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On time derivatives for <X^> and <p^>: formal 1D calculations

Sobre as derivadas com respeito ao tempo para <X^> e <p^>: cálculos formais em 1D

Abstracts

We present formal 1D calculations of the time derivatives of the mean values of the position (x) and momentum (p) operators in the coordinate representation. We call these calculations formal because we do not care for the appropriate class of functions on which the involved (self-adjoint) operators and some of its products must act. Throughout the paper, we examine and discuss in detail the conditions under which two pairs of relations involving these derivatives (which have been previously published) can be formally equivalent. We show that the boundary terms present in d{x}/dt and d{x}/dt can be written so that they only depend on the values taken there by the probability density, its spatial derivative, the probability current density and the external potential V= V9 (x) V = V(x). We also show that d(p)/dt is equal to -dv /dx=(FQ) plus a boundary term (Fq = aQ/ax)is the quantum force and Q is the Bohm's quantum potential). We verify that (fq) is simply obtained by evaluating a certain quantity on each end of the interval containing the particle and by subtracting the two results. That quantity is precisely proportional to the integrand of the so-called Fisher information in some particular cases. We have noted that fQ has a significant role in situations in which the particle is confined to a region, even if V is zero inside that region.

quantum mechanics; Schrödinger equation; probability density; probability density current; Bohm's quantum potential; quantum force


Apresentamos cálculos formais em 1D das derivadas com respeito ao tempo dos valores médios dos operadores da posição (x) e do momento linear (p) na representação de coordenadas. Chamamos esses cálculos formais porque não nos preocupamos com o tipo apropriado de funções sobre as quais devem atuar os operadores (auto-adjuntos) envolvidos e alguns de seus produtos. Ao longo do artigo, examinamos e discutimos em detalhe as condições em que dois pares de relações que envolvem essas derivadas (que foram previamente publicadas) podem ser formalmente equivalentes. Mostramos que os termos de fronteira presentes em d{x}/dt e d{x}/dt podem ser escritos de modo que eles só dependem dos valores a tomados pela densidade de probabilidade, sua derivada espacial, a densidade de corrente de probabilidade e do potencial externo V = V(x).. Também mostramos que d(p)/dté igual a -dv /dx=(FQ)mais um termo de fronteira ((Fq = aQ/ax)é a força quântica e Q é o potencial quântico de Bohm). Verificamos que (fQ)é obtido simplesmente através do cálculo de uma certa quantidade em cada extremidade do intervalo contendo a partcula e subtraindo os dois resultados. Em alguns casos particulares essa quantidade é justamente proporcional ao integrando da assim chamada informação de Fisher. Notamos que (fQ )tem um papel significativo em situações em que a partcula é confinada a uma região, mesmo se V é zero dentro dessa região.

mecânica quântica; equação de Schrödinger; densidade de probabilidade; densidade de corrente de probabilidade; potencial quântico de Bohm; força quântica


ARTIGOS GERAIS

On time derivatives for <X^> and <p^ > : formal 1D calculations

Sobre as derivadas com respeito ao tempo para <X^> e <p^ >: cálculos formais em 1D

Salvatore De Vincenzo1 1 E-mail: salvatore.devincenzo@ucv.ve.

Escuela de Fisica, Facultad de Ciencias, Universidad Central de Venezuela, Caracas, Venezuela

ABSTRACT

We present formal 1D calculations of the time derivatives of the mean values of the position (x) and momentum (p) operators in the coordinate representation. We call these calculations formal because we do not care for the appropriate class of functions on which the involved (self-adjoint) operators and some of its products must act. Throughout the paper, we examine and discuss in detail the conditions under which two pairs of relations involving these derivatives (which have been previously published) can be formally equivalent. We show that the boundary terms present in d{x}/dt and d{x}/dt can be written so that they only depend on the values taken there by the probability density, its spatial derivative, the probability current density and the external potential V= V9 (x) V = V(x). We also show that d(p)/dt is equal to -dv /dx=(FQ) plus a boundary term (Fq = aQ/ax)is the quantum force and Q is the Bohm's quantum potential). We verify that (fq) is simply obtained by evaluating a certain quantity on each end of the interval containing the particle and by subtracting the two results. That quantity is precisely proportional to the integrand of the so-called Fisher information in some particular cases. We have noted that fQ has a significant role in situations in which the particle is confined to a region, even if V is zero inside that region.

Keywords: quantum mechanics, Schrödinger equation, probability density, probability density current, Bohm's quantum potential, quantum force.

RESUMO

Apresentamos cálculos formais em 1D das derivadas com respeito ao tempo dos valores médios dos operadores da posição (x) e do momento linear (p) na representação de coordenadas. Chamamos esses cálculos formais porque não nos preocupamos com o tipo apropriado de funções sobre as quais devem atuar os operadores (auto-adjuntos) envolvidos e alguns de seus produtos. Ao longo do artigo, examinamos e discutimos em detalhe as condições em que dois pares de relações que envolvem essas derivadas (que foram previamente publicadas) podem ser formalmente equivalentes. Mostramos que os termos de fronteira presentes em d{x}/dt e d{x}/dt podem ser escritos de modo que eles só dependem dos valores a tomados pela densidade de probabilidade, sua derivada espacial, a densidade de corrente de probabilidade e do potencial externo V = V(x).. Também mostramos que d(p)/dté igual a -dv /dx=(FQ)mais um termo de fronteira ((Fq = aQ/ax)é a força quântica e Q é o potencial quântico de Bohm). Verificamos que (fQ)é obtido simplesmente através do cálculo de uma certa quantidade em cada extremidade do intervalo contendo a partcula e subtraindo os dois resultados. Em alguns casos particulares essa quantidade é justamente proporcional ao integrando da assim chamada informação de Fisher. Notamos que (fQ )tem um papel significativo em situações em que a partcula é confinada a uma região, mesmo se V é zero dentro dessa região.

Palavras-chave: mecânica quântica, equação de Schrödinger, densidade de probabilidade, densidade de corrente de probabilidade, potencial quântico de Bohm, força quântica.

1. Introduction

Almost any book on quantum mechanics states that the mean values of the position and momentum operators ( and ) satisfy, in a certain sense, the same equations of motion that the classical position and momentum ( and ) satisfy. This result, which establishes a clear correspondence between the classical and quantum dynamics is the Ehrenfest theorem[1, 2]:

Note that Eq. (2) contains the average value of the external classical force operator , rather than the own force evaluated at . As a result, we are clarifying the statement preceding Eq. (1).

When trying to prove Ehrenfest's theorem in a rigorous way, the difficulty arises that each of the operators involved ( and , which must be preferably self-adjoint) has its own domain, and some plausible common domain must be found in which Eq. (1) and/or Eq. (2) are/is valid, which is a non-trivial and complicated matter. To review some of the difficulties that may arise, as well as certain aspects of these domains, Refs.[3-6] can be consulted (Ref. [4], which was recently discovered by the present author, is especially important). For a rigorous mathematical derivation of the Ehrenfest equations (which does not use overly stringent assumptions), see Ref. [7]. For a more general (and rigorous) derivation, see Ref. [8]. For a nice treatment of this theorem (specifically, for the problem of a particle-in-a-box) that is based on the use of the classical force operator for a particle in a finite square well potential, which then becomes infinitely deep (effectively confining the particle to a box), see Ref. [9]. For a study of the force exerted by the walls of an infinite square well potential, and the Ehrenfest relations between expectation values as related to wave packet revivals and fractional revivals, see Ref. [10].

The usual formal (or heuristic) demonstration in textbooks of Eqs. (1) and (2) in the coordinate representation with appears to have no problem; however, it is known that the quantities and with (where is a finite interval) do not always obey the Ehrenfest theorem [3, 6]. This problem occurs because boundary terms that are not necessarily zero arise in the formal calculation of the time derivatives of and . To verify this result in this article, we carefully reexamine the formal traditional approach to the Ehrenfest theorem in the coordinate representation from the beginning. Hence, we do not consider the domains of the involved (self-adjoint) operators. Specifically, in this article, we do not care for the appropriate class of functions on which these operators and some of its products must act. In our study, the notion of self-adjointness of an operator (or strict self-adjointness) is essentially replaced by the hermiticity (or formal self-adjointness), which is known to be less restrictive. We believe that a formal study of this problem alone is worthy and pertinent; in fact, the strict considerations related to the domains of the involved operators and their compositions seem to be too demanding. In our paper, we also examine and discuss in detail the conditions under which two pairs of relations involving and (which were published in Refs. [5, 6]) can be formally equivalent.

We start with the position and momentum operators, and , for a non-relativistic quantum particle moving in the region (which may be finite or infinite). The inner product for the functions and (belonging at least to the Hilbert space , and on which and act) is , where the bar represents complex conjugation. The corresponding mean values of these operators in the (complex) normalized state are as follows

The operator is hermitian because it automatically satisfies the following relation

where and are functions belonging to . The time derivative of expressions (3) and (4) leads us to the following relations

and

In the last expression, we have used the commutativity of the operators and .

In non-relativistic quantum mechanics, the wave function evolves in time according to the Schrödinger equation

where is the Hamiltonian operator of the system and is the (real) external classical potential. By substituting in Eqs. (6) and (7) the time derivatives of and (which are obtained from Eq. (8) and its complex conjugate), we obtain and . As will be discussed in the next two sections, these derivatives always have terms that are evaluated at the ends of the interval . However, if these derivatives must be real-valued, certain mathematical conditions (which are, of course, physically justified) should be imposed on the boundary terms. We will show that these boundary terms can be written so that they can only depend on the values taken by the probability density, its spatial derivative, the probability current density and the external potential V at the boundary.

2. Time derivatives for

For example, the time derivative of the average value of specifically depends on the values taken by the probability density and the probability current density in these extremes. In fact, the following result can be formally proven (see formula (A.1) in Ref. [5])

where we use the notation here and in further discussion. The function is the probability current density

and is the probability density

These two real quantities (which are sometimes called "local observables'') can be integrated on the region of interest, and each of these integrals is essentially the average value of some operator. Indeed, the integral of j is

The integral on the right-hand side in this last expression is precisely the average value of the operator (see formula (4)). Finally, we can write

The integral of (which is a finite number only if the probability density is calculated for a state ) is precisely the mean value of the identity operator

It is important to note that the operator satisfies the relation

for the functions and belonging to . If the boundary conditions imposed on and lead to the cancellation of the term evaluated at the endpoints of the interval , we can write the relation as . In this case, is a hermitian operator. If we make = in this last expression and Eq. (13), we obtain the following condition (see formula (11))

Moreover,

. These last two results are consistent with Eq. (12).

Formula (9) was obtained from the following formal relation (formula (11) in Ref.[5] with ):

In the case where and , this equation is precisely Eq. (6) (compare the first equality in Eq. (15) with Eq. (6)). To check Eq. (9), formula (15) can be developed by first calculating the following two scalar products:

Before subtracting these two expressions, we develop the first integral in . Then, we use the relation

and the definitions of the probability current density (Eq. (10)) and the probability density (Eq. (11)). After identifying the terms that depend on and , we obtain the following result

which is substituted into Eq. (15), leading to formula (9). The average value of the commutator in formula (9) is calculated as follows:

By developing this expression, we obtain

Finally, substituting results (14) and (17) into formula (9), we obtain the following

In the writing of this formula, we used the condition (i.e., is a hermitian operator), but Eq. (18) is also consistent with the hermiticity of ().

It is convenient to mention here a result that pertains to the Hamiltonian of the system, . Indeed, this operator satisfies the following relation

for the functions and belonging to . If the boundary conditions imposed on and lead to the cancellation of the term evaluated at the endpoints of the interval , we can write the relation . In this case is a hermitian operator. If we make in this last expression, as well as in Eq. (19), we obtain the following condition (see formula (10))

Moreover,

. In formula (18), condition (20) is not sufficient to eliminate the term evaluated at the boundaries of the interval .

We can now compare result (18) with the result obtained in Ref. (see formula (17) in Ref. [6])

From the beginning, Ref. uses real-valued expressions for the temporal evolution of and . For example, Eq. (21) is obtained from the following

That is, Eq. (21) is consistent with the hermiticity of . In fact (as we observed after Eq. (15)), because , formula (6) can be written as follows:

Furthermore, because , Eq. (22) is obtained. As observed from the discussion that follows formula (12) in Ref. [6], is the probability density and is the velocity field, which is related to the probability current density as follows: . From this last formula we can write

Comparing Eq. (23) with formula (12) (after applying condition (14)), the relation is obtained. Returning to formula (21), it is clear that it is equal to formula (18), and the latter is equal to formula (9), provided that formula (14) is verified. We can then say that the time derivative of the mean value of the operator is not always equal to . For example, Ref. [3] shows a specific example that confirms the validity of Eq. (18).

In summary, the temporal evolution of the mean value of is given by Eq. (18) and also by Eq. (21). Assuming that (in addition to and ) the operator is hermitian, we can write the following expression:

(in which we used relation (20)). Only one boundary condition involving the vanishing of the boundary term in Eq. (13), but also leading to the vanishing of the probability current density at the ends of the interval , gives the equation . This scenario is clearly possible, for example, for the Dirichlet boundary condition . However, the same is not necessarily true for the periodic boundary conditions and [3] .

3. Time derivatives for

Next, we consider the momentum operator . The following result was formally proved in Ref. [5](see formula (A.2) in Ref. [5])

This formula was obtained from the following formal relation (formula (11) in Ref. [5] with )

In the case where y , this equation simplifies to Eq. (7) (i.e., in writing Eq. (26), no special condition has been imposed). If we want to verify the validity of Eq. (25), we can begin to develop formula (26). Thus, we first compute the following scalar products present there:

By integrating by parts the first integral in and then subtracting these two expressions, we obtain the following result:

which can be substituted into (26) to produce formula (25). Likewise, the mean value of the commutator in formula (26) can be explicitly computed using and calculating ; in fact,

By developing the derivative in the last integral above and simplifying, we obtain an expected result (see Refs. [1, 2], for example)

where we have also identified the external classical force operator . Finally, formula (25) can be written as follows

Note that formula (27) is obtained by making in relation (19). Thus, if the boundary term in Eq. (19) is zero because of the boundary conditions (and consequently, is hermitian), the boundary term in Eq. (29) does not necessarily vanish. An example of this scenario is provided by the Dirichlet boundary condition, . Indeed, with this boundary condition , is hermitian, but the boundary term in Eq. (29) is not zero. Within the case of the periodic boundary condition, and , the operator is also hermitian, but the boundary term in Eq. (29) does vanish (from the Schrödinger equation in (8) we also know that if the potential satisfies ). Similarly, in an open interval () the boundary term in Eq. (29) is zero if and its derivative, , tend to zero at the ends of that interval. Specifically, if a wave function tends to zero for , at least as , then its derivative also tends to zero there, and the boundary term in both in Eqs. (19) and (29) vanishes (as a result, we also have ). This result provides the formal argument for the cancellation of these two boundary terms. Clearly, if satisfies a homogeneous boundary condition for which is hermitian and satisfies the same boundary condition, the boundary term in Eq. (29) vanishes (this result seems to be very restrictive).

Consequently, result (25) was obtained from formula (26). Likewise, the following expression for was also obtained from formula (26) (see formula (19) in Ref.[6])

where (as we said before) and ; moreover, is Bohm's quantum potential,

Now let us verify and reexamine the validity of Eq. (30). Returning to result (26), it is clear that it can also be written as follows:

and, if the condition

is used, we can write

Therefore, the time derivative of is given by the following

which is automatically real-valued. It is important to note that the formula

is obtained by setting in relation (13). If the boundary conditions imposed on lead to the cancellation of the boundary term in Eq. (34), then formula (32) is verified; however, that same boundary condition can also cancel the boundary term in Eq. (13), with (the latter would imply that is hermitian). The spatial part of the boundary term in Eq. (34) is unaffected by the presence of the time derivative.

As is known, by substituting the polar form of the wave function in the Schrödinger Eq. (8) (i.e., ), where is essentially the phase of the wave function) and then separating the real and imaginary parts, we obtain (i) the quantum Hamilton-Jacobi equation

and (ii) the continuity equation

The probability current density j can also be written in terms of and S after replacing the polar form of in formula (10)

Formula (33) can be written as follows

and by substituting the relation in Eq. (38), we obtain the following result

By solving for and in Eqs. (35) and (36), respectively, and substituting them into Eq. (39), formula (30) is obtained (after some simple calculations).

The boundary term in formula (29) is real-valued if Eq. (14) is verified. To obtain this result, we first write that boundary term separately but in terms of and j (or ):

(Eq. (40) is, in fact, also valid without vertical bars, ). As we have observed before, the hermiticity of () requires that the probability density (for the state ) satisfies formula (14). Differentiating that formula with respect to time, we obtain the following:

Now, using the continuity equation (Eq. (36)), we obtain the condition

With this last result, the entire boundary term in Eq. (40) (and therefore in Eq. (29)) is real-valued (the first term in (40) is always real). Consistently, and are both real-valued quantities in Eq. (29).

In the proof of the formula (30), the condition given in Eq. (32) was used; thus, the results in Eq. (29) (or Eq. (25)) and Eq. (30) are not equivalent. However, from the expression for that is written after Eq. (31), we can write the following:

Now, instead of using Eq. (32), we use relation (34) (from which we solve for ). This process leads to the following expression

(in which we have used to write the boundary term in Eq. (42)). Indeed, formulas (29) and (42) are equivalent. The first term on the right-hand side of Eq. (42) is precisely the entire right-hand side of Eq. (30). Additionally, the boundary term in Eq. (42) can be rewritten using Eq. (8). In this way, we obtain the following result:

Now, we use the following (remarkable) relation:

(where we have made use of the definition of the Bohm's quantum potential given by Eq. (31)), to write

which leads us to the following result:

Finally, because the following relation is verified:

(see Eq. (40)), formula (44) is precisely result (29) (i.e., Eqs. (42) and (29) are equivalent).

Recapitulating, the temporal evolution of the mean value of is given by Eq. (29), but the boundary term must be real-valued if the mean value of is real. As we have demonstrated (see Eq. (40)), to accomplish this, it is enough that the boundary conditions satisfy Eq. (14), which implies that Eq. (41) is also satisfied because the continuity equation is verified. After substituting Eqs. (40) and (41) in Eq. (29), this formula (Eq. (29)) can be written as follows:

Formula (30) also gives us the average value of , but this equation must also be consistent with Eq. (14) (because is hermitian) and the boundary conditions should cancel the boundary term that appears in Eq. (34). This term is precisely

and because (as a result of the validity of Eq. (14)), we have that the vanishing of the left-hand side in Eq. (46) implies the following

Now, multiplying the quantum Hamilton-Jacobi equation (Eq. (35)) by and substituting the expression (Eq. (37) with ) and Eq. (47), the following relation is obtained (in this way, this result is also a consequence of the elimination of the left-hand side in Eq. (46))

Now, returning to formula (30) and substituting relation (43), we obtain the following result

Formula (49) becomes formula (45), as long as relation (48) is obeyed (this is an expected result!). Thus, Eqs. (29) and (30), together with the condition given by Eq. (14) (which is consistent with the hermiticity of ), give us identical results if the boundary term in Eq. (34) vanishes (which occurs if is hermitian); i.e., if Eq. (48) is verified (see the comment after Eq. (34)). In conclusion, Eqs. (49) and (29) show that the time derivative of the mean value of is always equal to a term evaluated at the ends of the interval containing the particle plus the mean value of the external classical force operator. However, as is shown in Eq. (49), the boundary term may depend only on the values taken at and by the probability density, its first spatial derivative, the probability current density and the external potential.

In agreement with the previous results (see the discussion that follows Eq. (29)), all of the boundary terms in Eq. (49) do not vanish for the solutions to the Schrödinger equation satisfying the Dirichlet boundary condition. In this case, both the density of probability and the probability current density vanish at the ends of the interval, i.e., and . Therefore, we have and . The latter result because (Eq. (14)) and (Eq. (20)). Moreover, we also know that , which is consistent with Eq. (48). Thus, we can write the following result

The boundary term in Eq. (50) can be written as follows:

and (in this case) it coincides with (this result comes from Eq. (43)). Also (in this case), the boundary term coincides with the following expression:

(see the relation that follows Eq. (44)). Consequently, the mean value of the quantum force

can be calculated by simply evaluating a quantity (which, in this case, only depends on and ) at and at and then subtracting these two results. Similarly, if we assign the following expressions to :

or

which are clearly distinct from each another and also from , the correct value for is obtained. However, an exact expression for can be obtained using the relation that precedes Eq. (43), in which . The result is the following

Clearly, is always equal to a boundary term. Formula (51) can be written without the explicit presence of Bohm's quantum potential. Indeed, by substituting the expression for Q (the expression to the right in Eq. (31)) in Eq. (51), we obtain the following

This last result has been known in hydrodynamic formulations of Schrödinger's theory; see, for example, the following recent [11](and further references therein).

Now, if we return to Eq. (50) and assume that the external potential is zero (), we can write the following

Consequently, the mean value of the force encountered by a free particle confined to a region and colliding with the two walls is precisely . Then, from Eq. (53), and because the formula that follows Eq. (44) (which is also valid without vertical bars, ) with is verified, we can say that the average force on the particle when it hits the wall at x=b is given by the following

At x=a, the expression for this force is obtained from Eq. (54) by making the following replacements: and . Let us now consider the example of the confined (free) particle moving between and , and in some of its possible stationary states

where , with (naturally, the corresponding probability density is independent of time). Using these results in Eq. (54) (in either of the two expressions), we can determine that the average force on the particle at is given by , and at it is given by ; therefore, (this same result was obtained in Ref. [12] following a procedure different from that shown here). However, if the state is a linear combination of the solutions (55) (and hence, the corresponding probability density is also a function of time), does not necessarily vanish (in this specific case, the average force on the particle at x = L is not always minus the value at ) . In Ref.[13] the issue of the average forces for a particle ultimately restricted to a finite one-dimensional interval, either because there exists an infinite potential or because we put the particle in the interval and neglect the rest of the line, has been recently treated.

Consistently with previous results (see the discussion following Eq. (29)), the entire boundary term in Eq. (49) vanishes for the solutions satisfying the periodic boundary condition. Indeed, we know that ; therefore, (provided that ) and (see Eq. (48)). Finally, because

all of the boundary terms in Eq. (49) vanish, and the result is reached. However, in this case, we also know that . This result occurs because the boundary term in Eq. (24) is not zero (because the probability current density does not vanish at the ends of ), and its derivative with respect to t does not vanish either. Clearly, this situation does not occur when the relation is obeyed (as in the case of the Dirichlet boundary condition).

Finally, as was explained before (see the discussion following Eq. (29)), the boundary term in Eq. (29) is zero in an open interval (), provided that appropriate conditions can be satisfied as (i.e., and its derivative should vanish at infinity). Equivalently, the boundary term in Eq (45) is also zero, as well as that in Eq. (49) (because Eq. (48) is satisfied). We can then conclude (from Eq. (43)) that ; therefore, . From Eq. (24), relation is also verified; consequently, .

4. Conclusions

We have formally calculated time derivatives of and in one dimension. Simultaneously, we have identified the conditions under which two pairs of these derivatives, which have been previously published, can be equivalent. When the particle is in a finite interval, we have observed that the Ehrenfest theorem is generally not verified. In fact, because of the large variety of boundary conditions that can be imposed in this case (and for which and are hermitian operators), the boundary terms that appear in and (which may depend only on the values taken there by the probability density, its spatial derivative, the probability current density and the external potential) do not always vanish. Particularly, if the boundary term in does not vanish, we generally know that . If the particle is at any part of the real line, but there is a very small chance for it to exist at infinity, the time derivatives of and obey the usual Ehrenfest relations, as expected. As we have demonstrated, is equal to , plus a boundary term, but we can also say that is equal to plus a boundary term. In the first formula, the respective boundary term is zero whenever the probability current density vanishes at the ends of the interval (see Eq. (24)). As a case in point, the same result is observed in the second formula when the probability density and current are zero there (see, for example, Eq. (45) conjointly with Eq. (43)).

If a free particle () is confined to a box, the quantum force (or rather, its mean value ) is the quantity that reports the existence of the box's impenetrable walls (at least for the Dirichlet boundary condition). In all cases, the average value of is simply obtained by evaluating a certain quantity at each end of the interval occupied by the particle and subtracting the two results (see Eq. (51)). That quantity is precisely proportional to the integrand of the so-called probability density's Fisher information, , in particular cases; for example, when at the ends of the interval. In effect, for a particle in an interval , we obtain the following (see, for instance, Refs. [11, 14]):

Clearly, in this case, we obtain by evaluating the integrand in (times ) at and (see Eq. (51)).

Recebido em 10/7/2012;

Aceito em 13/1/2013;

Publicado em 16/5/2013

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  • Publication Dates

    • Publication in this collection
      05 July 2013
    • Date of issue
      June 2013

    History

    • Received
      10 July 2012
    • Accepted
      13 Jan 2013
    Sociedade Brasileira de Física Caixa Postal 66328, 05389-970 São Paulo SP - Brazil - São Paulo - SP - Brazil
    E-mail: marcio@sbfisica.org.br