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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 of the mean values of the standard position, velocity and boundary condition), the ‰H; PŠ cannot be 2 momentum operators, for a relativistic Dirac particle in a one-dimensional de¢ned because PC2=Dom H†, moreover, the 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 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 hXiˆ C; XC† and hPiˆ 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 , 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 x†x2 (harmonic oscillator), U x†x (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 x†ˆ0 (free particle). mechanics, the 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=dt†hXi 6ˆ i=T†h‰H; XŠi and d=dt†hPi 6ˆ i=T†h‰H; PŠi. 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; XŠ 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 ˆ 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 hAiˆ h‰H; AŠi: 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- hXiˆ h‰H; XŠi ˆ hni; hniˆ h‰H; nŠi; gation of initially localized particles. This problem remains dt T dt T 5† d i with all position operators commuting with the sign of hPiˆ h‰H; PŠi 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 hXiˆÀ Im HC; XC†; current density j x; t†ˆC‡aC not zero at the walls: dt T d 2 j 0; t†ˆj L; t† 6ˆ 0. In Section 4, we study an example of hniˆÀ 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; t†ˆj L; t†ˆ0. In hPiˆÀ 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; AŠ 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 ˆ C x; t† is Let us consider a relativistic particle in a one-dimensional  d d @C @C box on the interval O ˆ‰0; LŠ. The of the sys- hAiˆ C; AC†ˆ ; AC ‡ C; A : 2 2 dt dt @t @t tem is H ˆ„L O†ÈL O†; with scalar product denoted by L ‡ ‡ C1; C2†ˆ 0 C1 C2dx,whereC is the adjoint of C. Taking into account the Dirac evolution equation, Using the Dirac representation we write iT @C=@t†ˆHC, one has  f d i i C ˆ ; hAiˆ HC; AC†À 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. f and w are respectively the so-called Since the operator A is self-adjoint: C; AHC†ˆ large and small components of the Dirac spinor. AC; HC†, and we write A necessary condition in order to have a relativistic ``free'' particle in O is that the probability current density be non d i i 2 zero at the walls: j 0†ˆj L† 6ˆ 0.Therelativisticphysical hAiˆ HC; AC†À AC; HC†À Im HC; AC†: 2† dt T T T momentum operator P in O,de¢nedbyPC x; t†ˆ With the above requirements on C x; t†,thisequationis ÀiT @=@x††C x; t†; has the domain [9],   always true and may be used to calculate the time derivative f of hAi: One might have Ran A†\Dom H†ˆ ,where Dom P†ˆ C ˆ C 2 H; a:c: on O; PC 2 H; w Ran A† is the range of A, in which case HA and ‰H; AŠ  7† are meaningless. C fulfils C L†ˆC 0† 6ˆ 0 However, if AC 2 Dom H†, the commutator ‰H; AŠ may be introduced, and Eq. (1) written as where hereafter a.c. means absolutely continuous functions.  d i i i The Hamiltonian operator for a ``free'' particle, H  HF, hAiˆ HC; AC†À C; HAC†‡ C; ‰H; AŠC : 3† is a function of P in the interval O,andisgivenby dt T T T 2 The Hamiltonian operator is self-adjoint, so HC; AC†ˆ HF P†ˆcaP ‡ mc b: 8†

# Physica Scripta 2000 Physica Scripta 61 398 Vidal Alonso and Salvatore De Vincenzo

The domain of HF is essentially induced by that of P. If C are belongs to Dom P†; then C belongs to Dom H † if F s0 1 PC 2 Dom a† and C 2 Dom b†. Since the domain of the 1 1 jE j‡mc2 n B C iknx matrices a ˆ sx and b ˆ sz is the whole space, all these un x†ˆp @ Tck Ae L 2jEnj n conditions are satis¢ed. So, if C 2 Dom P† then 2 jEnj‡mc C 2 Dom HF P††: Therefore the domain of HF is [9] for  

f q Dom HF†ˆ C ˆ C 2 H; a:c: on O; HFC 2 H; 2 2 2 2 4 w En ˆjEnjˆ T c k ‡ m c > 0;  9† n C fulfils C L†ˆC 0† 6ˆ 0 with 2np k ˆ ; n ˆ 0; Æ1; Æ2; ...; which is one of the self-adjoint extensions of the Ham- n L iltonian operator: periodic boundary condition [9,10]. and The operator X (multiplication by x) is the standard pos- s0 1 ition operator (or coordinate operator): XC x; t†ˆ ÀTckn 1 jE j‡mc2 xC x; t†; its domain is the whole space, that is n B 2 C iknx wn x†ˆp @ jEnj‡mc Ae L 2jEnj Dom X†ˆfgˆCjC 2 H; XC 2 H H: 10† 1

The standard velocity operator of a Dirac particle is de¢ned for En ˆÀjEnj < 0; by nC x; t†caC x; t†. ca is a bounded operator. In fact, Let us evaluate the mean values hXi; hni and hPi for the this matrix has the eigenvalues ‡c; Àc. So its domain is also Dirac wavepacket C x; t†. They are given by the whole space. s L L jE j‡mc2† jE j‡mc2† The equations (6) are satis¢ed by requiring, for all t hXiˆ À 2 1 2 4p jE2kE1j  C L†ˆC 0† 6ˆ 0; HFC† L†ˆ HFC† 0† 6ˆ 0: 11† Tck Tck jE j‡jE j†  2 À 1 sin 2 1 t ; 2 2 Since XC does not satisfy the periodicity condition in (9), jE2j‡mc jE1j‡mc T then XC2=Dom HF†: On the other hand nC† L; t†ˆ hniˆ0; nC† 0; t† 6ˆ 0; therefore nC 2 Dom HF†: Likewise, since 3pT H C 2 Dom H †: and bC† L; t†ˆ bC† 0; t† 6ˆ 0; it follows hPiˆ : F F L that aPC† L; t†ˆ aPC† 0; t† 6ˆ 0;, and therefore PC 2 In this example, the time dependent term in hXi is due to Dom HF†: So, the evolution equations for the mean values hXi; hni and hPi are interference between the positive and negative energy com- ponents in the wavepacket. d 2 This function describes the Zitterbewegung. In contrast, hXiˆÀ Im H C; XC†; dt T F the mean values hPi and hni do not oscillate in the box. d 2 i The time derivatives of hXi; hni and hPi, which coincide with hniˆÀ Im H C; nC†ˆ h‰H ; nŠi; 12† dt T F T F the right hand side of equations (12), are d 2 i s hPiˆÀ Im HFC; PC†ˆÀ h‰HF; PŠi; d jE j‡jE j†L jE j‡mc2† jE j‡mc2† dt T T hXiˆÀ 2 1 2 1 dt 4pT jE2kE1j where  Tck Tck jE j‡jE j†  2 À 1 cos 2 1 t ; i 2ic2 2i jE j‡mc2 jE j‡mc2 T h‰H ; nŠi ˆ hPiÀ hnH i; 2 1 T F T T F d 2ic2 2i d hniˆ hPiÀ hnH iˆ0; hPiˆ0 dt T T F dt since C 2 Dom P†; and i=T†h‰HF; PŠi ˆ 0; since in this case 2 HF commutes with P. thus, the three equations (5) cannot where hnHFiˆ 3pTc =L†: Clearly, d=dt†hXi 6ˆhni,so,the be applied, only two of them are valid. time derivative of the mean coordinate is not given by Let us consider, without loss of generality, an example equation (4) with A ˆ X. It is worth mentioning that in this where the wavefunction C x; t† is a linear combination of case the mean value hni is always constant (in our example four stationary states: two of them are positive energy sol- hniˆ0), therefore d=dt†hniˆ i=T†h‰HF; nŠi ˆ 0, in spite of utions and the other two are negative ones. This thefactthattheoperatorsHF and n do not commute. wavefunction satis¢es the periodic boundary condition: C L; t†ˆC 0; t† 6ˆ 0; i.e., 4. Con¢ned particle in a box  jE1j jE2j A necessary condition in order to have a relativistic con¢ned 1 Ài T t Ài T t C x; t†ˆ2 u1 x†e ‡ u2 x†e particle on O is j 0†ˆj L†ˆ0. Thus, let H  HD ˆ  ÀiTca @=@x†‡mc2b be the Hamiltonian operator with the ijE1jt ijE2jt ‡ w1 x†e T ‡ w2 x†e T Dirichlet boundary condition for the large component. This boundary condition is one of the self-adjoint extensions where the common orthonormal eigenfunctions of HF and P of the Hamiltonian operator for a Dirac particle in a Physica Scripta 61 # Physica Scripta 2000 Ehrenfest-TypeTheorems for a One-Dimensional Dirac Particle 399 one-dimensional box [9,10]. The domain of HD is given by with   f Np Dom H †ˆ C ˆ C 2 H; a:c: on O; H C 2 H; kN ˆ ; N ˆ 1; 2; ...; D w D L  and C fulfils f L†ˆf 0† 6ˆ 0 : 13† s0 1 r ÀiTckN 2 sin k x† 2 jEN j‡mc B 2 N C On the other hand, the standard velocity operator is given by gN x†ˆ @ jEN j‡mc A n ˆ ca with domain the whole space. Likewise, the operator L 2jEN j cos k x† X is de¢ned for every C 2 H. The domain of the momentum N operator P can be written as [9], for EN ˆÀjEN j < 0: Note that C x; t†2Dom n†\   Dom HD†: In fact, f L; t†ˆf 0; t†ˆ0: Moreover, f Dom P†ˆ C ˆ C 2 H; a:c: on O; PC 2 H; HDC x; t†2Dom n†: Since for any eigenfunction of HD it w is easy to see that w L†ˆ À1†N w 0†; it follows that 14† N N  C x; t†2=Dom P†: The mean values of X and n are C fulfils C L†ˆC 0† : s L L jE j‡mc2† jE j‡mc2† hXiˆ À 2 1 Note that in this case HD is not a function of P,sothe 2 2 p jE2kE1j domain of HD is not induced by that of P.  Tck2 Tck1 Since HD acts on a spinor C x; t† for which the upper  1 ‡ jE j‡mc2† jE j‡mc2† component satis¢es f L; t†ˆf 0; t†ˆ0, it can easily be 2 2 checked that xf† 0; t†ˆ xf† L; t†ˆ0, so, on this C x; t† jE jÀjE j†  cos 2 1 t the operators HDX and ‰HD; XŠ make sense. On the other T s hand, if nC 2 Dom HD†; the spinor C x; t† must additionally L jE j‡mc2† jE j‡mc2† satisfy w L; t†ˆw 0; t†ˆ0; for all t; then C L; t†ˆ ‡ 2 1 2 C 0; t†ˆ0, which is too restrictive for the solutions of 9p jE2kE1j  the [11]. Thus, in this case the commutator Tck Tck jE j‡jE j†  2 ‡ 1 sin 2 1 t ‰H ; nŠ is formally meaningless. Likewise, the equation 2 2 D jE2j‡mc jE1j‡mc T d=dt†hPiˆÀ 2=T†Im HDC; PC† is satis¢ed by requiring, for all t, the boundary conditions: f L; t†ˆf 0; t†ˆ0; s C L; t†ˆC 0; t† and H C† L†ˆ H C† 0†.Thislast c jE j‡mc2† jE j‡mc2† D D hniˆ 2 1 relation implies that: † @=@x†C 0†ˆ @=@x†C† L†: If p jE2kE1j PC 2 Dom H †; then f x; t† must additionally satisfy  D Tck Tck jE j‡jE j† † @=@x†f 0†ˆ @=@x†f† L†ˆ0: but this is not compatible  2 ‡ 1 sin 2 1 t jE j‡mc2† jE j‡mc2† T with the other requirements on C x; t†: So we write the time s21 derivatives of hXi; hni and hPi as c jE j‡mc2† jE j‡mc2† ‡ 2 1 3p jE2kE1j d 2 i  hXiˆÀ Im HDC; XC†ˆ h‰HD; XŠi ˆ hni; Tck Tck jE j‡jE j† dt T T  1 ‡ 2 1 cos 2 1 t : 2 2 d 2 jE2j‡mc jE1j‡mc T hniˆÀ Im HDC; nC†; 15† dt T From (15), the times derivatives of the mean values are d 2 hPiˆÀ Im H C; PC†: dt T D d hXiˆhni; Let us consider, without loss of generality, a wavepacket dt s 2 2 2 C x; t† consisting of a linear combination of positive and d 3c jE2j‡mc † jE1j‡mc † jE2jÀjE1j negative energy stationary states hniˆ dt L jE2kE1j jE2j‡jE1j   Tck2 Tck1 1 ÀijE1jt ÀijE2jt  1 ‡ C x; t†ˆ f1 x†e T ‡ f2 x†e T 2 2 2 jE2j‡mc † jE1j‡mc †   jE2jÀjE1j† ijE1jt ijE2jt  cos t ‡ g1 x†e T ‡ g2 x†e T T s c2 jE j‡mc2† jE j‡mc2† jE j‡jE j where the normalized HD-eigenfunctions are À 2 1 2 1 3L jE2kE1j jE2jÀjE1j 0 1  rs sin kN x† 2 Tck2 Tck1 jE2j‡jE1j† 2 jEN j‡mc B C  ‡ sin t : 2 2 fN x†ˆ @ ÀiTck A jE2j‡mc jE1j‡mc T L 2jEN j N cos k x† 2 N jEN j‡mc Note that the time dependent terms in the mean values for obtained above consist of two parts, one varies in times more rapidly than the other does. The rapidly varying part is q caused by the interference between positive and negative E ˆjE jˆ T2c2k2 ‡ m2c4 > 0; N N N energy states. This term is due to Zitterbewegung.

# Physica Scripta 2000 Physica Scripta 61 400 Vidal Alonso and Salvatore De Vincenzo

Another linear combination of positive and negative ary condition in (16) may be rewritten as energy stationary states is 0 g g 1  cos Àisin  jE j jE j f 0‡† B Tc Tc C f 0À† 1 Ài 2 t Ài 4 t C x; t†ˆ f x†e T ‡ f x†e T ˆ @  A 17† 2 2 4 w 0‡† g g w 0À† Àisin cos  Tc Tc ijE2jt ijE4jt ‡ g2 x†e T ‡ g4 x†e T : and corresponds again to the potential energy U x†ˆg x†, but comes from solving the Dirac equation for a general Note that in this case, C x; t†2Dom P†\Dom HD†: In sharply peaked potential and then taking the d-function limit fact f L; t†ˆf 0; t†ˆ0andC L; t†ˆC 0; t†; moreover of the potential [13,15]. This boundary condition seems to be HDC x; t†2Dom P†: So the mean values of X, n and P are the correct jump condition in the one-dimensional Dirac L equation with a local d-potential. Moreover, it is worth hXiˆ ; hniˆ0; hPiˆ0: 2 noting that the sign of the strength g, positive for repulsive potentials and negative for attractive ones, is not important Andfrom(15),thetimederivativesofthesemeanvaluesare as far as the existence of bound states is concerned, in d d d accordance with general results for bound states of the hXiˆhni; hniˆ hPiˆ0: dt dt dt one-dimensional Dirac equation [16]. In any case, both boundary conditions, (16) and (17), are self-adjoint Let us point out that in spite of having a vanishing probabil- extensions of Hd [17]. ity current density at the walls, the mean value of the For very small potential strength, these relativistic bound- ``quantum force'' d=dt†hPi vanishes. However, the evol- ary conditions have the same non-relativistic limit, in fact, ution of the probability density shows that the particle one obtains (see appendix II), ``interacts'' with each wall of the box. g f 0‡†  f 0À† À f0 0À†  f 0À†  f 0†; NR NR 2mc2 NR NR NR 5. Particle on a line with a hole: delta relativistic 2mg f0 0‡† À f0 0À†  f 0†: interaction NR NR T2 NR Let us consider a relativistic particle on the real line with the So the d relativistic point interaction (17), as well as (16), origin excluded, <Àf0g: The Hilbert space of this system is reduces to the well know non-relativistic d interaction. 2 2 H ˆ L < À f0g†„ È L < À f0g†;„with scalar product denoted For instance, for the ``relativistic one~dimensional hydrogen 0À ‡ ‡1 ‡ ‡ 2 À1 by C1; C2†ˆ À1 C1 C2dx ‡ 0‡ C1 C2dx; where C is atom'' [18] g ˆÀZe and we obtain m  2 tan Zafsc=2†† the adjoint of where afsc  1=137† is the ¢ne structure constant. For  Z  1wecanwritem  a . The results given by the two f fsc C ˆ : relativistic deltas, in this last case, become identical. w On the other hand, the standard velocity operator is 2 n ˆ ca, with domain the whole space. In this case, the Let H  Hd ˆÀiTca @=@x†‡mc b be the Hamiltonian operator with relativistic point interaction (d-type domain of the coordinate operator, which is again de¢ned by XC x; t†ˆxC x; t†, is not the whole Hilbert space, in relativistic potential). The domain of Hd is given by   fact, X is an unbounded self-adjoint operator with domain f Dom H †ˆ C ˆ C 2 H; d w Dom X†ˆfCjC 2 H; XC 2 Hg: 18†

a:c: in <Àf0g; HdC 2 H; C fulfils In analogy with the problem of the con¢ned particle in a box,  f 0‡† cos m isinm f 0À† the relativistic momentum operator P in <Àf0g; de¢ned by  ˆ PC x; t†ˆ ÀiT @=@x††C x; t†; has the domain [9], w 0‡† isinm cos m w 0À†   16† f Dom P†ˆ C ˆ C 2 H; a:c: on <Àf0g; w with  19† g PC 2 H; C fulfils C 0‡† ˆ C 0À† : m À2 tanÀ1 ; 2Tc where À1 < g < 0with0< m < p; and 0 < g < ‡1 with It is worth noting that, in the quantum system consisting of a p < m < 2p The boundary condition in (16) corresponds relativistic free particle in a one-dimensional box of length L, to the so-called d relativistic interaction with potential we can imagine bringing the extremities of the interval close energy U x†ˆgd x†. This boundary condition is obtained to each other, making it look like a circle with a hole. So the result of [9], with respect to (19), applies to this system by directly integrating„ the Dirac equation [12], making 0‡ 1 as well. use of the relation: 0À C x†d x†dx ˆ 2 ‰C 0‡† ‡ C 0À†Š[13]. This relation has been used because the relativistic The Hamiltonian operator Hd acts upon the spinors wavefunction is not continuous at x ˆ 0, in contrast with C x; t† for which the non-relativistic case. However, this last relation cannot  cos m isinm be imposed in general [13,14]. C 0‡; t†ˆ C 0À; t† isinm cos m In the limit g=2Tc††2  1, we obtain m À g=Tc†,if g < 0; and m  2p À g=Tc† if g > 0. In both cases the bound- is satis¢ed, with m ˆÀ g=Tc†; if g < 0; and m ˆ 2p À g=Tc† if

Physica Scripta 61 # Physica Scripta 2000 Ehrenfest-Type Theorems for a One-Dimensional Dirac Particle 401 g > 0; i.e., the boundary condition (17). It can be easily wavepacket C x; t† are given by checked that   T 2jEj jEj 2jEj  hXiˆ sin t ; hniˆ cos t ; cos m isinm 2mc T mc T XC† 0‡; t†ˆ XC† 0À; t†; isinm cos m hPiˆ0: which is trivially satis¢ed. On this C x; t†,therefore,the From (20), the time derivatives of the mean values are operators HdX and ‰Hd; XŠ make sense.  d d 2jEj2 2jEj On the other hand, if nC 2 Dom Hd†; the spinor C x; t† hXiˆhni; hniˆÀ sin t ; must satisfy for all t: dt dt mcT T  d cos m isinm hPiˆ0: nC† 0‡; t†ˆ nC† 0À; t†; dt isinm cos m which implies the boundary condition (16). Likewise, the 6. Summary and discussion equation d=dt†hPiˆÀ 2=T†Im HdC; PC† is satis¢ed by We have found some restrictions on the circumstances in requiring, for all t, the boundary conditions (16), which the time evolution of the mean values hXi; hni and C 0‡† ˆ C 0À† and HdC† 0‡† ˆ HdC† 0À†; which hPi hold, as they are usually written in the literature, for implies @=@x†C† 0‡† ˆ @=@x†C† 0À†: All these relations a free Dirac particle in a box, as well as for a free Dirac are very restrictive and are satis¢ed only if m ! 0; p,in particle on a line with a hole. A common feature of the which case d=dt†hPiˆ i=T†h‰Hd; PŠi; owing to PC 2 self-adjoint extensions of the Hamiltonian operators of Dom Hd†. Then the evolution equations for the mean values the former cases is such that, in general, d=dt†hAi 6ˆ hXi; hni,andhPi are i=T†h‰H; AŠi when A ˆ X; n or P. For a ``free'' particle in a box (that is, for a non-vanishing d 2 i hXiˆÀ Im H C; XC†ˆ h‰H ; XŠi ˆ hni; probability current density at the walls), equations (5) do not dt T d T d all hold because XC2=Dom H †; nevertheless, the time d 2 i F hniˆÀ Im HdC; nC†ˆ h‰Hd; nŠi; 20† derivatives of hXi; hni and hPi can always be calculated using dt T T (6). By considering an example of a Hamiltonian for a con- d i hPiˆÀ h‰Hd; PŠi; ¢ned particle in a box, i.e., with vanishing probability cur- dt T rent density at the walls, the laws of motion of the mean i 2ic2 2i values of X, n and P were obtained, and it was pointed where T h‰Hd; nŠi ˆ T hPiÀ T hnHdi; since C 2 Dom P†. Thus, the three equations (5) can be applied. out that these mean values do not satisfy equations (5). As an example, let us consider the normalized wavepacket However, in the problem of a Dirac particle with a delta C x; t†, which is a linear combination of the two bound relativistic interaction, which is simulated by means of states, one of positive energy and the other with negative boundary conditions, the three equations (5) can be applied. energy: In conclusion, the usual time evolution for an operator, hi which is proportional to the mean value of the commutator 1 ÀijEjt ijEjt between the Hamiltonian and the corresponding operator, C x; t†ˆp f‡ x†e T ‡ fÀ x†e T 2 is not always valid. In this paper we have chosen the usual Dirac represent- where the orthonomal Hd bound state wavefunctions are ation. As is well known, in the Foldy-Wouthuysen represen- 0 1 r 1 tation [19] the positive and negative energy states are p jEj‡mc2B C completely separated; therefore, this representation is more f x†ˆ k @ AeÀkjxj ‡ 2mc2 ÆiTck appropriate in order to take the [20]. jEj‡mc2 Moreover, in the Foldy-Wouthuysen representation the p wavepackets behave much more like a classical particle than for E ˆjEjˆ m2c4 À T2c2k2 > 0, with k > 0, and in the Dirac one. Recently, this was con¢rmed by examining 0 1 r ÇiTck the behaviour of wavepackets for the one-dimensional p jEj‡mc2B C version of the Dirac oscillator [21]. In a forthcoming note, f x†ˆ k @ jEj‡mc2 AeÀkjxj À 2mc2 the Ehrenfest theorem in the Foldy-Wouthuysen represen- 1 tation for a one-dimensional Dirac particle in a box, will be considered. for E ˆÀjEj < 0withk > 0. The upper sign refers to x  0‡ and the lower sign refers to x  0À. The magnitude of the energy is Acknowledgements "#ÀÁ The authors would like to express their gratitude to the referee for his interest, 1 À tan2 m jEjˆ ÀÁ2 mc2: comments and suggestions. This work was supported by CDCH-UCV under 2 m project No. PG 03-11- 4318-1999. 1 ‡ tan 2 Note that if C x; t†2Dom H †; then m ! 0; p and d Appendix I jEj!mc2: As in the boundary condition (17), m À g=Tc†,ifg < 0, in the limit g=2Tc†2  1, then when Let us consider a particle in the interval O ˆ‰0; LŠ: If the m ! 0 the mean values hXi; hni and hPi for the Dirac formal steps that yield to the Eq. (3) for the operator X

# Physica Scripta 2000 Physica Scripta 61 402 Vidal Alonso and Salvatore De Vincenzo are written out explicitly in coordinate representation, one and for the energy has 2 1 1 E ˆ mc ‡ ENR ‡ E1 ‡ E2 ‡ ... d L i c2 c4 hXiˆÀc xC‡aC† ‡ h‰H; XŠi 21† Substituting these expansions in (24) and comparing the dt 0 T termsofthelowerorder,thefollowingsystemisobtained where C 2 Dom ‰H; XŠ† for all t. For a particle on the real line with the origin excluded, it is enough to replace the inte- 2m E if0 ‡ w ˆ 0; iw0 ‡ NR f ˆ 0 gration limits: 0 ! 0À and L ! 0‡ If in addition NR T NR NR T NR C 2 Dom n†,then i=T†h‰H; XŠi ˆ hni. Since the correspond- where the primes denote di¡erentiation with respect to x. ing Hamiltonian operator is self-adjoint, the boundary term The connection between the components f and w of the in (2l) is automatically null and we obtain the Eq. (4) for Dirac spinor and the SchrÎdinger eigenfunction fNR is X. Likewise, we can write Eq. (3) for the operator n: obtained by keeping only the ¢rst term of the expansions, that is d L i hniˆÀc2 C‡C† ‡ h‰H; nŠi 22† f ! f ; w !Àlif0 dt 0 T NR NR where C 2 Dom ‰H; nŠ† for all t. If additionally where 2 C 2 Dom P†; then (i=T†h‰H; nŠi ˆ 2ic =T†hPiÀ 2i=T†hnHi: T As the corresponding Hamiltonian operator is self-adjoint, l ˆ : 2mc the boundary term in (22) is null and we obtain Eq. (4) for n: In the same way, Eq. (3) for the operator P is References 1. Ehrenfest, P., Z. Phys. 45, 455 (1927); Schiff, L. I., ``Quantum  L d @C i Mechanics'' (McGraw-Hill, New York, 1968); Messiah, A., ``Quantum hPiˆiTc C‡a ‡ h‰H; PŠi 23† dt @x T Mechanics''. Vol I. (North-Holland, Amsterdam, 1970); Schwabl, F., 0 ``'' (Springer, Berlin, 1992); Manfredi, G., Mola, where C 2 Dom ‰H; PŠ† for all t. This last condition is very S. and Feix, M. R., Eur. J. Phys. 14, 101 (1993). restrictive owing to the fact that C and @C=@x† must both 2.Hill,R.N.,Am.J.Phys.41, 736 (1973); Atkinson, W. A. and Razavy, M., Can. J. Phys. 71, 380 (1993); Rokhsar, D. S., Am. J. Phys. 64, belong to Dom H†,moreoverC L; t†ˆC 0; t† and 1416 (1996). @=@x†C† 0; t†ˆ @=@x†C† L; t† since C 2 Dom P† and 3. Carreau, M., Farhi, E. and Gutmann, S., Phys. Rev. D 42, 1194 (1990); HC 2 Dom P†: da Luz, M. G. E. and Cheng, B. K., Phys. Rev. A 51,1811(1995). Finally, as the corresponding Hamiltonian operator is 4. Alonso, V., De Vincenzo, S. and Gonza¨ lez-D|¨ az, L. A., ``Some remarks self-adjoint, the boundary term in (23) is null and we obtain on the Ehrenfest theorem in a one-dimensional box'' Preprint Universidad Central de Venezuela 1998, to be published. Eq. (4) for P. 5. Alon, O. E., Moiseyev, N. and Peres, A., J. Phys. A: Math. Gen. 28, When it is possible to write Eqs (21), (22) or (23) then, the 1765(1995); Eisenberg, E., Avigur, R. and Shnerb, N., Found. Phys. boundary terms are automatically null. In fact, for the 27, 135(1997). dynamicsstudiedinsections3,4and5wecanverifythis. 6. Sï eba,P.,Czech.J.Phys.36, 667 (1986); Albeverio, S., Gesztesy, F., HÖegh-Krohn, R. and Holden, H., ``Solvable Models in Quantum Mechanics'' (Springer, Berlin, 1987); Carreau, M., J. Phys. A: Math. Gen. 26, 427 (1993). Appendix II 7. Greiner, W., ``Relativistic Quantum Mechanics: Wave Equations'' (Springer, Berlin, 1997). The free Dirac equation for stationary states is given by 8. Thaller, B., ``The Dirac Equation'' (Springer, Berlin, 1992).  9. Alonso, V. and De Vincenzo, S., J. Phys. A: Math. Gen. 32, 5277 (1999). d 10. Alonso, V. and De Vincenzo, S., J. Phys. A: Math. Gen. 30, 8573 (1997). HC x†ˆ ÀiTca ‡ mc2b C x†ˆEC x†: 24† 11. Alonso, V., De Vincenzo, S. and Mondino, L., Eur. J. Phys. 18, 315 dx (1997). In the Dirac representation we write 12. Davison, S. G. and Steslicka, M., J. Phys. C 2, 12 (1969); Subramanian, R. and Bhagwat, K. V., Phys. Stat. Sol. B 48, 399 (1971); Sen Gupta. N.  D., Phys. Stat. Sol. B 65, 351 (1974); Steslicka, M. and Davison, S. G., f C ˆ ; Phys. Rev. B 1, 1858 (1970); Glasser, M. L. and Davison, S. G., Int. w J. Quant. Chem 3, 867 (1970); Fairbairn, W. M., Glasser, M. L. and Steslicka, M., Surf. Sci. 36, 462 (1973); Subramanian, R. and where f and w are respectively the spatial parts of the Bhagwat, K. V., J. Phys. C 5, 798 (1972); Roy, C. L. and Roy, G., so-called large and small components of the Dirac spinor. Physica 106B, 257 (1981); Sen Gupta, N. D., Ind. J. Phys. 49, 49 (1975). Assuming that the components satisfy f x; c†ˆf x; Àc†, 13. Calkin, M. G., Kiang, D. and Nogami, Y., Am. J. Phys. 55, 737 (1987). w x; c†ˆÀw x; Àc† and E c†ˆE Àc†, the functions 14. Grif¢ths. D. and Walborn, S., Am. J. Phys. 67, 446 (1999). 15. McKellar, B. H. J. and Stephenson. G. J., Phys. Rev. A 36, 2566 (1987). f x; Àc†; and w x; Àc† satisfy equation (24) with c !Àc; 16. Coutinho, F. A. B. and Nogami. Y., Phys. Lett. 124A, 211 (1987). consequently, we may write the following expansions in c 17. Falkensteiner. P. and Grosse. H., Lett. Math. Phys. 14, 139 (1987); for f x; c† and w x; c† Falkensteiner, P. and Grosse. H., J. Math. Phys. 28, 850 (1987); Benvegnu©. S. and Dabrowski. L., Lett. Math. Phys. 30, 159 (1994). 1 1 18.Lapidus.I.R.,Am.J.Phys.51, 1036 (1983). f ˆ f ‡ f ‡ f ‡ ...; NR c2 1 c4 2 19. Foldy. L. L. and Wouthuysen, S. A., Phys. Rev. 78, 29 (1950). 20. Costella, J. P. and McKellar, B. H. J., Am. J. Phys. 63, 1119 (1995). 1 1 1 21. Toyama, F. M., Nogami, Y. and Coutinho, F. A. B., J. Phys. A: Math. w ˆ wNR ‡ w1 ‡ w2 ‡ ... c c3 c4 Gen. 30, 2585 (1997).

Physica Scripta 61 # Physica Scripta 2000