**4. Interpretation of mathematical models**

#### **4.1 Fully relativistic quantum mechanics**

A fully relativistic description of excited atomic states specifies discrete fourdimensional field boundaries and continuous localized fields between the boundaries. Excited atomic states, electron, photon, and nucleus, are interpreted as a linear superposition of three field sources with respect to their electrostatic and electromagnetic field components. Recent experiments, referred to by the authors as "photon capture" and "photon storage," support the accuracy of this theoretical interpretation [8]. In these experiments light coherence is converted to atomic coherence and back again, so the photon in localized form must be present in excited atomic states from the time energy is absorbed until it is emitted. We describe the linear properties of atomic structure by introducing a wavelike field source ε, the localized photon, into our description of excited atomic states. The modified Hamiltonian is now given by

$$\mathbf{H} = \mathbf{T} + \mathbf{e} + \mathbf{V} \tag{4}$$

*Lagrangian Quantum Mechanics: A Fully Relativistic Theory of Atomic Structure DOI: http://dx.doi.org/10.5772/intechopen.90168*

and the Lagrangian is similarly given by

Solving for the action, we obtain solutions for localized energy or equivalently,

*Solid State Physics - Metastable, Spintronics Materials and Mechanics of Deformable…*

four-dimensional field boundaries, so they are also exact. In the case of time periods, this has been confirmed to the limits of experimental accuracy by atomic clocks that can operate for many billions of years without significant error [7]. Thus field energy from a laser is absorbed by the lattice of ytterbium atoms and localized within a four-dimensional field boundary. The emission and absorption energies E21 and E12 have also long been assumed to be exact by astronomers when employed for the measurement of distant star composition. Emission and absorption spectra, together with a red shift, often require billions of years before they are observed, thereby reflecting the precise role of energy and time in natural phenomena. From (2), a model of atomic structure may be constructed. The three field sources present in excited atomic states, electron, proton, and photon, superpose linearly and are momentarily stabilized in steady states. Although force is an unnatural concept in quantum mechanics, it may be interpreted with respect to the field boundaries that separate point sources by a careful consideration of (2). Thus the force on a bound electron due to the potential is equal to the continuously distributed excitation energy divided by the distance between field boundaries. In order for quantum mechanical forces to agree with relativity theory, we need only require that the action integral be invariant for all potentials both

Solutions of Equation (3) are determinations of energy and time between exact

Whereas the quantum mechanical force of bound states is due to well-defined field boundaries, the force due to instantaneous exchanges of momentum, such as occurs in the Compton effect, may be interpreted as a result of the encounter of a single, exact field boundary (xo,yo,zo,to) with a material point. Wave properties, on the other hand, occur in free space when field boundaries have no reference point, so they cannot be observed at all. Thus we interpret complementarity as the differ-

A fully relativistic description of excited atomic states specifies discrete fourdimensional field boundaries and continuous localized fields between the boundaries. Excited atomic states, electron, photon, and nucleus, are interpreted as a linear superposition of three field sources with respect to their electrostatic and electromagnetic field components. Recent experiments, referred to by the authors as "photon capture" and "photon storage," support the accuracy of this theoretical interpretation [8]. In these experiments light coherence is converted to atomic coherence and back again, so the photon in localized form must be present in excited atomic states from the time energy is absorbed until it is emitted. We describe the linear properties of atomic structure by introducing a wavelike field source ε, the localized photon, into our description of excited atomic states. The

H ¼ T þ ε þ V (4)

ent ways that fields and its field boundaries interact with matter.

**4. Interpretation of mathematical models**

**4.1 Fully relativistic quantum mechanics**

modified Hamiltonian is now given by

**22**

Eτ ¼ h (3)

photon creation; due to a transformation of field:

free and bound.

$$\mathbf{L} = \mathbf{T} + \mathbf{e} \text{-V} \tag{5}$$

where T represents the bound electron's energy, ε represents the energy of a "captured" photon, and V represents the potential energy due to the nucleus/proton. Each of the three field sources possesses a unique vector field, that is, a field with definite field geometry that is delimited from the others by field boundaries, where plus and minus signs indicate the linear superposition of delimited fields.

Eqs. (4) and (5) contain the essence of quantum mechanics as a three-body problem in real space as opposed to current descriptions based on a two-body system in abstract space. The use of an abstract space is necessary for nonrelativistic descriptions of atomic structure since the photon is not treated as an independent particle. The equations revert to their classical form when the influence of ε is negligible or equivalently when field boundaries are no longer determinable. To see whether the model accurately describes atomic structure, we shall compare it to the existing mathematical models.

#### **4.2 The path integral formulation**

From (5), it is postulated that the contribution of two energies is summed and one is subtracted to give the transition energy. In the path integral formulation, there are in fact two contributions that are summed, one determined by the paths and the other by the phase. In Feynman's words, "The paths contribute equally in magnitude, but the phase of their contribution is the classical action (in units of ђ)" [9]. The computation of the total energy is not complete, however, until contributions to the potential V due to self-energy are subtracted away by renormalizing. Therefore the mathematical structure of the Lagrangian in nonrelativistic quantum mechanics indicates the existence of three contributions and is in agreement with (5). It differs fundamentally from the fully relativistic Lagrangian method described here in its interpretation of space-time. Eq. (2) treats space and time equivalently as real parameters for both the integration limits and the region between them. On the other hand, the path integral formulation uses abstract forms of space and time to describe the region between the steady states since the paths follow all trajectories and for all times.

#### **4.3 Matrix mechanics**

In quantum mechanics, observables are determined by pairs of states, while in classical theory they refer to the same state. This is especially evident in matrix mechanics which describes the atom as a twofold infinite, denumerable array of virtual oscillators, where observables are vectors in Hilbert space whose magnitude defines a spectral line intensity, or transition amplitude, and whose direction corresponds to either an absorption or an emission. Although the physical model consists of two ideal particles, the virtual harmonic oscillators, the matrix elements include three field components: the fields of the two ideal particles and a spectral line intensity due to photon superpositions. All three components of the modified Hamiltonian given by Eq. (4) are present but with respect to classical space and time. In nonrelativistic theory, photons are singularities that correspond to pairs of states, whereas conformance with relativity theory requires that the photon be spatially and temporally extended. Fields are localized by exact four-dimensional

field boundaries separated from each other in space-time, while in nonrelativistic quantum mechanics, field boundaries do not exist.

continuous process that results from field superposition during the discrete time period τ. Therefore in a fully relativistic theory, interference effects are due to the instantaneous reinforcement and cancelation of superposed photons of the type described in (2), and the statistical nature of quantum mechanics that is observed in experiments such as double-slit interference is due to time averages. A fully relativistic optical theory will account for interference effects as they evolve in real

*Lagrangian Quantum Mechanics: A Fully Relativistic Theory of Atomic Structure*

*DOI: http://dx.doi.org/10.5772/intechopen.90168*

It has long been asserted that classical physics is inadequate for describing quantum mechanical phenomena. Consequently experimental results are explained by introducing complementarity and the correspondence principle. However, the problem is not that classical theory is deficient, but it is the insistence on using singularities in a nonrelativistic theory. If the photon's fields are singular, wave and particle properties seem to appear out of nowhere, and experimental results have an intrinsically defined uncertainty. But if the photon is instead described as a localization of fields, uncertainty and duality are accounted for by physical characteristics, fields and field boundaries, and complementarity has a classically derived meaning. A similar explanation is possible for the correspondence principle which specifies the point where a two-particle classical system must be replaced by a three-particle quantum mechanical system to explain what is observed. It may seem to be an acceptable practice to describe particles as singularities propagating and interacting continuously in time, but in a fully relativistic theory the photon cannot be singular. Rather it is a four-dimensional localization of fields defined symmetrically in space-time that determines electron behavior in bound states and also in

The mathematical framework surrounding quantum mechanics is precisely the type of description that is expected when a particle of zero mass is absorbed by a two-particle system. The particle properties of the photon are overwhelmed by the other two such that it is impossible to distinguish it independently of them. Sometimes the influence of its continuous properties is more evident (wave mechanics); at other times its discrete properties are prominent (matrix mechanics); and in path integral formulations, the exact field boundaries of bound states are manifested. Each of the three formulations of nonrelativistic quantum mechanics provides a unique perspective to atomic structure by emphasizing a different physical aspect of the three field sources. This may be compared to the simpler three-dimensional practice in architecture of providing three visual perspectives to a building. Each one provides a partial view, and when taken together they give an improved understanding of the structure as a whole. The "whole" of quantum mechanics is

free space by means of four-dimensional forces.

given of course by Lagrangian quantum mechanics.

**25**

space and time.

**6. Conclusion**

Because Heisenberg's uncertainty relations use a continuous time parameter, they are only valid when events are defined with respect to specific observers, but not in general for all observers. When interpreted according to Eq. (2) by a fully relativistic theory, we conclude that indeterminacy is due to measurements performed with a non-singular, spatially and temporally extended probe, the photon. This may be compared to the case in classical mechanics of measurements that are performed with a coarsely defined standard. In quantum mechanics, the standard of measurement is the photon, and no matter how high its energy, it cannot be used to localize a point particle more precisely than its wavelength. On the other hand, localizations in atomic clocks occur four-dimensionally with respect to *both* field boundaries, so they occur without measurable error.
