One-dimensional point interaction with Griffiths’ boundary conditions F.A.B. Coutinho, Y. Nogami, and F.M. Toyama Abstract: Griffiths proposed a pair of boundary conditions that define a point interaction in one dimensional quantum me- chanics. The conditions involve the nth derivative of the wave function where n is a non-negative integer. We re-examine the interaction so defined and explicitly confirm that it is self-adjoint for any even value of n and for n = 1. The interaction is not self-adjoint for odd n > 1. We then propose a similar but different pair of boundary conditions with the nth derivative of the wave function such that the ensuing point interaction is self-adjoint for any value of n. PACS Nos: 03.65.–w Résumé : Griffiths a proposé une paire de conditions à une limite qui définissent une interaction ponctuelle en mécanique quantique en une dimension. La condition implique la n e dérivée de la fonction d’ onde, où n est un entier positif. Nous ré- examinons l’interaction ainsi définie et confirmons explicitement qu’elle est self adjointe pour toute valeur paire de n et pour n = 1. Elle n’est pas self adjointe pour les valeurs de n > 1 impaires. Nous proposons alors une paire de conditions si- milaires, mais différentes, avec la n e dérivée de la fonction d’ onde telle que le point d’interaction qui s’ensuit est self adjoint pour toute valeur de n. [Traduit par la Rédaction] 1. Introduction In one-dimensional quantum mechanics, the simplest point interaction is the one that is represented by the potential V ðxÞ¼ Z 2 2m cdðxÞ ð1Þ where m is the mass of the particle under consideration, c is a real constant and d(x) is the Dirac delta function. In the fol- lowing we use units such that Z 2 =ð2mÞ¼ 1 throughout. This interaction is a special case of the point interaction that is re- presented by the following boundary condition imposed on the wave function j(x) and its derivative j′(x)=dj(x)/dx at x = 0: j 0 þ j þ ! ¼ U j 0  j  ! U ¼ a b d g  ! ð2Þ where j ± = j(±0) and j 0  ¼ j 0 ð0Þ [1–6]. It is understood that j(x) and j′(x) are discontinuous at x = 0 in general but j(x) is twice differentiable at x ≠ 0. The matrix elements a, b, g, and d are real constants that are subject to the condition ag – bd = 1. Here the interaction is assumed to be invariant with respect to time-reversal [5]. The delta function potential of (1) leads to (2) with a = –1, b = –c, g = –1, and d = 0. Boundary condition (2) guarantees that the probability cur- rent –i(j*j′– j′*j) is continuous across x = 0. The inter- action that is defined by (2) is self-adjoint. Starting with the interaction potential of the form of V ðxÞ¼ cd ðnÞ ðxÞ¼ cd n dðxÞ dx n ð3Þ where c is a real constant and n is a positive integer, Griffiths [7] introduced the following pair of boundary conditions: Dj 0 ¼ ð1Þ n cj ðnÞ ð0Þ ð4Þ and Dj ¼ ð1Þ ðn1Þ ncj ðn1Þ ð0Þ ð5Þ Here for any function f(x), by Df, and f(0) we mean Df ¼ f þ  f  f ð0Þ¼ 1 2 ðf þ þ f  Þ ð6Þ This notation also applies to the derivatives, that is, Df (n) = f ðnÞ þ  f ðnÞ  and f (n) (0) = ð1=2Þðf ðnÞ þ þ f ðnÞ  Þ. It is understood that, for any relevant values of n, j (n) (x)=d n f(x)/dx n is well defined for x ≠ 0. The dimensionality of c varies depending on n. Griffiths’ derivation of (4) and (5) is beset with a pro- blem as was pointed out in ref. 4 but we accept (4) and (5) as such as given boundary conditions. We are interested in the self-adjointness of the point inter- action that is defined by boundary conditions (4) and (5). More precisely we mean the self-adjointness of the hamilto- nian that consists of the kinetic energy operator –d 2 /dx 2 and the interaction. The domain of the hamiltonian operator con- Received 12 December 2011. Accepted 7 March 2012. Published at www.nrcresearchpress.com/cjp on 09 April 2012. F.A.B. Coutinho. Faculdade de Medicina, Universidade de São Paulo, 01246-903, SP, Brazil. Y. Nogami. Department of Physics and Astronomy, McMaster University, Hamilton, ON L8S 4M1, Canada. F.M. Toyama. Department of Computer Science, Kyoto Sangyo University, Kyoto 603-8555, Japan. Corresponding author: F.M. Toyama (e-mail: toyama@cc.kyoto-su.ac.jp). 383 Can. J. Phys. 90: 383–389 (2012) doi:10.1139/P2012-030 Published by NRC Research Press Can. J. Phys. Downloaded from www.nrcresearchpress.com by USP UNIVERSIDADE DE SAO PAULO on 11/15/13 For personal use only.