symmetry-12-01373-v2.pdf

symmetry

Article

Symmetry, Transactions, and the Mechanism of Wave

Function Collapse

John Gleason Cramer

* and Carver Andress Mead

Department of Physics, University of Washington, Seattle, WA 98195, USA

California Institute of Technology, Pasadena, CA 91125, USA; carver@caltech.edu

Correspondence: jcramer@uw.edu

Received: 17 June 2020; Accepted: 14 August 2020; Published: 18 August 2020

Abstract:

The Transactional Interpretation of quantum mechanics exploits the intrinsic time-symmetry

of wave mechanics to interpret the

and

* wave functions present in all wave mechanics calculations

as representing retarded and advanced waves moving in opposite time directions that form a

quantum “handshake” or transaction. This handshake is a 4D standing-wave that builds up across

space-time to transfer the conserved quantities of energy, momentum, and angular momentum in

an interaction. Here, we derive a two-atom quantum formalism describing a transaction. We show

that the bi-directional electromagnetic coupling between atoms can be factored into a matched pair

of vector potential Green’s functions: one retarded and one advanced, and that this combination

uniquely enforces the conservation of energy in a transaction. Thus factored, the single-electron wave

functions of electromagnetically-coupled atoms can be analyzed using Schrödinger’s original wave

mechanics. The technique generalizes to any number of electromagnetically coupled single-electron

states—no higher-dimensional space is needed. Using this technique, we show a worked example

of the transfer of energy from a hydrogen atom in an excited state to a nearby hydrogen atom in

its ground state. It is seen that the initial exchange creates a dynamically unstable situation that

avalanches to the completed transaction, demonstrating that wave function collapse, considered

mysterious in the literature, can be implemented with solutions of Schrödinger’s original wave

mechanics, coupled by this unique combination of retarded/advanced vector potentials, without the

introduction of any additional mechanism or formalism. We also analyze a simplified version of the

photon-splitting and Freedman–Clauser three-electron experiments and show that their results can

be predicted by this formalism.

Keywords:

quantum mechanics; transaction; Wheeler–Feynman; transactional interpretation;

handshake; advanced; retarded; wave function collapse; collapse mechanism; EPR; HBT; Jaynes;

NCT; split photon; Freedman–Clauser; nonlocality; entanglement

1. Introduction

Quantum mechanics (QM) was never properly finished. Instead, it was left in an exceedingly

unsatisfactory state by its founders. Many attempts by highly qualified individuals to improve the

situation have failed to produce any consensus about either (a) the precise nature of the problem,

or (b) what a better form of QM might look like.

At the most basic level, a simple observation illustrates the central conceptual problem:

An excited atom somewhere in the universe transfers

all

of its excitation energy to another single

atom, independent of the presence of the vast number of alternative atoms that could have received all

or part of the energy. The obvious “photon-as-particle” interpretation of this situation has a one-way

symmetry: The excited source atom is depicted as emitting a particle, a

photon

of electromagnetic

energy that is somehow oscillating with angular frequency

while moving in a particular direction.

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The photon is depicted as carrying a quantum of energy

, a momentum

, and an angular

momentum

through space, until it is later absorbed by some unexcited atom. The emission and

absorption are treated as independent isolated events without internal structure. It is insisted that the

only real and meaningful quantities describing this process are

probabilities

, since these are measurable.

The necessarily abrupt change in the quantum wave function of the system when the photon arrives

(and an observer potentially gains information) is called “wave function collapse” and is considered

to be a mysterious process that the founders of QM found it necessary to “put in by hand” without

providing any mechanism. [The missing mechanism behind wave function collapse is sometimes

called “the measurement problem”, particularly by acolytes of Heisenberg’s knowledge interpretation.

In our view, measurement requires wave function collapse

but does not cause it.

] [Side comments will

be put in square brackets]

Referring to statistical quantum theory, which is reputed to apply only to

ensembles

of similar

systems, Albert Einstein [1] had this to say:

“I do not believe that this fundamental concept will provide a useful basis for the whole of physics.”

“I am, in fact, firmly convinced that the essentially statistical character of contemporary quantum

theory is solely to be ascribed to the fact that this [theory] operates with an incomplete description of

physical systems.”

“One arrives at very implausible theoretical conceptions, if one attempts to maintain the thesis that the

statistical quantum theory is in principle capable of producing a complete description of an individual

physical system ...”

“Roughly stated, the conclusion is this: Within the framework of statistical quantum theory, there is

no such thing as a complete description of the individual system. More cautiously, it might be put

as follows: The attempt to conceive the quantum-theoretical description as the complete description

of the individual systems leads to unnatural theoretical interpretations, which become immediately

unnecessary if one accepts the interpretation that the description refers to ensembles of systems and

not to individual systems. In that case, the whole ’egg-walking’ performed in order to avoid the

’physically real’ becomes superfluous. There exists, however, a simple psychological reason for the

fact that this most nearly obvious interpretation is being shunned—for, if the statistical quantum

theory does not pretend to describe the individual system (and its development in time) completely,

it appears unavoidable to look elsewhere for a complete description of the individual system. In doing

so, it would be clear from the very beginning that the elements of such a description are not contained

within the conceptual scheme of the statistical quantum theory. With this. one would admit that,

in principle, this scheme could not serve as the basis of theoretical physics. Assuming the success of

efforts to accomplish a complete physical description, the statistical quantum theory would, within the

framework of future physics, take an approximately analogous position to the statistical mechanics

within the framework of classical mechanics. I am rather firmly convinced that the development of

theoretical physics will be of this type, but the path will be lengthy and difficult.”

“If it should be possible to move forward to a complete description, it is likely that the laws would

represent relations among all the conceptual elements of this description which, per se, have nothing

to do with statistics.”

In what follows we put forth a simple approach to

describing the individual system (and its

development in time),

which Einstein believed was missing from statistical quantum theory and which

must be present before any theory of physics could be considered to be complete.

The way forward was suggested by the phenomenon of

entanglement

. Over the past few

decades, many increasingly exquisite Einstein–Podolsky–Rosen [

] (EPR) experiments [

–

] have

demonstrated that multi-body quantum systems with separated components that are subject to

conservation laws exhibit a property called “quantum entanglement” [

]: Their component wave

functions are inextricably locked together, and they display a nonlocal correlated behavior enforced

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over an arbitrary interval of space-time without any hint of an underlying mechanism or any show of

respect for our cherished classical “arrow of time.” Entanglement is the most mysterious of the many

so-called “quantum mysteries.”

It has thus become clear that the quantum transfer of energy must have quite a different symmetry

from that implied by this simple “photon-as-particle” interpretation. Within the framework of

statistical QM, the intrinsic symmetry of the energy transfer and the mechanisms behind wave function

collapse and entanglement have been greatly clarified by the

Transactional Interpretation of quantum

mechanics

(TI), developed over several decades by one of us and recently described in some detail

in the book

The Quantum Handshake

[

]. [We note that Ruth Kastner has extended her “probabilist”

variant of the TI, which embraces the Heisenberg/probability view and characterizes transactions as

events in many-dimensional Hilbert space, into the quantum-relativistic domain [

] and has used

it to extend and enhance the “decoherence” approach to quantum interpretation [15]].

This paper begins with a tutorial review of the TI approach to a credible photon mechanism

developed in the book

Collective Electrodynamics

[

], followed by a deeper dive into the

electrodynamics of the quantum handshake, and finally includes descriptions of several historic

experiments that have excluded entire classes of theories. We conclude that the approach described

here has

not

been excluded by any experiment to date.

1.1. Wheeler–Feynman Electrodynamics

The Transactional Interpretation was inspired by classical time-symmetric Wheeler–Feynman

electrodynamics [

] (WFE), sometimes called “absorber theory.” Basically, WFE assumes that

electrodynamics must be time-symmetric, with equally valid retarded waves (that arrive

after

they

are emitted) and advanced waves (that arrive

before

they are emitted). WFE describes a “handshake”

process accounting for emission recoil in which the emission of a retarded wave stimulates a future

absorber to produce an advanced wave that arrives back at the emitter at the instant of emission.

WFE is based on electrodynamic time symmetry and has been shown to be completely interchangeable

with conventional classical electrodynamics in its predictions.

WFE asserts that the breaking of the intrinsic time-symmetry to produce the electromagnetic

arrow of time, i.e., the observed dominance of retarded radiation and absence of advanced radiation

in the universe, arises from the presence of more absorption in the future than in the past. In an

expanding universe, that assertion is questionable. One of us has suggested an alternative cosmological

explanation [

], which employs advanced-wave termination and reflection from the singularity of the

Big Bang.

1.2. The Transactional Interpretation of Quantum Mechanics

The Transactional Interpretation of quantum mechanics [

] takes the concept of a WFE handshake

from the classical regime into the quantum realm of photons and massive particles. The retarded

and advanced waves of WFE become the quantum wave functions

and

*. Note that the complex

conjugation of

* is in effect the application of the Wigner time-reversal operator, thus representing an

advanced wave function that carries negative energy from the present to the past.

Let us here clarify what an

interpretation

of quantum mechanics actually is. An interpretation

serves the function of explaining and clarifying the formalism and procedures of its theory.

In our view, the mathematics is (and should be) exclusively contained in the formalism itself.

The interpretation should not introduce additional variant formalism. [We note, however, that this

principle is violated by the Bohm–de Broglie “interpretation” with its “quantum potentials” and

uncertainty-principle-violating trajectories, by the Ghirardi–Rimini–Weber “interpretation” with its

nonlinear stochastic term, and by many other so-called interpretations that take the questionable

liberty of modifying the standard QM formalism. In that sense, these are alternative variant quantum

theories,

not

interpretations at all.]

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The present work is a calculation describing the formation of a transaction that was inspired by

the Transactional Interpretation but has not previously been a part of it. In Section 12 below, we discuss

how the TI is impacted by this work. We use Schrödinger’s original wave mechanics formalism

with the inclusion of retarded and advanced electromagnetic four-potentials

to describe and illuminate the

processes of transaction formation and the collapse of the wave function. We show that this approach

can provide a detailed mathematical description of a “quantum-jump” in which what seems to be

a photon is emitted by one hydrogen atom in an excited state and excites another hydrogen atom

initially in its ground state. Thus, the mysterious process of wave function collapse becomes just a

phenomenon involving an exchange of advanced/retarded electromagnetic waves between atomic

wave functions described by the Schrödinger formalism.

As illustrated schematically in Figure 1, the process described involves the initial existence in

each atom of a very small admixture of the wave function for the opposite state, thereby forming

two-component states in both atoms. This causes them to become weak dipole radiators oscillating at

the same difference-frequency

. The interaction that follows, characterized by a retarded-advanced

exchange of 4-vector potentials, leads to an exponential build-up of a transaction, resulting in the

complete transfer of one photon worth of energy

from one atom to the other. This process is

described in more detail below.

Figure 1.

Model of transaction formation: An emitter atom

2 in a space-antisymmetric excited

state of energy

and an absorber atom

1 in a space-symmetric ground state of energy

both

have slight admixtures of the other state, giving both atoms dipole moments that oscillate with the

same difference-frequency

−

. If the relative phase of the initial small offer wave

and

confirmation wave

∗

is optimal, this condition initiates energy transfer, which avalanches to complete

transfer of one photon-worth of energy ̄

2. Physical Mechanism of the Transfer

The standard formalism of QM consists of a set of arbitrary rules, conventionally viewed as dealing

only with probabilities. When illuminated by the TI, that formalism hints at an underlying physical

mechanism that might be understood, in the usual sense of the concept

understood

. The first glimpse of

such an understanding, and of the physical nature of the transactional symmetry,

was suggested

by Gilbert N. Lewis in 1926 [

], the same year he gave electromagnetic energy transfer the

name “photon”:

“It is generally assumed that a radiating body emits light in every direction, quite regardless of

whether there are near or distant objects which may ultimately absorb that light; in other words that it

radiates ’into space’...”

“I am going to make the contrary assumption that an atom never emits light except to another atom...”

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“I propose to eliminate the idea of mere emission of light and substitute the idea of transmission, or a

process of exchange of energy between two definite atoms... Both atoms must play coordinate and

symmetrical parts in the process of exchange...”

“In a pure geometry it would surprise us to find that a true theorem becomes false when the page

upon which the figure is drawn is turned upside down. A dissymmetry alien to the pure geometry of

relativity has been introduced by our notion of causality.”

In what follows, we demonstrate that the pair of coupled Schrödinger equations describing the

two atoms, as coupled by a relativistically correct description of the electromagnetic field, exhibit

a unique solution. This solution has exactly the symmetry of the TI and thus provides a

physically

understandable

mechanism for the experimentally observed behavior: Both atoms, in the words of

Lewis, “

play coordinate and symmetrical parts in the process of exchange

”.

The solution gives a smooth transition in each of the atomic wave functions, brought to abrupt

closure by the highly nonlinear increase in coupling as the transition proceeds. The origin of statistical

behavior and “quantum randomness” can be understood in terms of the random distribution of

wave-function amplitudes and phases provided by the perturbations of the many other potential

recipient atoms; no “hidden variables” are required. Although much remains to be done, these findings

might be viewed as a next step towards a physical understanding of the process of quantum

energy transfer.

We will close by indicating the deep, fundamental questions that we have not addressed, and that

must be understood before anything like a complete physical understanding of QM is in hand.

3. Quantum States

In 1926, Schrödinger, seeking a wave-equation description of a quantum system with mass,

adopted Planck’s notion that energy was somehow proportional to frequency together with deBroglie’s

idea that momentum was the propagation vector of a wave and crafted his wave equation for the time

evolution of the wave function

[22]:

−

m i

∇

q V

∂

(1)

Here,

is the electrical potential,

is the electron mass, and

is the (negative) charge on the

electron. Thus, what is the meaning of the wave function

that is being characterized? In modern

treatments,

is called a “probability amplitude”, which has only a probabilistic interpretation. In what

follows, however, we return to Schrödinger’s original vision, which provides a detailed physical

picture of the wave function and how it interacts with other charges:

“The hypothesis that we have to admit is very simple, namely that the square of the absolute value

is proportional to an electric density, which causes emission of light according to the laws of

ordinary electrodynamics.”

That vision has inspired generations of talented conceptual thinkers to invent solutions to technical

problems using Schrödinger’s approach. Foremost among these was Ed Jaynes who, with a number

of students and collaborators, attacked a host of quantum problems in this manner [

–

]. A great

deal of physical understanding was obtained, in particular concerning lasers and the coherent optics

made possible by them. The theory evolved rapidly and had an enabling role in the explosive progress

of that field. Indeed, the continued rapid technical progress into the present is due, in no small part,

to the understanding gained through application of the Jaynes way of thinking. A detailed review

of the progress up to 1972 was reported in a conference that year [

]. By then this class of theory

was called

neoclassical

(NCT) because of its use of Maxwell’s equations. While there was no question

about the utility of NCT in the conceptualization and technical realization of amazing quantum-optics

devices and their application, there was a deep concern about whether it could possibly be correct

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the fundamental level

—maybe it was just a clever bunch of hacks. The tension over this concern was a

major focus of the 1972

Third Rochester Conference on Coherence and Quantum Optics

[

], and several

experiments testing NCT predictions were discussed there by Jaynes [30].

He ended his presentation this way:

“We have not commented on the beautiful experiment reported here by Clauser [

], which opens up

an entirely new area of fundamental importance to the issues facing us...”

“What it seems to boil down to is this: a perfectly harmless looking experimental fact (nonoccurence

of coincidences at

◦

), which amounts to determining a single experimental point—and with a

statistical measurement of unimpressive statistical accuracy—can, at a single stroke, throw out a

whole infinite class of alternative theories of electrodynamics, namely all local causal theories.”

“...if the experimental result is confirmed by others, then this will surely go down as one of the most

incredible intellectual achievements in the history of science, and my own work will lie in ruins.”

The experiment he was alluding to was that of Freedman and Clauser [

], and in particular to

their observaton of an essentially zero coincidence rate with crossed polarizers. The Freedman–Clauser

experiment (see Section 13.3 below), with its use of entangled photon pairs, was the vanguard

of an entire new direction in quantum physics that now goes under the rubric of

Tests of Bell’s

Inequality

[

] and/or

EPR experiments

[

–

]. Both the historic EPR experiment and its analysis

have been repeated many times with ever-increasing precision, and always with the same outcome:

a difinitive violaton of Bell’s inequalities. Local causal theories were dead! [Much of the literature

on violations of Bell’s inequalities in EPR experiments has unfortunately emphasized the refutation

local hidden-variable theories

. In our view, this is a regrettable historical accident attributable to Bell.

Nonlocal

hidden-variable theories have been shown to be compatible with EPR results. It is

locality

that

has been refuted. Entangled systems exhibit correlations that can only be accomodated by quantum

nonlocality. The TI supplies the mechanism for that nonlocality.]

In fact, it was the manifest quantum nonlocality evident in the early EPR experiments of the

1970s that led to the synthesis of the transactional interpretation in the 1980s [

], designed to

compactly explain entanglement and nonlocality. This in turn led to the search for an underlying

transaction mechanism, as reported in 2000 in

Collective Electrodynamics

[

]. As we detail below,

the quantum handshake, as mediated by advanced/retarded electromagnetic four-potentials, provides

the effective non-locality so evident in modern versions of these EPR experiments. In Section 13,

we analyze the Freedman–Clauser experiment in detail and show that their result is a natural outcome

of our approach. Jaynes’ work does not lie in ruins—all that it needed for survival was the non-local

quantum handshake! What follows is an extension and modification of NCT using a different

non-Maxwellian form of E&M [

] and including our non-local Transactional approach. We illustrate

the approach with the simplest possible physical arrangements, described with the major goal of

conceptual understanding rather than exhaustion. Obviously, much more work needs to be done,

which we point out where appropriate.

Atoms

We will begin by visualizing the electron as Schrödinger and Jaynes did: as having a smooth

charge distribution in three-dimensional space, whose density is given by

∗

. There is no need for

statistics and probabilities at any point in these calculations, and none of the quantities have statistical

meaning. The probabilistic outcome of quantum experiments has the same origin as it does in all other

experiments—random perturbations beyond the control of the experimenter. We return to the topic of

probability after we have established the nature of the transaction.

For a local region of positive potential

, for example near a positive proton, the negative

electron’s wave function has a local potential energy (

) minimum in which the electron’s wave

function can form local

bound states

. The spatial shape of the wave function amplitude is a trade-off

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between getting close to the proton, which lowers its potential energy, and bunching together too

much, which increases its

∇

“kinetic energy.” Equation

(1)

is simply a mathematical expression of

this trade-off, a statement of the physical relation between mass, energy, and momentum in the form

of a wave equation.

A discrete set of states called

eigenstates

are standing-wave solutions of Equation

(1)

and have

the form

−

, where

and

are functions of only the spatial coordinates, and the angular

frequency

is itself independent of time. For the hydrogen atom, the potential

, where

is the positive charge on the nucleus, equal in magnitude to the electron charge

. Two of the

lowest-energy solutions to Equation (1) with this potential are:

100

−

√

−

210

r e

−

cos

(

)

√

−

(2)

where the dimensionless radial coordinate

is the radial distance divided by the

Bohr radius

≡

πe

0.0529 nm,

(3)

and

is the azimuthal angle from the North Pole (

axis) of the spherical coordinate system.

The spatial “shape” of the two lowest energy eigenstates for the hydrogen atom is shown in

Figure 2. Here, we focus on the excited-state wave function

210

that has no angular momentum

projection on the

-axis. For the moment, we set aside the wave functions

, which have

1 and

−

1 angular momentum

-axis projections. Because, for any single eigenstate, the electron density is

∗

−

, the charge density is not a function of time, so none of the other properties of

the wave function change with time. The individual eigenstates are thus

stationary states

. The lowest

energy bound eigenstate for a given form of potential minimum is called its

ground state

, shown on

the left in Figure 3. The corresponding charge densities are shown in Figure 4.

Figure 2.

Angular dependence of the spatial wave function amplitudes for the lowest (100,

left

) and

next higher (210,

right

) states of the hydrogen atom, plotted as unit radius in spherical coordinates

from Equation

(2)

. The 100 wave function has spherical symmetry: positive in all directions.

The 210

wave function is antisymmetric along the

-axis, as indicated by the color change. In practice,

the direction of the

-axis will be established by an external electromagnetic field, as we shall

analyze shortly.

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-

-

0.0

0.1

0.2

0.3

0.4

0.5

z

-

-

-

0.05

0.00

0.05

z

Figure 3.

Wave function amplitudes

for the 100 and 210 states, along the

-axis of the hydrogen atom.

The horizontal axis in all plots is the position along the

-axis in units of the Bohr radius.

-

-

0.00

0.05

0.10

0.15

0.20

0.25

z

-

-

0.00

0.01

0.02

0.03

0.04

0.05

0.06

z

Figure 4.

Contribution of

−

“slices” at position

of wave function density

∗

to the total charge

or mass of the 100 and 210 states of the hydrogen atom. Both curves integrate to 1.

In 1926, Schrödinger had already derived the energies and wave functions for the stationary

solutions of his equation for the hydrogen atom. His physical insight that the absolute square

∗

of the wave function was the

electron density

had enabled him to work out the energy shifts of these

levels caused by external applied electric and magnetic fields, the expected strengths of the transitions

between pairs of energy levels, and the polarization of light from certain transitions.

These predictions could be compared directly with experimental data, which had been previously

observed but not understood. He reported that these calculations were:

“...not at all difficult, but very tedious. In spite of their tediousness, it is rather fascinating to see

all the well-known but not understood “rules” come out one after the other as the result of familiar

elementary and absolutely cogent analysis, like e.g., the fact that

∫

cos

vanishes

unless

. Once the hypothesis about

∗

has been made, no accessory hypothesis is needed or

is possible; none could help us if the “rules” did not come out correctly. However, fortunately they

do [22,35].”

The Schrödinger/Jaynes approach enables us to describe continuous quantum transitions in

an intuitively appealing way: We extend the electromagnetic coupling described in Collective

Electrodynamics [

] to the wave function of a single electron, and require only the most rudimentary

techniques of Schrödinger’s original quantum theory.

4. The Two-State System

The first two eigenstates of the Hydrogen atom, from Equation

(2)

, form an ideal two-state system.

We refer to the 100 ground state as State 1, with wave function

and energy

, and to the 210 excited

state as State 2, with wave function

and energy

−

(4)

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where

/ ̄

, and

and

are real, time-independent functions of the space

coordinates. The wave functions represent totally continuous matter waves, and all of the usual

operations involving the wave function are methods for computing properties of this continuous

distribution. The only particularly quantal assumption involved is that the wave function obeys a

normalization condition

∫

∗

vol

(5)

where the integral is taken over the entire three-dimensional volume where

is non-vanishing.

[Envelope functions like

and

generally die out exponentially with distance sufficiently far from

the “region of interest”, such as an atom. Integrals like this one and those that follow in principle

extend to infinity but in practice are taken out far enough that the part being neglected is within the

tolerance of the calculation.].

Equation

(5)

ensures that the total charge will be a single electronic charge, and the total mass will

be a single electronic mass.

By construction, the eigenstates of the Schrödinger equation are real and orthogonal:

∫

vol

(6)

The first moment

〈

〉

of the electron distribution along the atom’s

-axis is:

〈

〉

≡

∫

∗

vol,

(7)

In statistical treatments,

〈

〉

would be called the “expectation value of

”, whereas for our

continuous distribution it is called the “average value of

” or the “first moment of z.” The electron

wave function is a

wave packet

and is subject to all the Fourier properties of one, as treated at some

length in Ref. [

]. Statistical QM insisted that electrons were “point particles”, so one was no longer

able to visualize how they could exhibit interference or other wave properties, so a set of rules was

constructed to make the outcomes of statistical experiments come out right. Among these was the

uncertainty principle

, which simply restated the Fourier properties of an object described by waves

in a statistical context. No statistical attributes are attached to any properties of the wave function in

this treatment.

Equation

(7)

gives the position of the center of negative charge of the electron wave function

relative to the positive charge on the nucleus. When multiplied by the electronic charge

, it is called

the electric dipole moment q

〈

〉

of the charge distribution of the atom:

〈

〉

∫

∗

vol.

(8)

From Equations (7) and (4), the dipole moment for the

th eigenstate is:

〈

〉

∫

∗

vol

∫

∗

z R

vol

∫

z d

vol.

(9)

Pure eigenstates of the system will have a definite parity, i.e., they will have wave functions with

either even symmetry [

(

) =

(

−

)

], or odd symmetry [

(

) =

−

(

−

)

]. For either symmetry,

the integral over

vanishes, and the dipole moment is zero. We note that, even if the wave function

did not have even or odd symmetry, the dipole moment, and all higher moments as well, would be

independent of time. By their very nature, eigenstates are stationary states and can be visualized as

standing-waves—none of their physical spatial attributes can be functions of time. In order to radiate

electromagnetic energy, the charge distribution

must change with time

The notion of

stationarity

is the quantum answer to the original question about atoms depicted as

electrons orbiting a central nucleus like a tiny Solar System:

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Why doesn’t the electron orbiting the nucleus radiate its energy away?

In his 1917 book,

The Electron

, R.A. Millikan [36] anticipates the solution in his comment about the

”. . . apparent contradiction involved in the non-radiating electronic orbit—a contradiction which

would disappear, however, if the negative electron itself, when inside the atom, were a ring of some

sort, capable of expanding to various radii, and capable, only when freed from the atom, of assuming

essentially the properties of a point charge.”

Millikan was the first researcher to directly observe and measure the quantized charge on the

electron with his famous oil-drop experiment, for which he later received the Nobel prize. Ten years

before the statistical quantum theory was put in place, he had clearly seen that a continuous, symmetric

electronic charge distribution would not radiate, and that the real problem was the assumption of a

point charge.

5. Transitions

The eigenstates of the system form a complete basis set, so any behavior of the system can be

expressed by forming a linear combination (superposition) of its eigenstates.

The general form of such a superposition of our two chosen eigenstates is:

−

(10)

where

and

are real amplitudes, and

and

are real constants that determine the phases of the

oscillations

and

Using

−

and

−

, the charge density

of the two-component-state wave

function is:

∗

(

−

)(

−

)

(

−

)

(

−

)

−

(

−

)

(

−

(

−

)

(

)

abR

cos

(

)

(11)

Thus, the charge density of the two-component wave function is made up of the charge densities

of the two separate wave functions, shown in Figure 4, plus a term proportional to the product of the

two wave function amplitudes. It reduces to the individual charge density of the ground state when

0 and to that of the excited state when

1. The product term, shown in green in

Figure 5, is the only non-stationary term; it oscillates at the transition frequency

. The integral of the

total density shown in the right-hand plot is equal to 1 for any phase of the cosine term, since there is

only one electron in this two-component state.

All the

∗

plots represent the density of negative charge of the electron. The atom as a whole is

neutral because of the equal positive charge on the nucleus. The dipole is formed when the center of

charge of the electron wave function is displaced from the central positive charge of the nucleus.

Symmetry

2020

, 1373

11 of 44

-

-

-

-

0.10

-

0.05

0.00

0.05

0.10

0.15

0.20

0.25

z

-

-

-

-

0.10

-

0.05

0.00

0.05

0.10

0.15

0.20

0.25

z

Figure 5. Left

: Plot of the three terms in the wave-function density in Equation

(11)

for an equal

(

√

)

superposition of the ground state (

, blue) and first excited state (

, red) of

the hydrogen atom. The green curve is a snapshot of the time-dependent

product term,

which oscillates at difference frequency

Right

: Snapshot of the total charge density, which is

the sum of the three curves in the left plot. The magnitudes plotted are the contribution to the total

charge in an

−

“slice” of

∗

at the indicated

coordinate. All plots are shown for the time such

that

cos

(

) =

1. The horizontal axis in each plot is the spatial coordinate along the

-axis of the

atom, given in units of the Bohr radius

. Animation here [37] (see Supplementary Materials).

The two-component wave function must be normalized, since it is the state of one electron:

∫

∗

vol

∫

(

−

)

(

−

)

vol

∫

vol

∫

vol

(

−

)

∫

vol.

(12)

Recognizing from Equations

(5)

and

(6)

that the individual eigenfunctions are normalized

and orthogonal:

∫

vol

∫

vol

∫

vol

(13)

Equation (12) becomes

∫

∗

vol

(14)

Thus,

represents the fraction of the two-component wave function made up of the lower state

, and

represents the fraction made up of the upper state

. The total energy

of a system whose

wave function is a superposition of two eigenstates is:

(15)

Using the normalization condition

1 and solving Equation (15) for

, we obtain:

−

(16)

In other words,

is just the energy of the wave function, normalized to the transition

energy, and using

as its reference energy. Taking

as our zero of energy and

−

Equation (16) becomes:

⇒

∂

(

)

∂

(17)