Experimental Test of E=mc2?

Let us seek experimental test for the cornerstone of modern physics Einstein's $E=mc^2$. We find on MIT News 2005: E=mc2 passes tough MIT test (as a celebration to Einstein's Annu Mirabilis 1905):

• MIT physicists report the most precise direct test yet of Einstein's most famous equation, E=mc2. And, yes, Einstein still rules. 
• The team found that the formula predicting that energy and mass are equivalent is correct to an incredible accuracy of better than one part in a million. That's 55 times more precise than the best previous test. Team member prof. Pritchard says:
• "In spite of widespread acceptance of this equation as gospel, we should remember that it is a theory. It can be trusted only to the extent that it is tested with experiments....If this equation were found to be even slightly incorrect, the impact would be enormous -- given the degree to which [it] is woven into the theoretical fabric of modern physics and everyday applications such as global positioning systems."
• (We meet here the common claim by physicists that GPS relies on relativity theory ($E=mc^2$), which is not the reason GPS works). 
• The mass loss was obtained at MIT by measuring the difference between the mass of the nucleus before the emission of a gamma ray and after. 
• Pritchard informs: "Determining the mass difference requires the individual masses to be measured with the incredible accuracy of one part in 100 billion -- equivalent to measuring the distance from Boston to Los Angeles to within the width of a human hair! This doesn't mean it has been proven to be completely correct. Future physicists will undoubtedly subject it to even more precise tests because more accurate checks imply that our theory of the world is in fact more and more complete."

Ok, detection of the mass loss required a measurement accuracy of one part in a billion (0.000000001) since the measurement was made on a single nucleus emitting a gamma ray. To measure mass defect it would seem to be better to involve a more easily measurable bulk of mass. To measure the mass/weight of a pile of sand, it would be senseless to measure the mass/weight of each grain of sand and then add up. It would be more sensible to seek to measure the mass defect in some nuclear reaction since it would require less precision, like the mass defect in an atomic bomb explosion...

In any case Pritchard admits that this does not suffice to be sure that $E=mc^2$ is exactly correct. And this is needed, because if $E=mc^2$ is not exactly correct, then the floor of modern physics collapses. If $E=mc^2$ is only approximately correct, then an abyss of questions opens: What is then missing? In my case, how much can I rely on $E=mc^2$? Which mass cannot be converted to energy and vice versa?  The suspicion from previous posts comes back again: Is $E=mc^2$ just and agreement, and as such always exactly correct, like the statement that there are 100 centimetres on a meter, for which experimental verification would be met with laugh.

A detailed inspection of many tests supporting the suspicion is given in Mass-energy equivalence not experimentally verified with punch line 

• It was not a verification of E = mc2, but just another experiment to deduce the mass of the neutron. To date, we have not measured the true mass of the neutron to any degree of accuracy; we only have a deduced estimate of the neutron mass based on the mass-energy equivalence of E = mc2.

We understand that the mass defect of the neutron, used as a proof of the correctness of $E=mc^2$, in fact is computed from $E=mc^2$ because the true mass of the neutron cannot be measured with enough accuracy. And so the mass defect is an agreement and not verified physical fact.  

Conservation of Mass/Energy vs Biggest Flaw of Modern Physics?

The previous post took a look at the proclaimed mass defect in chemical and nuclear reactions releasing energy: A mass defect or loss of mass exactly corresponding to the energy release is to be computed from $E=mc^2$. This showed a flavor of agreement/definition instead of actual real physics with mass defect in contradiction to conservation of mass (1) as a basic principle of physics. We also exhibited the origin of mass as reactivity to a gravitational potential. 

We may compare with conservation of energy (2) as the other basic principle of physics. Here energy is seen as potential to do work and can take the form of potential (e g chemical) energy or kinetic energy connected to motion. If now according to Einstein's $E=mc^2$  energy is "equivalent" to mass, then the two conservation laws (1) and (2) can be replaced by simply conservation of mass + energy (3), which is what Einstein had in mind, presumably: Mass can be converged to energy and vice versa, while the sum mass + energy remains constant. Fair enough. Clearly (3) follows from (1) + (2) and so cannot be disputed. 

But (1) and (2) do not follow from (3) in the presence of mass defect. If mass is really converted to energy,  then neither mass nor energy is conserved, only their sum mass+energy.

So here we stand. Two entities of different origin, mass as reactivity to gravitation, and energy as potential to do work, have been made “equivalent” as an expression of some deep modern physics expressing that mass+energy is conserved, but not both mass and energy separately. 

Is this a step forward to a deeper understanding of the Universe? This connects to the biggest flaw of modern physics which is to not offer a consistent theory including both gravitation and quantum mechanics, despite tremendous efforts by the sharpest minds over more than 100 years. Quantum Mechanics and General Relativity of Gravitation are Incompatible!! What can be the reason? After all, the Universe is built from the (a) quantum mechanics of atoms + (b) gravitation. How could (a) and (b) be incompatible? What would a Universe look like if being formed from incompatible physics? An Incompatible Universe?

Is it so that we if we insist that (a) and (b) are the same corresponding to energy being "equivalent" to mass according to $E=mc^2$, then we seem to be led into a dead end where everything is confused or "incompatible". It is like claiming than man = woman, a principle which is causing a lot of confusion in society. For sure there are shared aspects but if two different concepts are made "equivalent" by agreement/definition, then confusion is created. 

It may well be that if gravitation/mass is kept different from atoms/energy, then process can be made, while if confusion is allowed to reign, then no progress is possible. Ready to try?  Take a look at the listed labels on the blog: new quantum mechanics and extended Newtonian gravitation which are certainly compatible! 

In the next post I will investigate what concrete evidence there is that $E=mc^2$ is true physics, and not just an agreement. To prepare recall that Einstein somehow "derived" this relation in his Special Theory of Relativity to be a consequence of a postulate stating that all observes independent of inertial motion will have to measure the speed of light so that they get the same value named constancy of the speed of light. In other words, they have to use clocks and meter sticks to meet this end. More precisely, since 1983 all observers are demanded to use the SI Standard meter stick as the distance traveled by light over a certain length of time. The SI Standard thus commands all observers to agree on the same value of the speed of light: one light second/second = 1. 

But a command, or agreement if no observer objects, is just an agreement and as such is void of true physics. That the Earth is round is not an agreement, but a true physical fact. The agreement before was that the Earth is flat, and it is only recently that this agreement has evaporated. It would today be silly to say that Earth is round because we have agreed that it is. It is not agreement that makes the World go around. It goes around even if there is disagreement. But it is very difficult to get a modern physicist see the difference between agreement and physical fact, definition and theorem in mathematics.  

Modern physicists all agree that the Standard Model is correct, and so this is the way Nature is even if very strange...but understanding that real science boils down to showing that something is not so strange...

Here is an article leading into the next post: 103 years Later. Einstein Proven Correct. So it took modern physicist 103 years to come up with some real experimental evidence of $E=mc^2$. But the experiment is very tricky and so can be questioned. If $E=mc^2$ is indeed correct physics as the incarnation of modern physics, why has it been so difficult to verify experimentally? It would seem more likely that since it is so difficult to demonstrate, it cannot be true physics. Only an agreement that the Earth is flat, which you can argue is true in some sense agreed upon, but which is not really the whole story.  

Mass Defect vs Conservation of Mass

Here is the standard view connecting mass defect to $E=mc^2$.

Material bodies consist of matter formed by the elementary particles of protons and electrons, which do not ever decay and so express conservation of matter. The amount of matter is measured by the number of protons and electrons (and neutrons composed thereof). Matter is connected to gravitational mass manifested by free fall acceleration in a gravitational field scaling with the strength of the field, while being independent of the amount of matter/mass. As a result, all material bodies exhibit the same gravitational acceleration independent of composition. Conservation of matter translates to conservation of mass in the sense that mass does not change under gravitational free fall.

Assuming that inertial mass = gravitational mass, which is a cornerstone of both Newton's and Einstein's mechanics, it follows that all material bodies independent of composition react in the same way subject to gravitation or acceleration.

In particular, material bodies do not get ripped apart under gravitation, as the basic observation by Galileo. Further, a body in free gravitational fall will gain speed while loosing potential energy, while its mass will be conserved and not change. 

In chemical reactions the composition of molecules consisting of atoms consisting of protons and electrons change, but mass is conserved within measurement precision. So far so good. 

If we now bring in Einstein’s $E=mc^2$ as the golden apple of his Special Theory of Relativity SR, that is the idea that somehow mass $m$ is ”equivalent” to energy $E$ mediated by the factor $c^2$ with $c$ the speed of light, then things become strange. Einstein thus claims that in an exothermic chemical reaction releasing heat energy, there must be a corresponding mass defect with a loss of mass although the number of protons and electrons is the same after as before. However, this defect is too small to be measured, but in principle according to Einstein it is there, even if it cannot measured. This sounds like ghost science claiming ghosts to exist even if they cannot be detected.

But Einstein comes back in the case of nuclear reactions, claiming that the enormous release of energy in a nuclear explosion exactly corresponds to a mass defect according to $E=mc^2$, exactly. This is still under conservation of number of protons and electrons, since like in a chemical reaction only the composition (now of the nucleus) changes. And since the energy release is so big the mass defect is now measurable and then turns out to exactly match $E=mc^2$. Can anyone question this? In particular since the match is so precise! Exact!

Now the mass concept of SR is mysterious since it is supposed to change with velocity vs different observers using different reference frames moving with different velocities, and so be observer dependent, which appears to conflict with conservation of mass. If mass/matter is number of protons and electrons and this number does not change, then how can it be that different observers have different conceptions of mass? Are they not able to count?

This is a question which a child can pose, but it seems that physicists have no better answer than: Yes that is strange, but this is the way it is. Just accept it!

But it is easy for a suspicion to grow that in SR mass defect is computed from $E=mc^2$ to exactly fit with observed heat energy release. The mass defect would then not be the result of a measurement, which would be viewed to be superfluous, since anyway it is to be computed from $E=mc^2$. In other words, A = mass defect is claimed to be a consequence of B = $E=mc^2$. That is, B implies A or assuming B we obtain A. 

Now there is a lot confusion concerning the logic of implications, since often A implies B is viewed to be the same or at least a consequence of B implies A. But this is not correct logic. Nevertheless this incorrect logic is often used as argument to support that indeed $E=mc^2$ must be correct, since there is so much energy released in a nuclear explosion. But there is nothing in a nuclear reaction that has anything to do with the speed of light. 

Another thing to remember is that the mass-corrections from SR only become significant for speeds comparable to the speed of light. In the atom or the nucleus nothing is moving att such speeds and som the relevance of SR for atoms and nuclei is hard to motivate. You find as Many-Minds Relativity and analysis different from SR of measurements/perceptions of mass at velocities comparable to the speed of light. 

The net result is that conservation of mass and mass defect appear to be contradictory, and since there appears to be very good arguments for conservation of mass, we are led to conclude that mass defect is an illusion created by assuming $E=mc^2$ and computing mass defect from observing energy release. In particular, the argument that the existence of nuclear explosions proves $E=mc^2$, can be seriously questioned, even if modern physicists view it like undeniable evidence. 

Recall that $E=mc^2$ is derived from an assumption that all observers measure the same speed of light independent of inertial motion, which however is not an assumption (which can be wrong) but an agreement or definition (which cannot be wrong), since the unit of length (meter) is nowadays agreed to be measured in terms of the distance traveled by light over a certain length of time determined by the same atomic clock for all observers. This gives support to our  suspicion that $E=mc^2$ is also an agreement/definition from which mass defects are agreed to be computed in a certain way. The fact that computed mass defects exactly agree with $E=mc^2$ is a sign that we face an agreement/definition rather than physical fact. 

Nobel Physics Laureate: Quantum Mechanics is an Inconsistent Theory

Roger Penrose, who received the Nobel Prize in Physics 2020, makes the following statement in a recent  illuminating  discussion with Jordan B Peterson: 

• Quantum Mechanics is an Inconsistent Theory! 

In other words, the theory of quantum mechanics as expressed in terms of Schrödinger's equations, does not make sense as theory about physics of reality. Penrose here takes the same position as did Schrödinger himself. 

This is also the view I have arrived at after a fruitless search for the physical meaning of Schrödinger's equation. In this regard, quantum theory is similar to Einstein's special theory of relativity, which also lacks meaning in the sense of some form of real physics, a fact admitted by Einstein himself. 

The soul of modern physics is carried by the theories of quantum mechanics and relativity, and if these theories are empty of real physics, then this should pose a problem for modern physicists. But this does not seem to be the case. 

It connects to the tragedy of fluid mechanics by Nobel Chemistry Laureate Sir Cyril Hinshelwood captured as a split between 

• practical fluid mechanics (hydraulics) observing phenomena which cannot be explained, 
• theoretical fluid mechanics explaining phenomena which cannot be observed,  

thus a total collapse of science. 

In both cases scientists are helped out by the fact that the basic equations (Schrödinger's and Navier-Stokes equations) are uncomputable. Thus true comparison between theory and reality cannot be made and so it is possible to claim that true solutions to the equations always agree with reality, even if these solutions cannot be found. And so the spectacle can continue.

Finally Physics Laurate in Physics Speaks Out on Climate Alarmism

Climate alarmism can continue to bog down western societies into preindustrial state in a meaningless race to reduce CO2 emission as long as physicists do not speak out and tell that this has no physics rationale. Sadly to say, physicists have kept silent and so the destructive charade has been allowed to continue.

Now finally a Nobel Laureate, Prof John F. Clauser, Nobel Prize in Physics 2022, speaks out in unmistakably clear terms as reported by CO2 Coalition: 

• The popular narrative about climate change reflects a dangerous corruption of science that  threatens the world’s economy and the well-being of billions of people. 
• Misguided climate science has metastasized into massive shock-journalistic pseudoscience. In turn, the pseudoscience has become a scapegoat for a wide variety of other unrelated ills. 
• It has been promoted and extended by similarly misguided business marketing agents, politicians, journalists, government agencies, and environmentalists. 
• In my opinion, there is no real climate crisis. 
• There is, however, a very real problem with providing a decent standard of living to the world’s large population and an associated energy crisis. 
• The latter is being unnecessarily exacerbated by what, in my opinion, is incorrect climate science.

This should open the mouth of all physicists with some knowledge of basic physics, Nobel Laureates or not. Let us see this happen. In particular, the Royal Swedish Academy of Sciences handing out the Nobel Prize in Physics, now must retract its stated support to climate alarmism formed to serve a political agenda and return to science. I have reminded the Academy about its duty to serve science and not opportunistic politics and report the response.  

What is Equality in Physics?

In modern western society the concepts of equity or equality plays an important role. Equality can refer to equality of possibilities or equality of outcomes in a democratic system or equality of everybody in an Orwellian sense in a communistic society. Equity can refer to equality over different gender/sex and can connect to an idea of equality woman = man and man = woman.

We also find extreme forms of equality in 1984: War is Peace, Slavery is Freedom....

We understand that equality can express both a truth and serve as a disguise or cover-up of a truth. 

We also remind of the difference of the statement A=A expressing identity and B=A typically expressing that B inherits certain aspects of A (but not all). Often these aspects are not specified completely and the statement thus sometimes is open to confusion.  

The previous post discussed two different expressions of equality in physics: 

1. inertial mass = gravitational mass.
2. mass = energy, energy = mass 

Here 1. represents a truth resulting from viewing inertial mass to be an expression of (derived from) gravitational mass and thus the same. 

2. is Einstein's E=mc2, which is viewed to be the corner stone of modern physics. Since in physics energy is power to do work, it says that a body falling in a gravitational field doing work, such as the falling water in a Hydropower Plant, will loose energy (gravitational potential energy) and thus should loose mass according to 2. So there will be less water coming out than coming in to the plant, in contradiction to the physical principle of conservation of mass. A physicist would argue that the difference is too small to be noticed, but surely it is there, in principle. 

In any case a suspicious mind may wonder if that there is something fishy with the statement that energy = mass = energy (E=mc2). Ok, so more precisely Einstein does not say that mass and energy are equal, only that they are equivalent without specifying in what sense. But even so, as shown in the above example, a forced equivalence of mass and energy seems to lead into a strange contradiction.

It connects to Orwell's War = Peace which as identity is absurd, but as equivalence certainly has been used to justify a war by claiming that the aim is peace, that is Peace = War.  

We learn that equalities of the form B = A with B not identical equal to A, can (purposefully) be ambigious and need further specification to be meaningful. Seeking a specification of the meaning of energy = mass (E=mc2) may well lead to the conclusion that the equality does not make much sense. 

A contemporary physicist of course will claim that both chemical and nuclear exothermic reactions releasing energy suffers from a corresponding loss of mass according to E=mc2, which at least in the case of nuclear reactions is measurable and then shows to always exactly match E=mc2, nota bene if only properly measured. Any measurement not complying with a dogma of E=mc2 then must be corrected.

Phyisc is full of ambiguities as concerns equality. A first check is always to see if the units on both sides of the equality sign are equal. Sometimes this is also used to derive an equality by "dimensional" analysis.

New Newtonian Cosmology: Mass is Gravitational. Light is Not.

The basic postulate of New Newtonian Cosmology is that gravitational potential $\phi (x)$ is primordial from which mass density $\rho(x)$ is created through the Poisson differential equation:

• $\rho (x) =\Delta\phi (x)$                          (1)  

where $\Delta$ is the Laplacian differential operator acting on a Euclidean space coordinate $x$. The gravitational force $F(x)$ is equal to $-\nabla\phi (x)$ and scales with mass density $\rho (x)$:

• $F(x) = -\nabla\phi (x)\times \rho (x)$.    (2)

The action of $F(x)$ on $ \rho (x)$ can be viewed to be instant since both force and mass are created by the local operation of differentiation. 

In standard cosmology instead mass density$ $\rho (x) is assumed to be primordial from which the gravitational potential $\phi (x)$ is created as solution to the differential equation:

• $\Delta\phi (x) =/rho (x)$        (3)

corresponding to a summation/integration process which is global and so requires instant action at distance, which is a basic unsolved problem of cosmology.   

New Newtonian Cosmology thus offers a solution to a basic problem of standard cosmology. It also illuminates the nature of mass as basically gravitational mass "created" by (1), which is the origin of inertial mass as resistance to acceleration with necessarily 

• inertial mass = gravitational mass.     (4)

Einstein took (4) as the basic postulate/assumption of General Relativity picked from the air without need of justification as a deep secret of Nature beyond human comprehension.  In New Newtonian Cosmology (4) is a direct consequence of the nature of mass as created by a gravitational potential through (1) and thus ultimately of gravitational origin and nature. In other words, mass is gravitational mass and so any other expression of mass such as inertial mass, must be equal to gravitational mass. 

We also understand that what we view as matter is something having positive mass density with corresponding gravitational force appearing as weight. We thus see that the concepts of matter, mass and weight are basically the same as all being created by the same gravitational potential through (1).   

New Newtonian Cosmology based on (1) also may offer a solution to the mysterious dark matter as something with positive mass density, which does not emit/reflect light and so is invisible. The argument is that there is nothing in (1) that says that mass density must be visible. 

If now gravitational force acts only on positive mass density according to (2), it follows that something with zero mass density is not acted upon by gravitation. Or the other way around, if light has positive mass   

then light must be created by gravitation and so there can be no dark matter as something created by gravitation, which is invisible. We thus can argue that since there apparently is some dark matter out there in the Universe even if not detectable here on Earth, light must be massless. 

In other words, light cannot be affected by gravitation, since it does not have gravitational origin and so is massless. The apparent "gravitational lensing" of light thus must be caused by something else such as variable speed of light in a non-perfect vacuum (containing some form of "dust"). 

On the other hand, General Relativity states that light "bends" in a gravitational field just like the trajectory of any space object with mass. In other words, Einstein's energy = mass can be extended to also light = mass, and so everything is put into the pot as the same potato even if of fundamentally different nature. To make different concepts the same concept may seem as an advance but may also simply express confusion.

Notice that the idea that all mass is gravitational mass, is very different from the prevailing idea of modern physics expressed in the Standard Model, where mass is created by a force-carrying elementary particle of the Higgs field named Higgs boson granting mass to quarks and electrons, with unclear connection to gravitational potential.

So we have to views on mass and gravitation on cosmological scales: 

• New Newtonian Cosmology: Gravitational potential primordial giving mass to matter.
• Standard Model: Mass granted to matter by Higgs bosons as elementary particle. Gravitation a mystery.   

New Newtonian Cosmology connects to Mach's idea that the gravitational potential connects to the entire mass of the Universe, as a top-down process from large scale to small scale. The Standard Model offers a bottom-up process from extreme small scale to extreme large scale, which does not capture gravitation.

Which process would you vote for?  

New View on Strong Nuclear Force 2

RealQM model of Helium nucleus consisting of 4 protons held together by electron cloud of  2 electrons.

This is a continuation of the previous post. We consider a RealQM model of the nucleus of an atom consisting of Z protons (each with charge +1) and N neutrons with in the normal case N=Z. Since each neutron can be viewed as a proton + an electron (with charge -1) the RealQM nucleus model consists of 2Z protons and Z electrons. 

We compare with a model of an atom ion with kernel charge +2Z surrounded by Z electrons, thus of charge +Z. With the electrons (for simplicity) homogenised into a single charge density $\phi (x)$ around the kernel (then without internal repulsion), the corresponding Hamiltonian takes the following form (in atomic units of length) 

• $-\frac{1}{2}\Delta_x - \frac{2Z}{\vert x\vert}$     (1)

where $\Delta_x$ is the Laplacian differential operator acting on a 3d space variable $x$. Assuming spherical symmetry it follows from the corresponding 1d Schrödinger equation that the bulk of the charge density has a width scaling with $\frac{1}{Z}$. In other words, the charge $Z$ sets the physical scale of the electron charge in terms of atomic units of length from a balance of kinetic energy and potential energy. The total energy then scales with $Z^2$.

Let us now try a model for the nucleus with Hamiltonian 

• $H=-\frac{1}{2}\Delta_x - K(x)$                           (2)

where $K(x)$ is the sum of potentials of the form $ \frac{1}{\vert x -x_i\vert}$, where the $x_i$ for $i=1,..., 2Z$ represent the positions of the 2Z protons of the kernel, with again $H$ acting on a single electron charge density $\phi (x)$ (without internal repulsion). We can alternatively view (2) as model of a molecule ion consisting of 2Z protons held together by an electron cloud of charge Z acting like glue connecting the protons. We compare with (1) with an electron cloud of charge Z surrounding a single nucleon of charge 2Z. 

As a model of a nucleus (2) represents a physical scale which is a factor of at least 1000 smaller than when viewed as a model of a molecule ion, in which the former case the electron appears as being compressed "inside" the nucleus instead of being "outside" as in (1). 

We can now play with the model (2) and vary Z as an "effective charge" setting a scale where the size of the electron matches the distance between the protons so that it can act as glue keeping protons together even under proton-proton repulsion. This scale can be found by minimizing the total energy normalised by $Z^2$.  Here you can play case of Helium nucleus (2P+2N) and here with a (hypothetical) Beryllium nucleus (4P+4N). 

This analysis suggests that it may be possible to replace the strong force supposed to keep an atomic nucleus together, by the electromagnetic force between protons and electrons acting on a smaller nuclear scale. Quite a bit of simplification if true...

To sum up, we are led to explore the possibility that protons and electrons can interact on two different scales: 

1. Electrons on atomic/molecule scale surrounding smaller scale nuclei. 
2. Electrons on smaller nuclear scale "compressed" with protons as neutrons and then acting like a glue between protons just like in a molecular ion. (It is not clear if "compressed" electrons inside a nucleus are subject to repulsion).

The scale difference is around a factor 1000 with thus the energy dissociation of one electron increasing from eV for atoms to MeV for nuclei. 

Summary so far as concerns the composition of a nucleus (connecting to Platons ideal solids):

• Hydrogen: 1P
• Deuterium: 1P+1N, shape: 1d linear
• Tritium: 1P+2N, 2d triangle (compare with H3+ tritium cation stable)
• Helium: 2P+2N, 3d tetrahedron
• Lithium: 3P+3N, octahedron
• Beryllium: 4P+4N, cube
• ...
• Coal: 6P+6N, icosahedron 
• ...
• Neon 10P+10N, dodekahedron
• ...  

 

New View on the Strong Nuclear Force 1

Atomic nuclei consist of about the same number of protons and neutrons 

This is a continuation of a previous post on the possibility of viewing a Neutron consisting of a proton and electron as a smaller analog of a Hydrogen atom again consisting of a proton and electron. In particular it suggests that atomic nuclei consisting of nearly the same number of protons and neutrons is held together by a strong force which is an electromagnetic force acting on a smaller scale than the electromagnetic force keeping an atom/molecule together. 

To see the analog, consider the H2+ molecule consisting of two protons and one electron which acts like an glue keeping the two protons together into a molecule from the attractive force between the protons and the electron in between, which is stronger than the repulsive force between the protons and forms the H2+ molecule having a dissociation energy of about 1eV  at a distance of about 1Å. This is what the Schrödinger equation says. In atomic units the Schrödinger equation is parameter-free with unit length scale, and the physical dimension is set through Planck's constant and the mass/charge of the electron. You can view and play with the Schrödinger equation for H2+ here.

We next consider a nucleus consisting of a proton and a neutron, as the nucleus of deuterium as an isotop of hydrogen, thus two protons and one electron just as the H2+ molecule. The difference is the scale with the nucleus being much smaller than the molecule. It now appears to be conceivable that a nucleus is again held together by electromagnetic forces in the same way as the molecule, just on a smaller physical scale scale. In other words also the nucleus would be governed by the Schrödinger equation just on a smaller physical scale.  How much smaller physical scale?

 It follows from Schrödingers equation that energy scales like $d^{-2}$ with the dimension $d$. The decay energy of a neutron is around $10^6$ eV thus around $10^6$ times the dissociation energy of H2+ of about 1 eV. This indicates a scale factor of about 1000 between atomic and nuclear scale. 

The typical size of an atom is 1Å or $10^{-10}$ meter. The size of a nucleus is decided by shooting high energy alpha particles at the nucleus and measuring the point of closest approach to be about $10^{-14}$ meter, thus with a scale difference of 10.000. But it is possible that the effective size is bigger, and so the factor 1000 may capture reality.

We are thus led to the following

Question: Is the strong nuclear force in fact an electromagnetic force acting on small scale? 

Real Quantum Mechanics for Molecules

RealQM 3d H20 ground state (1-2d section)

Real Quantum Mechanics RealQM presents a new mathematical model of an atom as a system of one-electron wave functions with non-overlapping supports meeting at a free boundary with details presented on this website. RealQM boils down to a (parameter-free) classical continuum mechanics system of partial differential equations over a common domain in three space dimensions and as such is computable for atoms with any number of electrons. This is to be compared with the standard Schrödinger equation involving 3N space dimensions for a system with N electrons, which is uncomputable already for N>4.

RealQM has now been extended to molecules using a level set method on a fixed grid to capture the free boundary. An example computation of the ground state of a water molecule can be experienced here.

The road to complex molecules is hereby opened with further results to follow.