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The quantum telegraph equation

P. SANCHO(*)

Departamento de Física, Grupo de Física Estadística, Universidad Autónoma de Barcelona 08193 Bellaterra, Barcelona, Spain

(ricevuto il 7 Aprile 1995; approvato il 25 Giugno 1997)

Summary. — Diffusive equations used in physics, Fourier’s law, turbulent spread equation and Schrödinger’s equation, are parabolic equations with unbounded velocities of propagation. Hyperbolic equations extending in a local way Fourier’s law and the turbulent spread equation have been proposed in the literature. Here we present the complex generalization of these hyperbolic equations, the quantum telegraph equation, that extends in a local way Schrödinger’s equation.

PACS 03.65.Bz – Foundations, theory of measurement, miscellaneous theories (including Aharonov-Bohm effect, Bell inequalities, Berry’s phase).

PACS 02.30 – Function theory, analysis.

1. – Introduction

For a free nonrelativistic wave packet it is well known that it spreads instantaneously over all of space if it is localized in a bounded region at some initial time [1]. Nonlocality implies that one runs into conflict with Einstein causality (no propagation faster than the speed of light) as soon as one can localize particles in a bounded region. Even a relativistic theory faces very similar difficulties: a similar phenomenon occurs for the Newton-Wigner position operator and for discussions of localization and causality in a model-independent way [2]. This possible Einstein acausality must be seen more as a problem of the underlying theory than as an experimentally verifiable prediction. However, more subtle experimental consequences of the nonlocality could yet be expected.

The nonlocality problem is also present in other areas of physics. In thermodynamics Fourier’s law describing heat conduction is a parabolic nonlocal equation for the temperature. The equation is locally extended by the introduction of a relaxation term in the constitutive equation (Maxwell-Cattaneo’s equation) that leads to an hyperbolic equation, the telegraph equation for the temperature [3]. The

(*) Present address: GPV de Valladolid, Centro Zonal de Castilla y León, Avda Dr. Villacián s/n. 47071 Valladolid, Spain.

turbulent equation of spreading of heat and matter has also unbounded velocities and is replaced by a turbulent telegraph equation [4-5].

In order to make all the discussion about the nonlocality quantitative, it is essential to formulate some sort of local generalization of Quantum Mechanics (QM). This line of reasoning is very close to that developed by Weinberg in their study of precision test of the quantum-mechanical linearity [6]: a nonlinear extension of QM is used to show that the fraction of the energy of the9Be nucleus that could be due to nonlinear corrections is around 10221_{. Here, we restrict the discussion to Schrödinger’s equation, the} parabolic complex equation that rules the wave function behaviour. We show in the following that the telegraph equation can be extended to the complex field to obtain an hyperbolic generalization of Schrödinger’s equation.

In a different context, the telegraph equation has been applied to the determination of the tunneling time [7]. The authors apply the method to analyze the results of delay-time measurements in narrowed waveguides, performed as a test of tunneling time models. These results improve the agreement of the predictions of quantum-mechanical models with experiments.

2. – A local extension of Schrödinger’s equation

The formulation of the EPR argument, has resulted in increasing attention on the nonlocality problem in measurement theory [8-9]. However, very little work has been done on the related issue of nonlocality in Schrödinger’s equation, the equation that rules the system between two measuring processes.

The retarded propagator of a free particle is [1] Gr( r K 2, t2; r K 1, t1) 4u(t22 t1) e3 piO4Q

g

m 2 pˇ(t2_{2 t}1)

h

3 O2 exp

y

im( r K 22 rK1) 2 2 ˇ(t2_{2 t}1)

z

. (1)

When the particle is localized in rK0at t1c( r K

, t1_{) Ad( r}K_{2 r}K0), the solution for t2_{D t}1is c( rK, t2) AGr( r

K , t2; r

K

0, t1) and the wave function for any t2D t1differs from zero over all of space.

Physically, this nonlocality is based on the behaviour of high-frequency components. Any plane wave c 4c0exp [i( k

K

Q rK2vt) ] is a solution of the Schrödinger equation. The phase velocity of this wave is (ˇvO2m)1O2 _{that in the high-frequency limit} reaches an arbitrarily large value. According to Heisenberg’s principle, if the particle is localized spatially, the momentum will be completely undetermined. The general solution of the Schrödinger equation is a superposition over all momenta and the high-frequency components propagating at high velocities cause the wave function to differ from zero instantaneously over all of space.

If we consider the behaviour of a wave packet, the characteristic velocity is the group velocity, given by

vg4 dv dk 4

g

2 ˇv m

h

1 O2 (2)

and we observe again an unbounded behaviour at high frequencies.

Fourier’s equation, describing heat conduction, faces a similar problem. The phase velocity of the solution of the Fourier equation is vp_{4 ( 2 lOrc)}1 O2_v1 O2_{(with l , r and c} the thermal conductivity, the density and the specific heat, respectively) showing the

same problematic behaviour at high frequencies. Two solutions have been proposed in the thermodynamic literature [3]:

– A nonlinear diffusion equation with the thermal conductivity depending on the temperature in a given way.

– The addition of a relaxation term to the constitutive relation.

The first solution cannot be translated to QM without modifying one of its most important properties, the superposition principle (see Weinberg [6] for nonlinear generalizations of QM). In this paper only the second possibility will be considered.

In thermodynamics, Fourier’s equation is substitued by the following equation: t¯ 2_T ¯t2 1 ¯T ¯t 2 l rcDT 40 (3)

with T the temperature and t the relaxation time of the heat flux.

This is an equation of the telegraphist type, which is well known to be hyperbolic. The phase velocity of the solution of eq. (3) is

vp4

(

2(lOrc) v

)

1 O2

(

tv 1 (11t2v2)1 O2

)

21 O2.

Now in the high-frequency limit one has v_{pQ4 (lOrct)}1 O2_{, that is a finite value.} We propose the following complex generalization of the telegraph equation:

2tˇ¯ 2_c ¯t2 1 iˇ ¯c ¯t 4 2 ˇ2 2 mDc , (4)

t being a parameter with dimensions of time. By similitude with thermodynamics, it will be called the relaxation time. Probably t would be very short.

The solutions are plane waves with a phase velocity vp4

v

(

2 mv( 1 1tv)Oˇ

)

1 O2 . (5)

In the high-frequency limit v_{pQ4 (ˇO2 mt)}1 O2that is finite. For low frequencies tv b 1 , the result based on Schrödinger’s equation is recovered.

The knowledge of v_pQallows one to calculate the relaxation time

t 4 ˇ

2 mv2 pQ

. (6)

If the limiting velocity is identified with the light velocity, t 4ˇO2mc2

A 10221sgs for the electron. However, we are considering phase velocities. The special theory of relativity holds for material particles and says nothing about phase velocities; the relaxation time remains undetermined.

The group velocity of the extended theory is vg4

g

2 ˇ m

h

1 O2

₍

_{v( 1 1tv)}

₎

1 O2 1 12tv (7)

To end this section, we obtain the solutions of the extended equation. For the sake of simplicity we restrict the problem to only one spatial dimension, x:

2tˇ¯ 2 c ¯t2 1 iˇ ¯c ¯t 4 2 ˇ2 2 m ¯2c ¯x2 . (8)

This equation can easily be reduced to the form ¯2f

¯T2 1 f 4 ¯2f ¯X2 (9)

by the changes c 4f exp [itO2t], T4tO2t, X4 (mO2ˇt)1 O2_{x .}

The solution of eq. (9) is well known and can be found in any textbook on partial differential equations, f(X , T) 4 1 2F(X 1T)1 1 2F(X 2T)1 1 2



X 2T X 1T H(X , T , s) ds , (10) where H(X , T , s) 4 (11) 4 2iTF(s) J 80

(

2 [ (s 2 X) 2 2 T2]1 O2

)

[ (s 2X)2 2 T2]1 O2 1 G(s) J0

(

2 [ (s 2 X) 2 2 T2]1 O2

)

. The new functions F and G are the initial conditions, that is, f(X , 4) 4F(X); ¯_{fO¯Y(X, 0) 4G(X) and J}0 and J08 are the Bessel function of order zero and its derivative.

When the initial conditions F and G are different from zero only in a bounded region, the spread of the wave function out of this region is not instantaneous. For instance, if we take as initial conditions F(x) 4d(x) and G(x) 40, the evolution of the wave function is given by

f(x , t) 4 1 2d

gg

m 2 tˇ

h

1 O2 x 1 t 2 t

h

1 1 2d

gg

m 2 tˇ

h

1 O2 x 2 t 2 t

h

2 (12) 2 it 4 tJ 80

gg

2 mx2 2 tˇ 1

g

t 2 t

h

2

h

1 O2

h

Q

g

mx 2 2 tˇ 2

g

t 2 t

h

2

h

21 O2 . The last equation shows that in the extended theory the evolution of the wave function of a particle placed in x 40 at the initial time has a wave front x46(ˇO2mt)1 O2_{t . Out of} this front, the wave function is assumed to be equal to zero.

3. – Propagator and physical interpretation

It is easy to demonstrate that if the density of probability is defined by r 4cc*1it

g

c * ¯c

¯t 2 ¯c *

¯t c

h

, (13)

However, this density of probability is of no physical interest. Remembering eq. (9), we see that if the initial conditions are real functions the density is zero. On the other hand, if the initial conditions are taken as complex functions the density of probability becomes it(f * ¯tf 2f¯tf * ) that is not positive-definite.

Thus the function given by eq. (13) cannot be interpreted as a density of probability. The physical origin of this difficulty lies in the fact that the extended theory introduces a very short temporal scale, given by the relaxation time. When one is concerned with these very short temporal scales (it is neccesary to consider them if one wants to avoid the unbounded velocities of Schrödinger’s equation), according to Heissenberg’s principle t DE Aˇ [1], and the uncertainty in the energy can become comparable to the rest mass; the creation of new particles is now possible. Note that the origin of the particle creation phenomenon in extended and relativistic theories is different. In relativistic theories the velocity of the particle is comparable to the speed of light c, introducing a momentum scale p 4mc for a particle of mass m. In the quantum domain, the typical longitude scale corresponding to this momentum scale is ˇOmc, that is, the Compton wavelength. The analysis of the position of the particle with this accuracy can require, according to uncertainty relations, an energy-momentum of the same order of the rest mass. On the other hand, in the extended theory the particle can be nonrelativistic, arising the phenomenon when very short temporal scales, comparable to the relaxation time, are considered. The exact description of the particle creation is in terms of a many-body theory. In spite of these difficulty, the extended wave equation and its one-particle interpretation can be useful and physically sensible as long as we consider free particles or external forces which are slowly varying. They provide us with the first “local” corrections to the Schrödinger picture.

To clarify the physical meaning of the new equation it is necessary to calculate the propagator or Green’s function, which is given by the equation

g

2tˇ ¯ 2 dt2 1 iˇ ¯ ¯t 1 ˇ2 2 mD

h

G(x 2x 8) 4ˇd 4 (x 2x 8) . (14) The solution is G(x 2x 8) 4



d 3 k K dv ( 2 p)4 exp [i k K Q( rK2 rK8 ) ] exp [2iv(t 2 t 8 ) ] G * ( kK, v) , (15) with G * ( kK_{, v) 4}

u

_{v 1tv}2 2 k K₂ 2 m

v

21 . (16)

This solution is valid except in the points given by

v₆₄ 1 2 t

u

21 6

u

1 1 2 tˇ kK2 m

v

1 O2

v

. (17)

A rule for handling the two singularities in the denominator is necessary to complete the expression in eq. (16). This is determined by the retarded boundary condition. We add infinitesimal _{parts (v 2v1)(v 2v2) K (v2v11 ie)(v 2 v21 ie). Both} singularities lie below the real axis. For t 8Dt the contour is closed along an infinite

semicircle above the real axis. Both poles lie outside the contour, and the integral vanishes. For t Dt 8 the contour may be closed along an infinite semicircle below the real axis:



iu(t 2t 8) d3k

K

( 2 p)3(v_{12 v2}) Q (18)

Q exp [i kKQ( rK2 rK8 ) ]

[

exp [2iv1(t 2t 8) ]2exp [2iv2(t 2t 8) ]

]

. The last expression is equivalent to

iu(t 2t 8)



dk * c *₁( rK_{8 , t 8 ) c1}( rK_{, t) 2iu(t2t 8)}



dk * c *₂( rK_{8 , t 8 ) c2}( rK, t) , (19)

where the wave function is now c₆( rK_{, t) 4 (2p)}23 O2_{exp [i( k}K_{Q r}K

2v6t) ] and ˇ dk * 4 t( 1 1 [2tˇ kK2

Om] )21 O2d3K_{k. The propagator written explicitly as a function of the wave} functions uses non-Cartesian measures dk * in phase space.

Positive and negative frequencies represent, respectively, particles and anti-particles. Note that for the same choice of the momentum p, the absolute values of the frequencies of particles and antiparticles are different, that is Nv1( p

K

) NcNv2( p K

) N. This behaviour is in marked contrast with the usual particle-antiparticle theory, where NvR1( p

K

) N4N(m2c4_{1 p}K2c2)1 O2_{OˇN 4 Nv}R

2( p K

) N. In an extended theory we expect different behaviours for particles and antiparticles. The adequate conditions to observe the real existence of these differences would be the study of particles in external electromagnetic fields. This property of the extended equation imposes limits on the possible values of the relaxation time and can be used as a test of the nonlocality in Schrödinger’s equation.

A final comment to end this section. The quantum telegraph equation can recall the Klein-Gordon equation. As a matter of fact, the extended equation can be written as

e(itO2t)

g

¯ 2 ¯t2 2 ˇ 2 mtD 1 1 4 t2

h

e 2(itO2 t)_{c 40 .} (20)

If the limiting velocity is taken as the light velocity, the relaxation time is t 4ˇO2mc2 and eq. (20) becomes Klein-Gordon’s equation up to a phase factor. However, this phase factor introduces a time scale that is not present in the Klein-Gordon equation, modify-ing the physical results (different frequencies for particles and antiparticles ...).

4. – Symmetries and Galilei invariance

We start by considering the equation of motion for the expectation value of an operator A. After a straightforward calculation,

iˇ d dtaAbc4 iˇ

o

¯A ¯t

p

c 1

»y

A , H 1tˇ ¯ 2 ¯t2

z«c

. (21)

If the operator A has no explicit time dependence, then its expectation value is a constant of the motion if it commutes with the operator H 1tˇ¯2O¯t2. If A is independent of time, the constants of the motion of the extended theory are just the same of the usual theory. In particular, the expectation value of the energy of a stationary system is a constant of the motion. The momentum is also conserved. The only difference with the classical theory becomes apparent when the operator has explicit dependence on time.

To study the invariances of the extended equation we need the expression of the evolution operator U, Nc(t)b 4U(t, t0) Nc(t0)b. From now on we suppose the Hamiltonian to be independent of time. The substitution of the last expression on the evolution equation gives an equation for U, whose formal solution is

U(t , t0) 4A exp [2i(t2t0) N1Oˇ] 1 B exp [2i(t 2 t0) N2Oˇ] , (22)

where A and B are integration constants, and N₆are the two operator solutions of t

ˇN 2

1 N 4 H . (23)

The quantum evolution of a system is invariant under a transformation group G if the evolution operator commutes with every operator UG associated with G, that is, if

[U , UG] 40. In the classical theory this condition is equivalent to the commutation

with the Hamiltonian, [UG, H] 40. We shall demonstrate that every symmetry of the

classical theory is also a symmetry of the extended theory. The commutator of UG and

the extended evolution operator is

[U , UG] 4A

[

exp [2i(t2t0) N1Oˇ], UG

]

1 B

[

exp [2i(t2t0) N2Oˇ], UG

]

.

(24)

As exp [W ] 4

!

Wn

On! the above commutator is the sum of terms [N6n, UG]. All these

terms are zero. For instance, for N₁, and using the well-known formula [A , Bn_{] 4} [A , B] nBn 21_,

(25) 0 4 [H, UG] 4 [ (tOˇ) N11 N12 , UG] 4

4 (tOˇ)[N12, UG] 1 [N1, UG] 4

(

1 1 (2tOˇ) N1)[N1, UG]

that implies [N₁, UG] 40 except for singular points.

Finally, [U , UG] 40, and we recover all the symmetries of the classical theory.

Now we consider the Galilei invariance of the extended theory. A Galilean transformation in the x-axis with velocity v is given by x 84x1vt and t 84t.

The Schrödinger equation for a free particle is invariant if the wave function changes as c 8(x 8, t 8) 4exp

[

2im[nx 2 n2tO2]Oˇ

]

c(x , t). For interacting systems some specific conditions on the Hamiltonian operator are necessary to preserve Galilei invariance [1].

In the extended theory even the free particle is not invariant. The wave function will transform as c 8(x 8, t 8) 4exp [if(x, t) ] c(x, t), but now whatever the phase f(x, t), it is impossible to recover the extended equation as a function of the new variables; the free-particle equation is not Galilei invariant. This result can be understood on more intuitive grounds. For a Galilei-invariant theory it is known that the energy operator generates time translations; in the extended theory the time translations are

generated by the combined action of the operators N₆, violating Galilei invariance. Moreover, finite propagation speed is neither required nor expected for a Galilei-invariant theory. Then, if we want to construct a (nonrelativistic) theory with an upper bound on the propagation speed, this theory cannot be Galilei invariant.

It is interesting to note, however, that the extended theory preserves the Galilei invariance in a restricted sense. As remarked earlier if the Hamiltonian is time independent, every symmetry of the classical theory is also a symmetry of the extended theory. Thus, although the fundamental equation is not invariant, when the Hamiltonian (provided to be time independent) generates Galilei-invariant solutions in the classical theory, the solutions of the extended theory (differing from the classical ones) are also invariant.

5. – Discussion

A complex version of the telegraph equation has been presented. This hyperbolic equation extends in a local way the parabolic equation of Schrödinger. The new equation is interesting in several aspects:

1) It provides a more unified view of diffusive equations. Hyperbolic equations have been proposed to generalize the parabolic equations of diffusion of heat (Fourier’s law) and turbulent spreading of heat and particles. The generalization of Fourier’s law has given rise to a new and rich thermodynamic formalism, the Extended Irreversible Thermodynamics [3].

2) The quantum telegraph equation is not a substitute of Schrödinger’s equation; rather it must be seen as an intermediate equation between the classical and relativistic levels. At low frequencies the results of Schrödinger’s equation are recovered. Moreover, if the operators considered are independent of time, the symmetries and invariants of both theories are just the same. On the other hand, the presence of a second-order derivative on time introduces some characteristics of relativistic theories such as bounded velocities or the existence of antiparticles. Reflecting these intermediate levels, the theory is neither Galilei nor Lorentz invariant.

3) The analysis of the theory provides us with an interesting prediction, the different absolute values of the frequencies of particles and antiparticles. This result can be an experimentally verifiable prediction of the theory, for instance, in the presence of external electromagetic fields. Therefore, it imposes stringent limits on the possible values of the relaxation time and, indirectly, is a test of the nonlocality of Schrödinger’s equation (provided that it is a linear equation. If we admit nonlinearities, other possibilities must be considered).

* * *

I am grateful to D. JOU for reading the manuscript. This work has been partially supported by the DGICyT of the Spanish Ministry of Education and Science under grant no. CLI-95-1867.

R E F E R E N C E S

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[3] JOUD., CASAS-VA´ZQUEZJ. and LEBONG., Extended Irreversible Thermodynamics (Springer-Verlag) 1993.

[4] MONINA. S. and YAGLOMA. M., Statistical Fluid Mechanics (MIT Press) 1971. [5] SANCHOP. and LLEBOTJ. E., Physica A, 205 (1994) 623.

[6] WEINBERGS., Phys. Rev. Lett., 62 (1989) 485.

[7] MUGNAID., RANFAGNIA., RUGGERIR. and AGRESTIA., Phys. Rev. Lett., 68 (1992) 259. [8] BELLJ. S., Physics, 1 (1965) 195.