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(V4) Increasing Exclusion: The Pauli Exclusion Principle and Energy
Conservation for Bound Fermions Are Mutually Exclusive
By Jonathan Phillips
Distinguished National Lab Professor, University of New Mexico
ABSTRACT
A review of those forms of standard quantum mechanics that include the Pauli Exclusion Principle as it is applied to atomic species, (that is versions of quantum that are multi-electron and multi-orbital) shows they are not consistent with energy conservation. Particular focus is given to helium in which it is shown that energy conservation is not consistent with current models. If the two electrons in the ground state, per current theory, are at the same energy as the ionization energy, it is demonstrated that according to the standard theory approximately 30 eV are lost during ionization, or alternatively, about 30 eV of energy are created during ionization/electron attachment. The same issue of energy loss during ‘relaxation’ of energy levels following ionization is shown to exist for all atomic species, thus demonstrating that the Pauli Exclusion Principle (PEP) and energy conservation are not consistent for any atomic species for current forms of distinguishable electron forms of quantum theory. Only that form of quantum that has a single orbital, and for which only one ionization energy can be computed (that is the original Schrödinger form), is consistent with an energy balance. However, this form is not consistent with the most common spectroscopy results, and, it is shown that the PEP has no meaning in this form of quantum theory. In contrast, a new model of quantum mechanics, Classical Quantum Mechanics (CQM), invented by R. Mills, is shown, following modification, to be consistent with all spectroscopy and energy conservation for bound electron systems. This new model is based on the validity of Maxwell’s Equations and Newton’s Laws at all scales. Detailed, and remarkably simple, computations for determining the ‘ground state’ energy levels in one and two electron systems using CQM are presented. Excellent agreement with data is found.
Keywords: Quantum mechanics, Pauli Exclusion Principle, Energy conservation,
helium, Classical Quantum Mechanics
INTRODUCTION
This article is intended to support two hypotheses. First, that standard quantum mechanics is inconsistent with known ionization energies. The ionization energies of the simplest multi-electron system, helium, are used to illustrate the point. In particular, we will endeavor to show that all forms (three classes are identified) of standard quantum fail, because the proposed ‘relaxation’ of electron energy levels, an inherent consequence of the PEP, in atoms following ionization is inconsistent with the conservation of energy. To emphasize: the failing is an inevitable consequence of assuming the Pauli Exclusion Principle (PEP) as generally understood. (The PEP as generally understood: No two electrons/fermions can have the same quantum numbers, and hence an orbital is ‘full’ when two electrons, identical other than having different spin directions, are in that orbital.) Second, we will show that the description of two electron systems based on a new paradigm of quantum mechanics, Classical Quantum Mechanics (CQM), developed by Randell Mills (1), but with a simple modification introduced herein, is consistent with energy conservation. Hence, it is concluded that modified CQM is a superior theory for the range of bound electrons. Extension to free electrons is left for future essays. In order to provide support for the above dramatic hypotheses, we include sections normally not found in scientific articles. First, we find it expedient to include a description of the scientific process as it applies to physical sciences. Second, we find it absolutely necessary to offer a new means to classify standard quantum theory. Specifically, a novel classification scheme is presented to help ‘untangle’ the net of quantum mechanics theories. It is argued there exist two distinct ‘standard’ quantum mechanics, the one ‘expository’ or ‘descriptive’ version employed by chemists and the quantitative version. Finally, there are the theories ‘between’ which are shown not to be true quantum in their very nature, indeed, they do not even employ Hamiltonians. These various versions of quantum are not compatible. Once we show that ‘quantum’ is not even a coherent single theory, and that the inclusion of the PEP in any quantum mechanics model makes the model inconsistent with energy conservation, we argue it is scientifically rational, in fact scientifically obligatory, to discuss alternatives to standard quantum theories. Only at this point in the presentation do we address the second hypothesis of the paper. To wit: It is shown that a new theory of quantum mechanics based solely on classical physics, particularly Maxwell’s equations and Newton’s Laws, and arguably one postulate regarding the strength of the interaction between two bound electrons, does give a totally satisfactory, algebraic, closed form, model of helium photo ionization with no variable parameters. Not only is it completely quantitatively consistent with all observations, but unlike standard quantum mechanics, the model is simple, self-consistent and consistent with energy conservation.
SCIENTIFIC PROCESS
system model of electrons/atoms ultimately failed, generally because they could not predict the correct g-factor (13), they indicate that many scientists have always understood that the current quantum model is far from satisfactory and that alternatives should be considered.
REVIEW OF STANDARD QUANTUM THEORY APPLIED TO BOUND FERMIONS
It is important in the context of the core of this paper to recognize that prior to the acceptance of the quantum model (QM) in its earliest form about 80 years ago, physicists believed that Newton’s Laws and Maxwell’s Equations applied at all size scales. However, no physicist succeeded in finding a means to explain the ‘quantum’ nature of stable energy levels in atoms, discovered via spectroscopy (e.g. Balmer series), using these laws (13). Hence, a new theory, herein called standard quantitative quantum mechanics (SQQM), was born that produced results quantitatively consistent with the quantum behavior of hydrogen. In time it was realized that this new theory, SQQM, was totally inconsistent with classical physics laws at size scales ‘less than h-bar’. For example, according to SQQM bound electrons, in single electron systems, are ‘point particles’ that jump randomly, obeying only a statistical description of their position- not forces, not conservation of momentum, etc., hence in violation of Newton’s Laws. Another clear example of the violation of Newton’s Laws: In those cases in which the point electron in a one electron system is far from the nucleus, allowed and expected according to the probability distribution used to describe the electron, it has nearly zero potential energy. At those positions in order to preserve the total binding energy, that is the sum of the potential and kinetic energy, the electron is presumed to have negative kinetic energy. This is clearly not consistent with Newtonian mechanics, even if it is argued these states only last a tiny fraction of a second. The violations of Maxwell’s equations are equally clear. The random travels of the electron in the single electron system anticipated on the basis of the probabilistic model, result in instantaneous jumps into/out of the nucleus. In a classical sense this is accelerated motion, which by Maxwell’s equations should require the emission (jump toward nucleus) or absorption (jump away from nucleus) of energy. Yet according to SQQM these processes occur without the emission or adsorption of energy, hence SQQM is a clear violation of Maxwell’s equations. To make SQQM match the older classical theories the community, under the guidance of Bohr, adopted the notion of the ‘Correspondence Principle’ more than 70 years ago. At some size scale, around h-bar, the two theories give identical predictions. Still, there are many who never were satisfied with this explanation. Einstein was very unhappy with the statistical description of the behavior of sub-atomic particles given in quantum theory, as succinctly put in his now famous comment, ‘God does not play dice with the universe.’ The above issues are well rehearsed even in basic quantum theory courses. However, the problems with SQQM are far deeper for multi-electron systems, e.g. helium, than they are for single electron systems. In particular: the multi-electron wave function multiplied by its complex conjugate, a single scaler value at every point in 3N (N is the number of electrons) phase-space, has no physical meaning. This is discussed in more detail later.
Another issue is the notion that anything regarding SQQM is proven. Indeed, is the, above described, division of physics at a size scale of h-bar proven? No. A theory shown to be consistent with all known phenomenon is still not proved, just shown to be ‘likely’. After all, science is empirical, hence not susceptible to absolute proof. Although proving a theory is difficult, disproving one is simple. Indeed, all that is required to disprove or at least limit the applicable range of a theory is to show that the theory fails to accurately describe a single ‘objective scientific observation’. Indeed, this is the rationale given for restricting the domain of applicability of Newton’s Laws and Maxwell’s Equations to scales ‘greater than h-bar’. It is still possible that Newton’s Laws or Maxwell’s equations do apply to matter on all scales. It has not been proven that they do not, all we know is that earlier efforts failed! Thus, the argument at the core of this manuscript, that a single epiphany regarding the correct geometrical description of elementary particles ‘allows’ correct (i.e. matches objective scientific observations) values to be obtained using classical laws of physics on all scales, cannot be summarily rejected as ‘unscientific’. If such a claim is made, it must be tested. In our case we provide a first test of the CQM physical description of bound electrons. We show that the new ‘physical’ model of the bound electron is quantitatively consistent with all presently known objective scientific phenomena. Will the physics community recognize the primacy of the scientific process and not be guided by the weight of tradition? Will they even consider questioning SQQM? The sections below amplify many of the themes introduced above. This is done with respect to the three major classes of standard quantum, only one of which can be the correct quantum. Discussions of standard quantum, including defenses of standard quantum, must necessarily begin with the identification of the class of quantum theory at issue. That is, those arguing ‘for’ quantum theory must select, and identify, only one of the three theories below as the ‘quantum’ they are defending. These three theories, it is argued below, are distinct and incompatible.
THE INTUITIVE VERSION: DQM One major difficulty in a critique of quantum mechanics is the difficulty of ‘pinning down’ the core theory. There are actually at least three categories of ‘parallel’ quantum theories. The most intuitive is herein called ‘Descriptive Quantum Mechanics’ (DQM). It is often erroneously conflated with SQQM. In fact, DQM is the theory generally taught to scientists and engineers. In the DQM theory, electrons are quasi-independent particles that independently occupy (hydrogen like) ‘orbitals’ and have energies/spins and other properties associated with the orbital they occupy. They can move between orbitals without significantly disturbing electrons in other orbitals. Each electron in a multi- electron system has a particular energy, associated with the quantum level it occupies. The total energy of the electron system, say helium, is the sum of the energies of the individual electrons. The total spin of the system is the sum of the ‘spins’ of the individual electrons. Thus, for example, helium in the ground state has two identical electrons at approximately - 25 eV each. One important feature of DQM is that in the language of DQM the PEP actually has meaning. (In contrast, in the mathematical language of SQQM the PEP cannot be expressed in a meaningful manner. More below.) A second positive feature of DQM is that it permits a simple means of keeping track of angular momentum and thus can account for selection rules on the basis of conservation
Fig. 1 - Classic Energy Level Diagram for Helium. If both electrons in the ground state are not at
- 24.5 eV, what are their energies? (From Ref. 16)
A third aspect of DQM of relevance is that after/during ionization the remaining electron in He+ ‘falls’ to a lower energy state. The measured value of ionization for the electron in He+ is - 54.4 eV, a value in close agreement with the ionization calculated using Schrodinger’s equation for the ground state of a one electron atom with a two proton nucleus. Although it is not generally explicitly stated, in virtually any standard text (15-17), it is an inescapable implication of the model, and the only interpretation consistent with energy level diagrams (see Figure 1) and the PEP. Thus, the electron that remains after He is ionized ‘falls’, according to this theory, from approximately - 24.7, to
- 54.4 eV The fourth and final aspect of DQM of relevance is the inclusion in the theory of the Newtonian law of energy conservation. There is no ‘scale’ at which this general principle does not apply.
The reader, armed with the four ‘aspects’ of DQM discussed above can demonstrate to himself the fact that DQM cannot provide a satisfactory explanation for the simple process of photionization of helium. Thus, one concludes that the nice physical picture of the atom at the heart of DQM is not valid, and hence DQM is not a scientific theory. Specifically, the reader can show that DQM is not valid simply by consideration of the following questions. Remarkably, there is no need for math any more complex than basic algebra to disprove DQM.
- During/after ionization the electron that is not ionized falls in energy from - 24.5 to - 54.4 eV. Where do the nearly 30 eV lost by the remaining bound electron in this process go? Since chemical reactions are basically the movement of electrons between states, this conundrum can be readily written in the symbolism of chemical reactions:
He (-49.0 eV) + hν (+24.5 eV) He+(-54.4 eV) + e-^ (0 eV); ΔHrxn= - 29.9 eV (1)
Where the energy of the species, assuming the Pauli Exclusion Principle is correct, are provided. (Incidentally, the identification of the measured ionization energy, 24.5 eV for helium, and ‘energy level’ are universal for this form of quantum.) The nearly 30 eV unaccounted for is an enormous energy. In contrast, the re-organization of hydrogen and oxygen to make water produces about 1.2 eV /H atom, and this is associated with a lot of sensible heat. (One plank of the U.S. national energy plan is to capture this very significant energy directly as electricity.) Moreover, it is clear the 1.2 eV is simply the net released by electrons moving from one set of stable orbitals to a new set of orbitals of lower energy that become available for occupancy upon the formation of water molecules. In precisely the same fashion the 29.9 eV of energy released (according to DQM), per Equation 1, represents the net change in the stable energy levels of the electrons in helium during ionization.
- During electron attachment to He+ to re-create atomic helium, where does the energy come from to raise the electron energy from - 54.4 to - 24.5 eV? In fact, it is an objective scientific observation that during electron attachment to helium almost 24.5 eV of energy is released as a photon. The release of energy during the process of electron attachment to He+ is in fact required to maintain ‘microscopic reversibility’.
He+^ (-54.4 eV) + e-^ (0 eV) He (-49.4 eV) + hν (+24.7 eV); ΔHrxn= +29.7 eV (2)
- Why is electron attachment spontaneous if the total energy of two bound electrons in atomic helium is lower than that of ionized helium? The total energy of the electrons in atomic helium according to DQM theory (2*-24.5) is - 49.0 eV. In contrast, for ionized helium, the total energy of the free electron (0 eV) and the one bound electron is - 54. eV. Clearly ionized helium is in a lower energy state than helium. Accordingly, helium should spontaneously ionize.
He + hν(+25 eV) He +^ + e-^ + hν(+30eV) ; ΔH=+5eV (4)
And then the helium ion recaptures an electron:
He +^ + e- He + hν(+25eV); ΔH=+25 eV (5)
Adding 5 and 6 together yields no change in the system (helium in, and helium out!) except for the net release of nearly 30 eV of energy! Another of the ‘missed emission’ modifications of SQM is that the ionized electron and the remaining He+^ absorb the energy in ‘recoil’. However, as in the co-ordinate system of the original atom momentum must be conserved, the electron, being almost 8000 times lighter than the ion, will have virtually all of the released energy. Is it possible the physics community never noticed 30 eV electrons produced during photo ionization? In any event, the Perpetual Motion argument applies to this model as well. The ‘stored energy’ models all suffer from a lack of precision and quantification. Where is the energy stored? Is it in the form of mass? Which particle in the ion gains mass during photo ionization? Is this even possible given that the rest mass of fundamental particles is immutable? The conversion of mass to energy involved entire particles, not parts thereof. How is the mass reconverted to energy during electron attachment? If the energy is postulated to be stored in ‘fields’ the question is: What fields? Such a model must be quantified, or it simply has no credibility. Beyond the Literature: A Proposed Modification to DQM- It is possible to imagine a modified DQM that is consistent with both energy conservation and the Pauli Exclusion Principle. Warning I: This modified version is not found in the literature (e.g. 14-17). Warning II: the modified version is difficult to test. In respect to its consistency with an energy balance the modified model is superior, however the modified model arguably contains some metaphysics as it is not possible to compare the computed electron energy values by any direct process. In particular, for these models, the measured ionization energy and energy levels of the excited states, do not correspond to measured values of ionization energy or excitation. ‘Abridged’ /‘schematic’ energy levels for all of the possible new models of helium are illustrated in Figure 2. In all three cases to make the PEP and an energy balance work at the same time, we must introduce the concept of ‘boost’. This is postulated to be a process in which the energy released as the unionized electrons fall to there post- ionization levels, the so-called ‘relaxation energy’, help ‘boost’ the ionized electron out of the atom. There are three versions for helium. In the ‘NON INTERACTING’ model the two electrons do not interact, and the ‘unexcited’ electron remains at its initial energy level until ionization. The key features are i) both electrons are initially at ~-39.5 eV (agrees with total observed electron energy of - 79 eV), ii) None of the energy levels of excited states are the same as in the ‘Standard Model’, iii) spacings between energy levels are the same as in Figure 1, iv) no energy level corresponds to - 24.5 eV, v) there is a gap of about 15 eV between the highest excited state and the vacuum energy and vi) the
jump to the vacuum level occurs when the ‘unexcited’ electron falls to its final energy of nearly - 55 eV. In the ‘INITIAL DROP’ model, the electrons also do not interact, but the ‘unexcited’ electron falls to its final energy state under any and all excitation processes. Thus the first excitation process puts the energy of the non-ionizing electron ‘drop’, all the way to - 54.4 eV, into the excited/ionized electron. After the first step the non- ionizing electron remains at its ‘ionized’ He level though all subsequent excitation or ionization processes. In this model, the excited state energy levels of the ‘excited/ionized’ electron match those of the ‘STANDARD MODEL’ and there is no large energy gap between excited states and the vacuum. In the ‘STAGED DROP’ model, it is assumed that there is electron-electron interaction energy. The total energy of each electron is - 39.5 eV, but that energy is made up of negative energy of binding with the nucleus and some positive, repulsive, ‘bond’ energy coming from the interaction between the electrons. For example, there could be a (repulsive) total ‘bond’ energy of +10 eV, and each electron could be bonded to the nucleus with an energy of - 44.5 eV. As the two electrons are identical, it is rational to assign half of the ‘bond’ energy to each electron. Thus, the net energy of each electron is
- 39.5, as shown. (This assignment of half the bond energy to each electron is certainly consistent with standard quantum which always assumes the existence of a repulsive force, that is positive energy, between electrons, yet assigns specific energy levels to the electrons.) With each photon adsorption/excitation process there is a concomitant drop in the energy of the unexcited electron. Thus the ‘unexcited’ electron occupies an entire set of energy levels, one for each excited state. Other notable features of this model: All of the excited states are at lower energies than those given in the ‘Standard Model’, and all of the energy spacings between excited energy levels are larger than those of the standard DQM model. All of the features of all of the energy level models present in Figure 2, EXCEPT THOSE OF THE STANDARD MODEL, are consistent with the PEP, and an energy balance. That is, the energy required for ionization in the standard model is - 24.5 eV, exactly the observed input energy during the first ionization of helium. Thus, there is no accounting for the energy lost when the ‘unionized’ electron falls nearly 30 eV in energy following the first ionization. In contrast, the other models all account for the energy lost by the ‘unionized’ electron when it looses, in all the new models, about 15 eV. That energy, plus the input energy of 24.5 eV are required to boost an electron initially at - 39.5 eV to the vacuum level. The ‘Staged Drop’ model is unique in that the positive repulsive interaction energy is given up in a series of steps, however, all energy is accounted for in this model. Given present information it is impossible to tell which, if any, is correct. It is abundantly clear that no current quantum model of the energy levels in helium agrees with any of them: No current model starts with the two electrons in the ground state at - 39.5 eV, and none provide ANY information regarding the energy state of the ‘unexcited/unionized’ electron during the excitation process.
electron from - 24.7 to - 54.4 is not accounted for. Hence, there is approximately 30 eV of energy ‘lost to the universe’ during the ionization process as it is described in the Standard Model.
THE REAL DEAL: SQQM
SQQM, is the actual quantitative quantum theory developed by Heisenberg, Schrödinger, Bohr, etc. As applied to atoms and ions SQQM does produce a model of photo ionization of helium that is consistent with energy conservation, but it must ‘give up’ all the physical content of DQM, in fact all physical meaning, in order to do so. Note: SQQM is a far more sophisticated mathematical ‘theory’ than DQM and hence different arguments must be brought to bear against it than those employed against DQM. Thus, the arguments given below, particularly those regarding the inherent inconsistency of the Pauli Exclusion Principle and Energy Conservation, as per the title of the manuscript, are distinct from those employed in the DQM section. The core arguments against multi-electron SQQM are: i) the PEP is technically meaningless as applied to SQQM, ii) SQQM does not have physical meaning for multi- electron systems, iii) SQQM is mathematically inconsistent because it switches back and forth between phase space and real space, iv) SQQM is self-inconsistent because for one electron the wave function is a probability map in real space, whereas for multi-electron systems there is no real space map and no probability distribution (acceptable theories are consistent theories), and v) SQQM arbitrarily drops some elements of classical physics (Newton’s Laws, Maxwell’s equations, magnetic force), keeps others (electrostatic potentials) and invents others (correlation energy). In SQQM applied to multi-electron systems, there are no ‘individual electrons’. Indeed, there is no such quantity as ‘the energy of an electron’ or the ‘spin of an electron’ or the ‘angular momentum of an electron’, or even the ‘probability distribution of an electron’. In SQQM one develops a single ‘wave function’, from a single Hamiltonian, not a number of Hamiltonians, that ‘number’ being set equal to the number of electrons. That would be the number required to obtain wave functions for each ‘particular’ electron. The Hamiltonian is intended to express all the energies arising from forces. Surprisingly, it never includes a magnetic interaction between electrons. For two electron systems, the Hamiltonian is written in non-operator form ( 18 ):
H = p^1
2 2 m +^
p 22 2 m!^
Ze^2 r 1!^
Ze^2 r 2 +^
e^2 | r 1! r 2 |^ (6)
Where p is the momentum, Z the nuclear charge and m is the reduced mass. The generalization to many electron systems is clear. One adds an additional kinetic and nuclear electrostatic term for each electron, and the number of two electron repulsive terms is one less than the number of electrons. This leads to the generation of a complex wavefunction that is most compactly expressed in the form of a Slater determinant ( 18 ),
which automatically yields the anti- symmetric (independent of the labels on the electrons) form appropriate for Fermions (fractional spin particles). In particular for a two electron system the wave function is most easily expressed as a product-sum of ‘single particle’ wave functions and has this general form: ! = 12 [ " 1 ( A ) " 2 ( B ) ± " 1 ( B ) " 2 ( A )] (7)
Where the subscripts represent the different ‘single particle’ wave functions (φ), and the letters in parentheses are the electron identifiers. The above is actually multiplied by a ‘spin function’, and the nature of the spin function determines the identity of the +/-. A minus sign is correct if the spins are parallel and positive if they are anti- parallel, to insure total antisymmetry of the wavefunction. Note that mathematically all electrons are ‘indistiguishable’. That is, if the electrons are interchanged, the wave function is not changed. The mathematically indistinguishable nature of the electrons is presumed to reflect the ‘real’ world. In other words, the bound electrons of an atom are sort of ‘one big electron’, hence describable by one big Hamiltonian. Multi-Electron Physical Meaning- The above discussion raises a fundamental question: What is the ‘meaning’ of a multi-electron wave function in SQQM? The answer, NONE. That this answer is correct is found from a consideration of three issues: normalization (phase space), computation of electron-electron interaction (real space), and examination of the values of the wave function ‘mapped’ from phase space into real space. It is argued that once properly understood simultaneous consideration of these three issues shows that SQQM is not even a valid scientific theory, and the wave function for multi- electron systems has no physical meaning. Indeed, it is clear that even the mathematics employed in the first two standard processes are inconsistent. This violates one of the fundamental requirements of a scientific theory: it must be self consistent. Another violation of the requirement of consistency is the nature of the wave function. Single electron wave functions can be plotted in real space. For single electron wave functions each point in real space has a single value and that value has a clear (?) meaning: The probability that the electron will be at that point. As shown below, each point in real space for a multi-electron wave function has an infinite set of value. As a consequence, multi-electron wave functions cannot be graphed. The first issue is the method employed to ‘normalize’ the multi-electron wave function. Of particular concern is the use of 3N (N is the number of electrons) ‘phase space’ to make the value of this function integrated over all ‘space’ equal one. Standard practice (19) is to formulate the wave function from a set of orthonormal orbital wave functions. Examination of Eq. 7 shows that any wave function composed of orthonormal component functions, multiplied by its complex conjugate, and integrated over ‘phase space’, has a normalized value of one. Indeed, the value for the Slater determinant antisymmetric wave function (18) for any number of electrons is one. The reader can show that the wave function multiplied by its complex conjugate actually becomes the sum of four terms, each having four of the original one electron/hydrogen type wave functions. Two of these terms integrate to zero because the wave functions are orthonormal in both 3D phase spaces. Two of these terms integrate to one, because over both phase spaces the integrated values of the wave function and its
infinite (e.g. see discussions of electron self-energy (22)). There may be a mathematical rationalization for the final expressions, however, there is clearly no physical explanation. This is simply one more example supporting the earlier offered conclusion that there is no physical meaning associated with standard quantum theory as applied to multi-electron systems. How is the difficulty of the singularity in real space avoided? The approach generally taken is to propose that electron motion is correlated. For example, in a two electron system, if one electron is at Point R1, then the other electron has ‘correlated motion’ such that the probability that it is also at Point R1 is ‘low’ (zero?) and in fact it is ‘probably’ not in the general region of Point R1 at all. Those who insist that the multi-electron wave function is ‘physical’ indicate that the wave function is really defined by this correlation. The wave function is a measure of the probability of a particular geometry of electrons. Thus, for a two electron system the value of the wave function at any point in space (R1) is the probability that one electron is at R1 and the second electron is at R2. This can be a small value if R2 is in the vicinity of R1, or a relatively large value if R2 is far from R1. But, there is some value for every geometry. To continue: The wave function at R1 must also be a measure of the probability that one electron is at R1 and the second electron is at R3. The wave function at this point must also be a measure of the probability that one electron is at R1 and the second at R4, etc., etc., etc. Clearly, the consequence of accepting this interpretation of the wave function is that the wave function must have an infinite set of values at every point in real space. This is because if one electron is at this point, there are an infinite number of positions that the other electron can be found, some more probable than others. There must be a probability value for each of these possibilities associated with Point R1, hence there are an infinite number of values at point R1 in real space. This is not only ‘ungraphable’, it is not a physically plausible interpretation. Alternatively the two electron wave function can be described using a ‘phase space’ picture. For every position Electron 1 takes in its ‘phase space’, there is a matching probability distribution, encompassing every point in the ‘phase space’ of Electron 2. (Perhaps, it can be argued, every change in Electron 1 position changes the Hamiltonian of Electron 2 in its separate phase space.) Since Electron 1 can be an infinite number of positions, there are an infinite number of Electron 2 distributions in Electron 2 phase space. The real space description given before, in which each point in space has an infinite number of values, can be considered a 3D ‘mapping’ of 6 D phase space. What about the element lead, for which the electron ‘phase space’ has 246 dimensions? At a minimum, it is clear that the multi-electron wave function does not have a ‘physical meaning’ even remotely similar to that of the one electron wave function. SQQM is not a self-consistent theory. Is there any other possible physical meaning to the multi-electron wave function? The first difficulty is attempting any type of mapping of the fully anti-symmetric 3N dimensional wave function into three dimensions (see above discussion of normalization). But, if it were possible, what would be the meaning of the scaler value, at every point in real space? Could this value be assigned any meaning? It cannot be a probability. For example, if it were the sum of the probabilities of various electrons occupying a position in space, then the normalized value of the wave function would equal the total number of electrons, yet as discussed above it is always one. If it were the
product of probabilities, the normalized value would be less than one. Indeed, the normalized probability of any one electron occupying a particular position is less than one. Clearly, if the individual electron probabilities are normalized, the products of position occupancy will be lower than that of any individual electrons, hence the integrated over space (normalized) value of their products would be less than one. The above arguments lead to the following conclusion: The multi-electron wave function has no physical meaning. It is merely a mathematical construct to be employed for computing a very limited number of measurable values, as described below. It is interesting to note in this regard that even the most thorough discussions of the physical meaning of quantum theory only discuss the one electron case (13). The argument that wave functions relate to probability are plausible in the one electron case. It is not plausible in the multi-electron case. The above interpretation shows that in essence there is no information about single electrons available in multi-electron SQQM wave functions. Thus, the multi-electron wave function can be considered a description of a group ‘average’ behavior, or perhaps is better understood as the behavior of one big electron with a total charge equal to that of the number of electrons in the wave function. It is also clear that the single energy value that can be calculated from the multi-electron wave function is not that of a single electron. It is the sum of the energy of all the electrons. The average radius is that for all the electrons, etc. This non-intuitive nature of SQQM, the remarkable limitations of the theory are widely understood. Careful physicists understand that the energy values computed using SQQM are only total system values. Thus, for example, in the NIST tables the excitation energies of helium are listed relative to the ground state (23). They are not listed as absolute values, nor is a value provided for either electron independently. Also, in careful computations of the helium energy, no single electron values are provided for either the ground state electrons, nor the ‘excited electron’. Only a total value, a single energy, is provided for the system of electrons (24). SQQM, as applied to multi-electron systems, is not consistent with the notion of multiple energy states, one for each electron in the atom. The failure to provide energy levels of electrons in multi-electron atoms is not only non-intuitive, it is inconsistent with the correlation of one-electron SQQM computations with measured ionization energy for one electron systems (H, He+, Li++, etc.). It is clear that in SQQM that if one were to provide the electron energies for multi- electron systems, they could not be equal to the measured ionization energies, or the ‘perpetual motion’ conundrums expressed in Equations 1-5 would apply to SQQM just as they do for DQM. The term ‘precise energy level’ as applied to any electron in an SQQM world is meaningless. All energy level diagrams for electrons, molecules, etc. are meaningless in this version of quantum. And exactly where is phase space? This analysis of the ‘meaning’ of a multi-electron wave function leads to an additional puzzle regarding physical understanding of the Pauli Exclusion Principle. This puzzle can be expressed in terms of a three electron atom: If electrons are indistinguishable AND only two can occupy a particular orbital (with opposite and paired spins), which two? In other words, as applied to SQQM the PEP as generally understood is not meaningful. In SQQM there are no independent identifiable electrons, there is only one orbital, one energy, one average radius, for any number of electrons. It doesn’t make
