ON THE WAVE-PARTICLE DUALITY IN QUANTUM PHYSICS Vu B Ho Advanced Study, 9 Adela Court, Mulgrave, Victoria 3170, Australia Email:
[email protected]
Abstract: Experiments in fluid dynamics have shown that the wave-particle duality can manifest at the macroscopic scale that can be explained by using the equation of continuity in addition to a wave equation. In this work, as illustrations, we show that these dual characters can be obtained for Einstein’s general relativity for the case when and Maxwell’s equations of classical electrodynamics. We also discuss the possibility to interpret the waveparticle duality in quantum physics by considering a wave equation in quantum mechanics as a constraint to the position of a particle, where the motion of the particle is considered to be in a confined geometry.
Recent experiments conducted by Couder et al [1-3] in fluid dynamics have shed some light on one of the most peculiar concepts in quantum mechanics, the wave-particle duality. In these experiments, the equation of continuity of the form plays a crucial role in determining the dual character of the dynamics of the fluid, in which the physical entity that is subjected to the equation of continuity is also subjected to the wave motion. In this work we will consider whether similar equations of continuity that are related to the electric, magnetic and gravitational fields can be formulated so that the wave-particle duality of the electromagnetic field and the gravitational field can be interpreted in the same manner. It should be mentioned here, as de Broglie pointed out, that the electromagnetic energy can be treated as a virtual fluid moving in space [4]. De Broglie also pointed out that it is not possible to represent elementary particles within the frame of the 3-dimensional spatial continuum and this perception of space-time structure at the sub-quantum level has led to attempts to extend the normal 3-dimensional space to describe the electric and magnetic fields [5]. We will show in this work that an equation of continuity of the form can be obtained by introducing a time component for the electric field and a time component for the magnetic field . An equation of continuity of a similar form can also be derived for the gravitational field for the case . It is noted that it is possible to interpret the wave-particle duality in quantum physics by considering a wave equation in quantum mechanics as a constraint to the position of a particle, where the motion of the particle is considered to be in a confined geometry. Despite the fact that quantum mechanics also relies on classical concepts such as particle and wave in order to describe the dynamics of a physical system, there is an essential difference when these concepts are applied to the dynamics of a single particle. How can a single particle manifest itself as a wave? In order to answer this question we need to re-examine
how the dynamics of a particle is formulated in classical mechanics and in quantum mechanics. In classical physics, the dynamics of a particle of an inertial mass that is under the influence of a force occurs in a 3-dimensional Euclidean continuum and the trajectory of the particle is determined by Newton’s second law of motion [6]
Besides Newton’s second law as expressed in Equation (1), the path of the particle is also determined by other conditions, such as the initial condition and constraints. When the motion of the particle is constrained, the constraints are normally described by scalar equations written in the form
For example, in classical dynamics, the equation of motion of a single pendulum of mass with position vector in the plane is written in the form of Equation (1) as with two holonomic constraints written in the form of Equation (2) by and , where is the tension. On the other hand, in quantum mechanics, if we consider Bohr’s postulate of the quantisation of angular momentum as constraints then Bohr’s planar model of a hydrogen atom is seen as being described by the equation with the holonomic constraint and the nonholonomic constraints [7]. Despite the fact that Bohr’s model of a hydrogen atom is relatively basic, its dynamics in terms of Newton’s second law of motion and constraints may shed light on the meaning that can be given to wave equations in quantum mechanics in general. In quantum mechanics, the dynamics of a quantum particle is determined by Schrödinger wave equation [8]
where is the wavefunction of the particle, which has no physical meaning except for the widely accepted interpretation that it plays the role of a probability amplitude. The probability of finding the particle at the position and time is proportional to . The main information that can be obtained from the wavefunction is the probabilistic distribution of positions of the particle. For example, the stationary states of a hydrogen atom with the Coulomb potential is described by the time-independent Schrödinger equation as follows
where is the reduced mass of the electron and the nucleus. Since the Coulomb potential is spherically symmetric, Equation (4) can be written in the spherical polar coordinates as
where the orbital angular momentum operator
is given by
Solutions of Equation (5) are found using the separable form
where is a radial function and is the spherical harmonic. In terms of from Equation (7), Equation (5) is reduced to the system of equations
and
The normalised radial eigenfunctions of the bound states can be found as
where and bound state energy eigenvalues given by
is the associated Laguerre polynomial with the
If we assert that there is a real dynamics that underlies the probabilistic description provided by a wavefunction then the positional wavefunction may be seen as special constraints applied to the position of the electron when it orbits the nucleus. As in the case of the single pendulum, where the particle can only move in an arc of the circle of radius in the plane, the electron of a hydrogen atom can only move in the constraint region of the 3-dimensional space determined by the solutions of the Schrödinger wave equation. These constraint relations imposed on the position of the electron are determined by all possible interactions between the electron and the nucleus. In the case of the Schrödinger wave equation of a hydrogen atom, the constraint relationship is no other than the law of conservation of energy . It is noted that if examined microscopically the constraint imposed on the single pendulum is in fact also due to interactions between electrons and nuclei. It is obvious that the unknown dynamics that underlies the statistic description of wave mechanics may be caused by other laws in addition to Newton’s second law of motion, such as temporal laws of motion, and other forces in
addition to Coulomb’s force of electrostatics, such as short-lived time-dependent forces or forces that are derived from different forms of potentials [9]. We now discuss the wave-particle duality that may be related to the equation of continuity for the electromagnetic field and the gravitational field. As has been shown in our previous work that the principle of least action has a dual character in the sense that by specifying an appropriate relationship between a mathematical object and a physical entity both classical and quantum dynamics can be formulated from it [10]. This result may pave the way for the possibility to interpret the wave-particle duality in quantum physics in a deterministic manner by determining functions that characterise physical processes and satisfy both a classical dynamic equation and a wave equation. For example, let , where , be a function which is used to represent a physical process, and we assume that the function is also used to describe both the particle dynamics and the wave dynamics of a quantum object. If is a differential equation that describes the classical dynamics of the particle, which admits the function as a solution, then we have
where and . On the other hand, if is a differential equation that describes the wave dynamics of the particle then we should also have
It seems that in order to establish the required functions and differential equations we would need to know more about the nature of quantum objects. Even though recent developments in physics have shown that geometrical and topological methods in classical and quantum physics have played an important role in identifying physical entities, especially quantum observables, with mathematical objects, these obstacles still remain [11-13]. Therefore, in the following we will only discuss the possibility to apply the above suggestive procedure into the electromagnetic field and the gravitational field with the equation of continuity will play the role of the equation . In the present formulation of physics, in order to describe the classical dynamics of a particle we apply Newtonian mechanics. For example, the equation of motion of a charged particle in an electric field and a magnetic field is given by the Lorentz force law [14]
On the other hand, the quantum dynamics of the charged particle is described by Schrödinger wave equation as a probabilistic wave [15]
where
and
are written in terms of the potentials
and
as
It is seen from Equations (14) and (15) that these two physical characterisations of the charged particle are irreconcilable. In classical electrodynamics, an equation of continuity that involves the electric charge density and the electric current density accompanies the Maxwell’s field equations. These equations are given by
It is observed that the equation of continuity of the form for the electric field and the magnetic field can be obtained if we introduce a time component for the electric field and a time component for the magnetic field by the defining relations
With the time component , Maxwell’s equations for the electric field with the corresponding equation of continuity can be re-written as
Similarly, with the time component , Maxwell’s equations for the magnetic field with the corresponding equation of continuity are given by
The above wave equations can be simplified further if the electric current density can be written in the form . By introducing the time components for the electromagnetic field, the concepts of an electric charge and a possible magnetic charge are reduced to that of
the rate of change with respect to time of other physical entities which may be considered as components of the electromagnetic field itself. It is seen from Equations (24-27) that the electric field and the magnetic field each follow a particle dynamics and a wave dynamics. It is seen from Equation (23) that the time component of the magnetic field does not depend explicitly on time. On the other hand, if the charge density of an elementary particle is written in the form , where is the position vector of the charge , then is the Heaviside step function with respect to time. Maxwell’s equations can be generalised further if a magnetic charge is assumed. In this case the field equations of the electromagnetic field can be written in a more symmetric form as follows
In this case Equation (23) is re-written as with similar wave equations for the electric field and the magnetic field. For a complete description, it should be mentioned here that by using the four-potential , the field equations of electromagnetism with a magnetic charge and the Lorentz force law can be formulated in terms of the electromagnetic tensor and the dual electromagnetic tensor as [14]
Despite the fact that the above discussions for the electromagnetic field can be applied to the field equations of linearized gravity, in the following we will derive an equation of continuity for general relativity for the case when , which admits a solution that also satisfies a wave equation. The field equations of general relativity are written in the form
where is the energy-momentum tensor, is the metric tensor, is the Ricci curvature tensor, is the scalar curvature and is the cosmological constant [16]. It is shown that Equation (35) can be derived through the principle of least action , where the action is defined as
where
characterises matter fields [17]. It is shown that when
we have the identity
, and in this case the field equations given by Equation (6) reduces to [10]
It is indicated from Equation (37) that a physical entity can be directly identified with a mathematical object. This interesting feature can be seen as an underlying principle for quantum physics. As long as the energy-momentum tensor is directly identified with the metric tensor through the relationship given by Equation (37) then for the case when we have
, therefore the resulting equations given in Equation (35) from
the principle of least action are satisfied by any metric tensor . The most general form of a line element of the 2-dimensional space-time manifold is given as
By a time transformation, Equation (38) can be re-written in the form
where inverse
and
[18]. With this line element, the metric tensor
and its
are
Using Equations (37) and (40), the energy-momentum tensor found as follows
and its inverse
are
With the metric tensors given in Equations (40), the non-zero elements of the affine connection defined by
are found as follows
If the energy-momentum tensor is defined through the relation verified that the following local conservation of energy-momentum is satisfied
, then it is
Since the functions and are arbitrary functions of space and time, in order to formulate a classical dynamics for the gravitational field we need to establish a dynamical equation. As mentioned above, due to the identity
it is not possible
to obtain the required dynamical equation for the gravitational field by applying Equation (35). In this case, however, a dynamical equation can be obtained if we assume that the energy-momentum tensor also satisfies the following equation of continuity
With this equation of continuity, using Equation (41) we have
If we assume further
, then we have
This equation has the general solution of the form
where is an arbitrary function of . Equation (47) is the transport equation, or the equation of continuity, in classical dynamics which describes a flow or movement of mass, charge, energy or momentum at a constant rate in the negative direction of . Even though Equation (47) does not have the status of Newton’s dynamical equation to describe the dynamics of a particle, it represents a particle motion in a deterministic manner. On the other hand, the general solution given in Equation (48) is also a general solution of the following wave equation
From these results it can be concluded that the function can be used to describe the dynamics of either a particle or a wave, and the observable result will depend on the setup of an experiment. It is worth to mention here that in quantum mechanics we also have a similar situation for Schrödinger’s wave-functions. If the wavefunction is a solution of Schrödinger’s wave equation
then the probability density defined by density defined by
and the probability current satisfy the continuity equation
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[14] L.D. Landau and E.M.Lifshitz, The Classical Theory of Fields (Pergamon Press, Sydney, 1987). [15] L.D. Landau and E.M.Lifshitz, Quantum Mechanics (Pergamon Press, Sydney, 1987). [16] A. Einstein, The Principle of Relativity (Dover Publications, New York, 1952). [17] R. M. Wald, General Relativity (The University of Chicago Press, London, 1984). [18] P. Collas, Am. J. Phys. 45 (1977).
