The structural foundations of the Principle of Indetermination
In 1927, Heisenberg submitted to the members of the Copenhagen School a strange development in quantum theory that appeared to limit on principle the possibility of knowing the parameters of energy and time simultaneously for any particular particle in space.
After careful consideration, the positivist Copenhagen School and Bohr declared that the impossibility of knowing a particle’s position and energy simultaneously was due to the fact that “the same particle could not physically at the same time possess both a position and an energy that were well-defined”.
This proposition, that raised an experimental impossibility to the level of a philosophical principle, quickly became very important in the description of the properties of the microphysical world. It was elevated to the rank of leading physical principle and subsequently baptized the Indetermination Principle by Eddington.
On this basis, many indeterministic hypothesis were confidently asserted in quantum physics, which opened the door to the wildest intellectual fantasies and even revived the ancient ghost of the theory of solipsism that had long been buried.
Conveniently installed on Quantum Mechanics’ chariot, the Indetermination Principle tourned the world and took root firmly in the minds of all physicists who from then on judged every phenomenon and any theory designed to explain it in terms of whether or not it supported the new principle.
But the beautiful red apple was hiding a deadly worm.
To subscribe fully to the Indetermination Principle implied
giving up the Principle of Causality.
For an eighteenth-century physicist, even the slightest doubt about the principle of causality would have been unacceptable. Even for many modern physicists, the possibility that it must be abandoned remains inconceivable. But no one today denies that the Indetermination Principle is hiding something mysterious that is basic in nature.
Eddington said about this:
Every scientific authority appears to agree that at or very close to the root of everything in the world of physics we find the mysterious formula:
q p – p q = i h / 2π
We do not understand it yet; perhaps if we could understand it, we would not find it as fundamental. The mathematical test tube has the advantage of being able to make use of it […]. Not only does it lead us to those phenomena described by earlier laws governing quanta such as rule h, but also to many related phenomena, for which the old formulas had proved useless. In the second part of the formula, alongside h (the atom of action) and the purely numerical factor 2 π, we find “i” (square root of -1), that may appear rather mysterious.
But this is no more than a rather well-known subterfuge; very early in the last century, physicists and engineers were well aware that the presence of the square root of –1 was a kind of signal to warn them of waves or oscillations.
(Was this prophetic, or simply logical?)
The second part contains nothing unusual; it is the first part that practically overwhelms us. We call p and q certain coordinates and quantities of motion and borrow thereby our vocabulary from the world of space and time and from other rough experiments; but this sheds no light on their nature nor does it explain why (the product of) q x p is so ill-mannered as to not be equal to p x q. It is obvious that q and p cannot represent simple numerical measures, as in that case qp – pq would equal zero.
For Schrödinger, p is an “operator”. Its “momentum” is not a quantity, but a signal that allows us to carry out a certain mathematical operation on certain quantities that follow it. For Born and Jordan, p is a “matrix”; it is not a quantity or even numerous quantities but an infinite number of quantities arranged in a systematic order.
It is clear from what Eddington wrote that the Indetermination Principle was indeterminate from the very beginning.
Following an idea by Condon, the american, Robertson, finally found an intuitive interpretation of the quantities q and p later.
He came very close to the physical significance of the Indetermination Principle when he found that between the precision with which we can determine the position of a particle and the precision with which we can simultaneously know its momentum, there is a formula in which the product of errors possible in the measurements of the position and momentum is at least as great as the Planck constant divided by 4p.
As a result, the formula Δ p Δ q = h /2π must apply generally.
In it, the deltas (Δ) associated with p and q establish that we must take into account intervals of values for probable errors in “measurement”.
A famous intellectual experiment by Heisenberg demonstrates the intuitive part of the Indetermination Principle and at the same time its epistemological solidity.
We will now describe the experiment underlying our current knowledge of the wave-like nature of photons and electrons.
- Take a microscope and establish the position of an electron by means of observation.
- Even if the microscope’s resolution allowed us to identify an electron, to see it we would somehow have to illuminate it with a beam of light, which would first strike the electron and then return to strike the rods in our eye.
- Thus, to be able to determine the direction from which the light is coming best, we would illuminate the electron with a single photon.
This is the same experimental condition we found for the Compton effect and, as we have already seen, the electron diffracts the photon and gives it a momentum determined by the initial energy of the incident photon and the angle at which it is diffracted.
The experiment with the microscope is too coarse, however, and the microscope’s lenses are unable to determine the angle of diffraction with sufficient precision.
We must therefore refine our intellectual experiment.
- Suppose our entire ideal experiment takes place within an enormous hollow sphere, within which we postulate a hypothetical absolute vacuum.
- Its interior walls should be covered with sensors that are as small as possible and able to signal the arrival of a photon or an electron such as hydrogen atoms at a temperature very close to absolute zero.
- As each atom is struck by a photon or an electron, it will emit an electron or a photon that will be captured on the outside of the sphere by one of the billions of photoelectric cells required to cover every possible emission, one series for each atom our instrument will create.
- Force an electron to the center of the sphere with a repulsive electrical field and strike it with a photon.
- Removing the field in the same instant, record where the photon strikes the hydrogen atom and where the electron subsequently collides with another atom.
Picture 73. Ideal experiment to test the Indeterminacy Principle using a hollow sphere and the trajectories of a photon and an electron.
- On the basis of the angle that can be recorded between the photon’s starting point, the center of the sphere and its arrival point, we can calculate the fraction of its energy that it transmits to the electron and then confirm this data in terms of the electron’s arrival point.
In the wave model of the atom we now have in mind, we can establish the experimental uncertainty about where the photon will arrive as corresponding at most to the length of the diameter of the electron’s orbit in the excited hydrogen atom.
Thus, as we can calculate the uncertainty of the value of the angle made by the photon following diffraction, we can establish the degree of uncertainty in the excited electron’s momentum.
We also have a further uncertainty involving the energy level and thus the wavelength of the incident photon that cannot be absolutely monochromatic. This uncertainty could be overcome by comparing the linked angles of the trajectories of the photon and the electron a posteriori, as they would in any case remain linked to the size of the electron’s orbit within the hydrogen atom.
It is on principle possible to reduce the residual error that has remained.
The overall error in the angles can be restricted to ever smaller values, if we modify our ideal measuring equipment and enlarge the diameter of the sphere 1,000, 10,000 or a million times and increase the number of indicator atoms on its internal surface by the square of its radius.
It would appear possible to reduce the definition of the error at will in this way, although it would never be possible to reduce it to zero.
Would it therefore be possible to reduce at will the indeterminacy between the value of a particle’s position and its momentum?
No! There is still one more peculiar uncertainty that, while it exists in the calculations and formulas of the Indetermination Principle, it has never been possible to attribute to a precise physical model and for which it is only now possible to describe with physical evidence using the Wave Theory of the Field.
- Let us examine the spherical model of the electron and the structure of the evolving spherical plane that develops on the plane of the electron’s resonance orbit.
- This structure is characterized by the fact that it does “not” possess central symmetry. It can come ever closer to a condition of central symmetry as it distances itself from the resonance orbit without ever achieving absolute symmetry.
- On the other hand, in the immediate vicinity of the orbit of the electron’s waves that created it, this structure is very definitely eccentric.
- In the Compton interaction, the parts of the electron’s structure that most influenced the incident photon were those farthest inside the evolving structure.
- This means the wave surfaces closest to the resonance orbit, as they have a smaller radius of curvature. And it is precisely here, close to the resonance orbit, that the spherical wave field assumes its most pronounced condition of “non-” symmetry.
- The magnitude of this “non-symmetry” or eccentricity can be at least a wavelength and we can see the reasons for this by observing a wave in the immediate vicinity of a resonance orbit in picture 74.
Picture 74. Anyone wishing to localize the Involvent plane in space must take into account the uncertainty involved in determining its position due to the experimental impossibility of determining the position of the center of the resonance orbit.
- The static figure depicts a specific point in the resonance orbit from which the wave front emerges that will ultimately develop into the evolving spherical plane. In a real electron, however, the evolution of the evolving spherical plane is dynamic.
- The resonant wave front rotates at the speed of light on the resonance orbit just as the resulting eccentricity in the wave system itself also rotates at the speed of light.
- We will never be able actually to see in what zone or at which point the eccentricity can be found when the wave system interacts with the incident photon, nor can we know it physically in any way.
This ignorance on our part forces us to introduce an element of uncertainty into the determination of the electron’s position and as a result in that of the point at which it interacts with the photon. We therefore can only determine the effect of the Principle of Relative Symmetry of the photon on the wave source by introducing an inevitable error of the order of a quantum of action for its momentum and one wavelength for its position:
Δ p Δ q ≥ h / 2 π.
The impossibility of knowing the position of the wave front circulating on the resonance orbit can now be understood as the real physical and experimental impossibility of determining the position of the center of the particle’s resonance orbit.
Indetermination is now a direct consequence of the existence and structure of the physical model of the spherical Involvent.
This does not mean that there is not at any particular instant a real and effective location of the position of the center of the electron’s resonance orbit in space and time. But it is true that as we will never know on which side we should locate the eccentricity of the electron wave system there will necessarily be an intrinsic indeterminacy in positioning the center of the particle in space.
To this indeterminacy based on our ignorance of its position, we must add an indeterminacy involving our initial knowledge of the photon’s energy. And there is also indeterminacy involving the photon’s action on the electron expressed as the transmission of momentum as a function of the Principle of Relative Symmetry.
All the indeterminacies we have described derive from our ignorance and inability to know the precise parameters of the wave phenomena and provide no philosophical disproof of the possibility that the actual parameters could really exist in space and time.
Even the fact that the electron’s field can interact with the fields of the material and radioactive parts of the tools used to measure it provides banal evidence in favor of the Wave Theory of the Field.
The opinion that is frequently expressed with extreme ease in Quantum Mechanics but then proven only with a small degree of material evidence that uncontrollable interactions become involved between the instruments and the entities being measured in the act of measurement is swallowed wholesale.
It is true that an unpredictable interaction does exist between the instrument and the object of measurement, if only within a precise interval of indeterminacy.
Some adherents of the Copenhagen School seemed to concern themselves stubbornly with the philosophical implications of these facts for Quantum Mechanics as the basis of physics. But they avoided the philosophical implications involved for the reality of the outside world. They meant to include the observer as part of the experiment. The observer was thus also interfering simply by observing given the uncertainty of the experimental result.
The whole issue could depend on the confusion between the realistic and probabilistic interpretations of the wave functions.
The question here is very different, however.
We are now talking about real fields and real waves and not just about probabilistic functions.
The issue of the objectivity of experimental phenomena is part of the platform necessary for the new way of looking at the world’s reality in terms of waves. It is clear that the new explanatory tools must exclude the observer’s subjectivity from the investigation as the factor that determines causal effects.
Although we are aware that fields of matter, by their very expansive nature, interact with the entire universe, when we describe or interpret an experiment, we must seek to limit the investigation to local wave interactions.
We presume that the effects of interactions over great distances are limited.
We find confirmation of this by observing that the effects of nuclear interactions, the most efficient interactions in nature, also decrease significantly with the increase in the radius of the spherical waves originating in the elementary components of matter.
The concepts of wave location and interaction can exist simultaneously in physics if we posit a “reasonable” decrease in the effects of the interaction as the distance increases.
Our understanding of the phenomena that is essential in this context can free itself finally of doubts that the instrument may be interacting with the photon.
No doubt remains. Both photons and electrons “certainly” interact physically with the wave field originating in the instrument.
Any experiment that seeks to describe a quantum phenomenon must take into account most of the wave fields involved with the phenomenon. But they must also carefully note that the effectiveness of the interaction with wave fields decreases significantly as the radius of the waves originating in the sources of the fields increases.