Physica Scripta.Vol. 61, 396^402, 2000 Ehrenfest-Type Theorems for a One-Dimensional Dirac Particle Vidal Alonso* and Salvatore De Vincenzo** Escuela de F|¨ sica, Facultad de Ciencias, Universidad Central de Venezuela, A.P. 47145, Caracas 1041-A, Venezuela Received June 2, 1999; revised version received October 11, 1999; accepted October 28, 1999 pacs ref: 03.65.Pm, 03.65.Ca, 03.65.Ge Abstract current density at the walls (for example, with the Dirichlet The time evolution of the mean values of the standard position, velocity and boundary condition), the commutator H; P cannot be 2 momentum operators, for a relativistic Dirac particle in a one-dimensional de¢ned because PC2=Dom H, moreover, the operator P box, as well as for a free Dirac particle on a line with a hole, are studied. in the Hamiltonian is not really de¢ned as P Á P [4,5]. By considering the cases of ``free'' particle, con¢ned particle and particle with However, the time evolution of the mean values of the a delta function interaction, it is shown that the Ehrenfest-type theorems for operators X and P, in a box, may be easily obtained just these operators are not always valid. by being careful with the involved operators domains. Another non-relativistic model problem where the poten- tial may become in¢nite in some region or points is that 1. Introduction of the point interaction potential. These potentials may be used to approximate, in a simple way, more structured In non-relativistic quantum mechanics the Ehrenfest and more complex, short~ranged potentials. Calculations theorem [1] gives the law of motion of the mean values involving point interaction potentials, usually represented hXi C; XC and hPi C; PC of a quantum system. In fact, these equations of motion are formally identical as delta-function potentials, are greatly simpli¢ed. The gen- eral point interaction in one dimension is obtained by con- to the Hamilton equations of classical mechanics, except sidering the self-adjoint extensions of the Hamiltonian of that the quantities which occur on both sides of the classical a free particle moving on a line with the origin excluded. equations must be replaced by their corresponding operator It is found that there is a four-parameter family of self- mean values. Certainly, if there exists a potential energy, adjoint Hamiltonians that can be characterized by a the mean values hXi and hPi follow the laws of classical four-parameter family of boundary conditions imposed mechanics; but this holds rigorously only when U x is a on the wavefunction [6]. One of these point interactions cor- polynomial of at most second degree in x, for example, with U xx2 (harmonic oscillator), U xx (charged particle responds to the familiar delta-function potential. In the framework of one-particle relativistic quantum in a constant electric ¢eld) and when U x0 (free particle). mechanics, the observables are associated with operators Otherwise U x mustvarysu¤cientlyslowlyoveradistance which do not mix positive and negative energy states. Such of the order of the extension of the wavepacket, in which case operators are called even operators [7]. In that case, at least thetimederivativeofthemeanvalueofthemomentum in three-dimensional problems with potentials which are operator is almost equal to the mean value of a local ``force''. bounded at in¢nity, the relativistic quantum mechanical In non-relativistic model problems, for example, in the operator equations and the corresponding relativistic classi- in¢nite potential well, the Ehrenfest theorem has been studied [2], however, some important aspects were not prop- cal equations should be similar. This means that the mean value of any operator complies with the classical equations erly considered. The usual requirement that the wave- (Ehrenfest's theorem). function vanish at the two walls (Dirichlet boundary In this paper we study the time evolution of the mean condition) is not the most general boundary condition. In values of the operators representing position (which we fact, there is a four-parameter family of boundary con- may call ``coordinate operator''), velocity, and momentum, ditions (or, equivalently, di¡erent self-adjoint extensions in the Dirac representation, for a free Dirac particle in a of the free Hamiltonian) each of which leads to unitary time one-dimensional box, as well as for a free Dirac particle evolution [3]. It has been shown that the Ehrenfest theorem does not hold as is usually written in the literature for some on a line with a hole. As we shall see below, the domains of the operators in the equations of motion, in the of the above mentioned self-adjoint extensions of H [4], SchrÎdinger picture, must be treated with care. Clearly, in particular, for the operators X and P we have in none of these cases can we obtain the relativistic classical d=dthXi 6 i=ThH; Xi and d=dthPi 6 i=ThH; Pi. relations for the mean values, because we are not restricted For example, for a ``free'' particle in a box, that is, for a to the even part of each operator. non-vanishing probability current density at the walls We consider the standard position operator X because of (periodic boundary condition), the Ehrenfest theorem does its simplicity, but this operator mixes up positive and nega- not hold because XC2=Dom H, so the commutator H; X cannot be de¢ned. On the other hand, since tive energy states. As is well known, this e¡ect is the origin of the Zitterbewegung. By choosing the even part of the H P Á P=2m is a function of P, they commute [4]. For X operator, which leaves invariant the positive and negative a con¢ned particle in a box, i.e., with vanishing probability energy subspaces, we would obtain the quantum analog *e-mail: [email protected] of the classical relativistic relation between energy and vel- **e-mail: [email protected] ocity without Zitterbewegung (with well-de¢ned domains Physica Scripta 61 # Physica Scripta 2000 Ehrenfest-TypeTheorems for a One-Dimensional Dirac Particle 397 of the involved operators). However, in three dimensions C; HAC, and hence there is a di¤culty with the even part of the position operator d i because its components do not commute, so for this operator hAi hH; Ai: 4 the notion of localization in a ¢nite region has no clear dt T meaning (see [8] and references therein). As is well known, It is worth pointing out that equation (4) makes sense only in there is another position operator without these problems. those cases where AC 2 Dom H. In particular, for the This is the so-called Newton Wigner position operator operators X, n and P we have (which is just multiplication by x in the Foldy-Wouthuysen d i d i representation), but then we will have an acausal propa- hXi hH; Xi hni; hni hH; ni; gation of initially localized particles. This problem remains dt T dt T 5 d i with all position operators commuting with the sign of hPi hH; Pi energy [8]. Thus there are several possible choices for pos- dt T ition operators, each having attractive features but also dis- wherewehaveassumedthatXC; nC and PC belong to advantages. In conclusion, all this indicates that for Dom H. These three equations constitute the usual particles with positive energy we have no relativistic notion equations of motion for the mean values hXi, hni and hPi of position analogous to the non-relativistic one and (see Appendix I). satisfying the causality requirements of a relativistic theory. On the other hand, if XC; nC and PC do not belong to In Section 2, we present in the SchrÎdinger picture the time Dom H, the time derivative of the mean value of the evolution equations of the mean values of the coordinate, coordinate, standard velocity and momentum operators standard velocity and momentum operators. The results take must be written as into account the involved operators domains. In Section 3, d 2 we consider a ``free'' particle in a box, i.e., with probability hXiÀ Im HC; XC; current density j x; tCaC not zero at the walls: dt T d 2 j 0; tj L; t 6 0. In Section 4, we study an example of hniÀ Im HC; nC; 6 boundary conditions that describes a con¢ned particle, i.e., dt T d 2 with vanishing current at the walls: j 0; tj L; t0. In hPiÀ Im HC; PC: Section 5we consider a particle on the x-axis with a hole dt T at the origin, which simulates a delta relativistic point inter- In the following sections, this theorem is considered in the action potential. A summary is given in Section 6. cases of a ``free'' particle and a particle con¢ned in a box, as well as for a relativistic particle with a delta function interaction. The commutator H; A will be introduced 2. The evolution of the mean values wherever possible. In the SchrÎdinger picture, the time derivative of the mean value of a time independent self-adjoint operator A in the 3. ``Free'' particle in a box normalized state C C x; t is Let us consider a relativistic particle in a one-dimensional d d @C @C box on the interval O 0; L. The Hilbert space of the sys- hAi C; AC ; AC C; A : 2 2 dt dt @t @t tem is H L OÈL O; with scalar product denoted by L C1; C2 0 C1 C2dx,whereC is the adjoint of C. Taking into account the Dirac evolution equation, Using the Dirac representation we write iT @C=@tHC, one has f d i i C ; hAi HC; ACÀ C; AHC 1 w dt T T where for all t we must have C 2 Dom A\Dom H and which denotes a two-component spinor depending upon HC 2 Dom A: x 2 O and upon time.