# Quantum ground states as equilibrium particle–vacuum interaction states

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## Abstract

A remarkable feature of atomic ground states is that they are observed to be radiationless in nature, despite (from a classical viewpoint) typically involving charged particles in accelerated motions. The simple hydrogen atom is a case in point. This universal ground-state characteristic is shown to derive from particle–vacuum interactions in which a dynamic equilibrium is established between radiation emission due to particle acceleration, and compensatory absorption from the zero-point fluctuations of the vacuum electromagnetic field. The result is a net radiationless ground state. This principle constitutes an overarching constraint that delineates an important feature of quantum ground states.

## Keywords

Quantum ground states Vacuum fluctuations Particle–vacuum interaction states Zero-point fluctuations Harmonic oscillator quantum ground state## 1 Introduction

One of the apparent paradoxes of quantum theory that students often query is the radiationless nature of atomic ground states. The paradox lies in the fact that, classically, radiation that might be anticipated to occur from accelerated charged-particle motions in atomic ground states is not observed. In the hydrogen atom, for example, the orbiting electron does not radiate its energy away and spiral into the nucleus. The fact that during decades of successful application of quantum theory we have come to take for granted the radiationless feature of these special “bottom-rung” stationary states does not in any way detract from this remarkable property. Fortunately, a rapprochement between classical and quantum viewpoints with regard to ground states is possible.

When addressed in analytical detail it becomes clear wherein the resolution to this apparent paradox lies. It is that one must properly take into account how charged-particle ground-state motions interact with the vacuum, specifically the zero-point fluctuations (ZPFs) of the vacuum electromagnetic field. Although such considerations are not usually invoked in the day-to-day application of quantum theory to ground-state specification, the argument that follows clarifies that the standard quantum formalism with its associated radiationless ground states has its genesis in the dynamics of underlying particle–vacuum interactions and that the vacuum field is in fact formally necessary for the stability of atoms in quantum theory. As summarized in an earlier paper addressing spontaneous emission processes: “The crucial role of the vacuum fluctuations emerges in the ground state of matter. The stability of the ground state (i.e., the fact that it does not radiate) is purely a quantum effect which is due to the vacuum fluctuations [1].”

Setting aside a full quantum mechanical treatment, it is sufficient for heuristic purposes to treat such problems semiclassically on the basis of point particles interacting with a random, classical radiation field whose spectral characteristics are those of the known quantum vacuum ZPF distribution, an approach known as stochastic electrodynamics (SED). Early detailed pedagogical papers by Boyer [2, 3] provide a broad foundation for the SED approach as applicable to many problems generally considered to require quantum mechanical treatment. The further development of SED in the decades following is well-summarized in [4]. The treatment in this paper extends earlier work by the author on the quantum ground state of hydrogen [5] to address the case for quantum ground states in general. The SED ansatz takes advantage of the fact that SED derivations of vacuum-fluctuation-driven phenomena based on multipole/radiation-field interactions parallel Heisenberg-picture derivations in standard quantum electrodynamics (QED) [6, 7]. Specifically, despite shortcomings in the SED approach as a proposed alternative to quantum theory [8], in the cases considered here calculations in SED are analogous to QED calculations with a symmetric ordering of photon creation and annihilation operators [9]. Before considering application on the basis of a general formalism, let us apply the central argument in detail to the simple one-dimensional harmonic oscillator.

## 2 Nonrelativistic harmonic oscillator

**,**the (nonrelativistic) Abraham–Lorentz equation of motion for a point particle of mass

*m*and charge

*q*, including radiation damping, is given by [10]

*x*component of the vacuum ZPF electric field and here we neglect the force contribution from the magnetic field (see Sect. 3, however).

*x*and its time derivatives. Thus,

*k*takes the form

## 3 Generalized approach

Having derived the above relationship between absorbed and radiated powers for the harmonic oscillator’s ground state, we now inquire as to whether this balance is specific to the harmonic oscillator by virtue of its simple linear restoring force, or can be extended to more general cases (e.g., nonlinear oscillator, hydrogen atom, particle in a box, etc.).

*V*a time-independent confining potential. Multiplication of Eq. (25) by \({\dot{\mathbf {r}}}\), taking into account mathematical simplifications (e.g., \({\dot{\mathbf {r}}}\cdot ( {{\dot{\mathbf {r}}}\times {\mathbf {B}}^\mathrm{zp}})\equiv 0, \quad \left( {1/2} \right) \mathrm{d}/\mathrm{d}t\left( {{\dot{\mathbf {r}}}^{2} } \right) = {\dot{\mathbf {r}}} \cdot {\ddot{\mathbf {r}}} \), \( \mathrm{d}V/\mathrm{d}t = \partial V/\partial t + {\dot{\mathbf {r}}} \cdot \nabla {\mathbf {V}} \rightarrow {\dot{\mathbf {r}}} \cdot \nabla {\mathbf {V}} \) for a time-independent potential, etc.), followed by collection of terms, leads to

^{1}As a consequence the remaining terms constitute the condition that reveals itself to be that the average power radiated due to accelerated motion (Larmor radiation) is balanced by the average power absorbed from the vacuum fluctuations. Substituting the definition of \(\Gamma \) from Eq. (8) we obtain

*V*and is thus applicable to a wide range of problems.

## 4 Concluding remarks

Addressed is the seeming paradox that even though quantum ground states typically involve charged particles in accelerated motions, such states are nonetheless observed to be radiationless in nature. Though this feature is overlooked in everyday application of quantum theory to ground-state description, nonetheless this remarkable property is worthy of some discussion and clarification. In detail, it is the recognition that ground-state atomic structures are not isolated entities in an empty background, but are perforce immersed in a background of vacuum fluctuations. With regard to the behavior of charged particles, the primary component of interest is that of the vacuum electromagnetic ZPFs. Atoms (and other quantum systems), therefore, constitute open systems engaged in dynamic interactions with the underlying vacuum states. Specifically, the on net radiationless characteristic of the ground state is seen to derive from particle–vacuum interactions in which a dynamic equilibrium is established between radiation emission due to particle acceleration, and compensatory absorption from the ZPFs of the vacuum electromagnetic field.^{2} Thus, employing an SED approach, we have shown here in detail that, as argued in [1], the vacuum field is formally necessary for the stability of atomic structures, and this underlying principle, therefore, constitutes an important feature of quantum ground states.

## Footnotes

- 1.
For the trivial case of a perfectly circular orbit this term vanishes even before averaging, given that the kinetic and potential energies are constant and the velocity and acceleration vectors are orthogonal. Beyond that, SED modeling attempts involving lengthy numerical simulations for hydrogenic atoms to verify the vanishing of the time-dependent term on the lefthand side of (26) have to date only been marginally successful, self-ionization of the atom often being the outcome [14, 15]. Under consideration are the incorporation of additional factors such as expansion beyond the dipole approximation, spin-orbit coupling, relativistic effects and so forth; thus simulation studies remain a work in progress.

- 2.
Further development of the foundational nature of the radiationless state for quantum mechanics in general can be found in [16], which is an update to the material presented in [4]. In this later compendium additional constraints on the SED approach are incorporated to match more closely the requirements of QED (e.g., detailed balance of energy). In the updated approach, labeled LSED (linear SED), it remains the case that “the ZPF is seen as the source of the quantum behavior of matter.”

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