum Mechanics. San Francisco: Holden-Day, [3]E.Merzbacher,Quantum Mechanics, Third Edition. New York: Wiley, [4]R.P.FeynmanA.R.Hibbs,Quantum Mechanics Path Integrals. New York: McGraw-Hill, [5] B. R. Holstein, The Harmonic Oscillator Propagator. Am. J. Phys. v. 66, p , [6] F. U. Chaos L. Chaos, Comment on The Harmonic Oscillator Propagator, by Barry R. Holstein Am. J. Phys. v. 67, p. 643, 1999.