Hydrogen Spectrum as Classical Physics

This is a continuation of the previous post on the necessity to give up classical physics for quantum physics in its text book form of Standard Quantum Mechanics StdQM. We ask to what extent RealQM as a form of classical physics can explain the observed spectrum of the Hydrogen atom as expressed in stimulated radiation. We thus will compare

• StdQM: Spectral line appears from superposition of wave functions of eigenstates. 
• RealQM: Spectral line appears from oscillation between charge distributions of eigenstates.

In both cases the frequency of the spectral line scales with the difference of energy between eigenstates, but with different explanations: 

• StdQM: Spectral frequency appears as beat frequency of wave functions with eigenfrequency variation in time according to the Schrödinger's equation. Connection between frequency and energy is secondary. Radiation can appear to be spontaneous.
• RealQM: Spectral frequency is not beat frequency, but simply the frequency $\nu=\frac{\Delta E}{h}$ which matches the energy difference $\Delta E$ between eigenstates with $h$ Planck's constant in an assumed coupling between frequency and energy. There is an active exchange of energy between atom and radiation field with frequency matching the jump in energy. The atom is forced to respond to radiation of certain frequency as a dipole. The radiation is not spontaneous. 

We see that StdQM offers an explanation in terms of time-dependent quantum mechanics without realism, while RealQM relies on the formal coupling between matter and radiation expressed by $E=h\nu$ appearing in blackbody radiation. Compare with this post on the physical meaning of $E=h\nu$.

We see that, at least in the case of stimulated radiation, the spectrum of an atom can be given a RealQM semi-classical explanation. It is not clear that StdQM offers something more enlightening. Or?

This discussion connects to quantum computing discussed in recent posts with StdQM supposed to support delicate superposition of wave functions free of forcing performing complex analog computations, while RealQM brings forward the aspect of forcing in terms of classical physics. 

PS Here is a chatGPT comment.

RealQM vs StdQM and DFT for H2

A chemist's idea of a Hydrogen molecule H2 agreeing with RealQM.

Let us compare RealQM as a new model of quantum mechanics with StdQM as the accepted model, for the formation of a Hydrogen molecule H2 from two Hydrogen atoms H each with one electron around a proton kernel approaching each other to find a minimum of total energy $E$ as the sum of kernel-kernel repulsion potential energy $K$, electron-kernel attractive potential energy $EK$, electron-electron repulsive potential energy $EE$ and electron kinetic energy $EKIN$. 

We first note that RealQM and StdQM are different mathematical models of an atom/molecule as a collection of atomic kernels and electrons sharing basic assumptions:

• A1 Electrons/kernels interact with electrons/ kernels by Coulomb potentials to form $EK$ and $EE$. 
• A2 Electrons contribute $EKIN$ as a measure of spatial variation.                                      

RealQM has the form of classical deterministic continuum physics as a free-boundary problem for a system of non-overlapping one-electron densities: A non-linear system of partial differential equations in 3 space dimensions of classical form with direct physical meaning as ontology. No mystery. Low cost computation allowing many electrons. A new assumption is introduces as a Bernoulli free boundary with

• A3 Continuity of electron density across a common boundary combined with vanishing normal derivative on each side of the boundary.                                                                               

StdQM has the form of a Schrödinger equation as a linear multi-dimensional partial differential equation in wave-function depending on $3N$ spatial dimensions for $N$ electrons with statistical meaning as epistemology as the essential novelty of modern physics. Lots of mystery coming from the interpretation of the square of the wave function as probability of electrons-as-particles configurations. Computational cost prohibitive for many electrons.

Although sharing the physics of A1+A2, RealQM and StdQM have very different mathematical form: 

• RealQM is classical non-linear continuum physics including A3 with direct physical meaning. 
• StdQM is modern/new linear multi-dimensional physics with only statistical meaning
• RealQM describes the atomic world as classical deterministic physics. 
• StdQM describes the atomic world as particles playing roulette.

RealQM describes H2 in terms of a common wave function depending on a 3d spatial varaiable $x$

• $\Psi (x)=\Psi_1(x) + \Psi_2(x)$   

as the sum of a one-electron wave function $\Psi_1(x)$ associated with proton 1 and a similar $\Psi_2(x)$ associated with proton 2, with non-overlapping supports thus dividing 3d space meeting at a Bernoulli free boundary as a plane midway between the kernels orthogonal to the line between the kernels. The electron charge densities are given by $\Psi_i(x)^2$ with total charge $\int\Psi_i(x)^2dx =1$ for $i=1,2$. The electron density $\rho (x)$ is given by 

• $\rho (x) =\Psi_1(x)^2 +\Psi_2(x)^2$.

The energies are with $D$ the distance between the protons located at $X1$ and $X2$ read:

• $K = \frac{1}{D}$
• $EK =-\sum_{i,j=1,2}\int \frac{\Psi_i(x)^2}{\vert x-Xj\vert }dx$
• $EE=\int\int\frac{\Psi_1(x)^2\Psi_2(y)^2}{\vert x-y\vert }dxdy$        (1)
• $EKIN=\frac{1}{2}\sum_{i=1,2}\vert\nabla\Psi_i(x)\vert^2dx$. 

Minimum energy is reached through a gradient method producing the functions $\Psi_1(x)$ and $\Psi_2(x)$ from some rough initial charge distributions, see this code. We see that $EK$ and $EKIN$ can be expressed in terms of a common electron density $\rho$, while $EE$ depends on the spatial partition of $\rho$ into $\Psi_1$ and $\Psi_2$. 

Minimal total energy $E=-1.17$ is reached for $D\approx 1.4$ (atomic units). The physics of RealQM modulo $EK$ consists of attractive and repulsive Coulomb potentials as classical physics, while $EK$ is new quantum physics giving an electron extension in space as a form of wave or rather extended charge density, and not particle without spatial extension. 

Setting the kernel distance to zero and eliminating $K$ we get a Helium atom with the two electrons separated into two half spaces meeting at a plane through the double kernel with a Bernoulli free boundary condition as continuity and zero normal derivative. RealQM gives $E=-2.903$ in accordance with observation (giving the proton a small positive diameter as a free parameter) see this code.

StdQM describes H2 in terms of a new form of wave function $\psi (x,y)$ depending on two 3d space variables $x$ and $y$ altogether 6d, typically of anti-symmetric Slater determinant form satisfying the Pauli Exclusion Principle:

• $\psi (x,y)=\frac{1}{\sqrt{2}}(\psi_1(x)\psi_2(y)-\psi_1(y)\psi_2(x))$

with both $\psi_1(x)$ and $\psi_2(y)$ having global support over respective 3d space with  $\int\psi_i(x)^2dx =\int\psi_2(y)^2dy =1$. All energies come out the same, except $EE$ which takes the following different form with the new contribution $EC$ named exchange-interaction energy or electron-correlation energy:

• $EE= \int\int\frac{\psi_1(x)^2\psi_2(y)^2}{\vert x-y\vert }dxdy+EC$        (2)
• $EC=-\int\int\frac{\psi_1(x)\psi_2(x)\psi_1(y)\psi_2(y)}{\vert x-y\vert }dxdy$   (3)

$EC$ is viewed as a new genuinely quantum mechanical (mysterious) effect offered by StdQM not present in classical physics of attractive and repulsive Coulomb potentials forming RealQM. We see that $EC$ decreases the electron repulsion from overlapping support. Let us now compare.

RealQM:

• The one-electron wave functions $\Psi_1(x)$ and $\Psi_2(x)$ depend on the same 3d space variable $x$ and have disjoint supports. Electron densities do not overlap and meet at a Bernoulli free boundary in this case a plane orthogonal/midway to the axis between the kernels. Pauli Exclusion Principle trivially satisfied. 
• Electron-electron repulsive potential energy is given by (1).
• Total energy minimisation corresponds to classical problem of continuum mechanics in 3D with computational cost scaling with $h^{-2}$ with $h$ spatial resolution (not Planck's constant). 

StdQM:

• The total wave function $\psi (x,y)$ depends on two 3d space coordinates $x$ and $y$.
• Electron-electron energy is given by (2) with electron-correlation energy correction by (3). 
• Minimization is performed over some variation of $\psi_1(x)$ and $\psi_2(y)$ depending on altogether 6 spatial dimensions with high computational cost.
• Results in agreement with observations can be reached under sufficient variability of wave functions by trial and error.

We understand that the electron-correlation energy is zero for RealQM, because $\Psi_1(x)$ and $\Psi_2(x)$ have disjoint supports so that $\Psi_1(x)\Psi_2(x)=0$ for all $x$ . 

We see that $EC$ can be seen as a negative correction to electron-electron repulsion potential energy $EE$ compensating overlap of supports of $\psi_1$ and $\psi_2$. 

For Helium StdQM gives $E=-2.85$ with $EC$ and $E=-2.75$ without choosing $psi_1$ and $psi_2$ to be two overlapping Hydrogen wave functions, with more complex functions (e g Hylleraas) needed to reach $E=-2.903$.  

RealQM can be seen as a very special form of Density Functional Theory DFT with one-electron wave functions with disjoint support directly expressing electron density satisfying Pauli Exclusion Principle. 

But standard DFT is rather seen as a reduced variant of StDQM with overlapping one-electron wave functions, and so meets a difficulty transforming the electron-correlation into dependence on common density. RealQM can be seen as a "localised" version of DFT with electron-correlation of obvious form with vague connection to ”partition density functional theory”.

Let us sum up:

• StdQM depends on 6 spatial dimensions and introduces new physics in the form of the exchange-interaction term $EC$. Electrons do not have individuality and are both everywhere and nowhere.  
• RealQM depends on 3 spatial dimensions and does not require new physics beyond the kinetic energy $EKIN$ eliminating possibility of electron particle nature. Electrons have individuality by occupying specific regions in space in accordance with chemist's idea of H2, see picture above. 
• StdQM needs anti-symmetric wave functions for electrons of same spin to satisfy Pauli Exclusion Principle, which is automatically satisfied by RealQM. 
• RealQM gives efficient computation in 3d. 
• STdQM requires heavy computation in 6d.
• Both RealQM and StdQM can give correct total energy. 
• RealQM can be seen as form of DFT without its difficulties in standard form. 
• RealQM shares aspects of StdQM and DFT, but has a distinct new quality of non-overlapping one-electron densities meeting at a Bernoullli free boundary. 
• RealQM has the form of classical continuum physics in 3d with free boundary.  
• The computational cost increases polynomially with number of electrons for RealQM, and exponentially for StdQM.
• Altogether RealQM appears to give more for the money with less new physics than StdQM and DFT.
• It is reasonable to expect that RealQM can reach an audience, but tradition going 100 years back is strong. Thousands of scientists have contributed to StdQM/DFT, only one to RealQM so far...

What is the Difference between a Hydrogen Atom and a Neutron?

This is a follow up of the previous post on Real Quantum Mechanics applied to atomic kernels.

A Hydrogen atom is composed of a small positive proton kernel and a surrounding large negative electron charge density cloud held together by Coulombic attraction. The binding energy is 13.6 eV. 

A neutron decays into a proton and an electron (and an antineutrino) releasing 0.78 MeV based on the rest masses of the neutron, proton and electron. We can thus view a neutron to be composed of a proton and an electron with a binding energy of 0.78 MeV,  thus with the same components as a Hydrogen atom with a binding energy of 13.6 eV, with a scale factor of about $10^5$.

Thus the same components but vastly different energies, how come? The neutron must be composed in a different way from a Hydrogen atom. The only possibility is to switch the roles between proton and electron and view a neutron to be composed of a very small electron kernel surrounded by a small proton cloud. 

A Hydrogen atom and a neutron will then be described by the same Schrödinger equation, with only a change of spatial scale with some factor $S$, and then with ground state energies also scaling with $S$.  

With an energy scale factor of $S=10^5$, we would thus expect a neutron to be about $10^5$ times smaller than a Hydrogen atom, which is confirmed by observation. 

We thus find experimental support to an idea of viewing a neutron to be composed of a very small electron kernel surrounded by a small proton cloud as an explanation of its very large binding energy compared to a Hydrogen atom. 

Nucleosynthesis into heavier elements would then start by transformation of Hydrogen=proton+electron into neutron=electron+proton under very high pressure and temperature, followed by proton+neutron synthesis. Synthesis of proton+proton into 2proton would then not be needed, and in fact is not observed. But electron+electron into 2electron seems to be needed.

Hydrogen as Two-Density Schrödinger Equation

This is a continuation of the previous post with a Hydrogen Atom modeled according to Real Quantum Mechanics RealQM  in terms of two spatial charge densities, $\phi (x)$ for the proton $\psi (x)$ for the electron as functions of a Euclidean space coordinate $x$, assuming $\phi$ and $\psi$ have non-overlapping supports filling space meeting at a common boundary $\Gamma$ with some boundary conditions to be specified, starting for simplicity with zero charge density for both proton and electron. 

We start with the ground state with the proton occupying a fixed sphere of diameter $d$ with the electron filling the exterior volume. We characterise the ground state as the state of minimal total energy 

• $E(\phi ,\psi ) = PE(\phi ,\psi ) + KE(\phi ,\psi)$

where

• $PE(\phi ,\psi ) = -\int\frac{\phi^2(x)\psi^2(y)}{\vert x-y\vert}dxdy$

 is mutual potential energy, and  

• $KE(\phi ,\psi )=\int\frac{d^2}{2}\vert\nabla\phi (x)\vert^2dx+\int\frac{1}{2}\vert\nabla\psi (x)\vert^2dx$

is the sum of proton and electron compression energies (also referred to as "kinetic energies") under the normalisation 

• $\int \phi^2(x)dx =1$ and $\int \psi^2(x)dx =1$.

Here the coefficient $d$ sets the size of the proton compared to the electron of unit size and we assume that $d$ is small. 

The proton charge density is given by a spherical harmonic as a "blob" of diameter $d$ centered at $x=0$, while for small $d$ the electron charge distribution is close to the standard Hydrogen ground state with $\psi (x)\sim\exp(-\vert x\vert )$. The total energy comes out as the standard electron compression/potential energy $-\frac{1}{2}$ plus proton compression energy as a constant independent of $d$. Letting $d$ tend to zero and neglecting the proton energy makes the proton into a point source as in the Born-Oppenheimer approximation in terms of only electron density, compare PS below.  

Eigenstates of higher energies emerge as stationary points of $E(\phi ,\psi )$ in a variational setting. 

We see that for small $d$ the two-density model thus reduces to the standard electronic wave function with a constant shift of total energy, which does not affect the spectrum since it corresponds to energy differences. Thus the two-density model for Hydrogen may be seen as a trivial extension of the electronic one-density model, but allowing $\Gamma$ to be a free boundary included in the variational setting may open new views on the interaction between proton and electron. One can then ask if the presence of the electron around the proton affects the proton density, as well as ponder different boundary conditions. 

We note that in RealQM the combined density comes out as the sum of proton and electron densities in 3-dimension physical space, while in standard QM it is the product in 6-dimensional configuration space, which is not physical. 

Altogether, we find that RealQM naturally can be extended beyond electronic interaction. One can then address the question why the proton appears to be so much smaller than the electron in e.g. the Hydrogen atom. It reflects that the proton has a much smaller "resistance to compression" than the electron, which can be accepted as a physical fact asking for deeper analysis.   

Note that it is more natural to connect the compression/kinetic energies to spatial size rather than mass, since the quantum mechanical model concerns electromagnetic interaction without effects of inertia/gravitation. More precisely, the coefficient $\frac{d^2}{2}$ in RealQM corresponds to $\frac{1}{2m}$ in stdQM with $m$ the proton inertial/gravitational mass, which is strange since standard QM primarily concerns electromagnetics. Only in molecular modeling using the Born-Oppenheimer approximation with kernel dynamics treated by classical mechanics, does kernel masses enter. In any case, $d$ appears to scale with $\sqrt{m}$ which with the table value $m=1836$ gives $d\approx 0.02$ to be compared with 1 as atom size. 

Returning to the idea of a neutron as an "inverted Hydrogen atom" with the electron at the center surrounded by a proton of size $d$ will give a large increase of electron compression energy which can be released when the neutron decays observed to be around 1 MeV, which suggests an electron size of $10^{-3}$ which may again suggest a proton size $d\approx 0.02$  

Note that we here speak about "electromagnetic" size, which may be different from a smaller inertial/gravitational size as measured in collision experiments. 

PS1 The article On the hydrogen atom beyond the Born–Oppenheimer approximation considers a two-density model in the spirit of stdQM with a combined wave function as a product of proton and electron densities. Model computations suggest that in RealQM one can assume both proton and electron densities to vanish on the common boundary. 

PS2 The two-density model in the above form contains one parameter $d$ which connects proton mass to electron charge/mass with a direct coupling to the non-dimensional fine structure constant $\alpha\approx\frac{1}{137}$ as expressed here.  

Neutron as Inverted Hydrogen Atom?

Is this a proton charge density surrounded by an electron charge density. Or is it the other way around? 

The Hydrogen atom consisting of a positively charged proton and a negatively charged electron can in Real Quantum Mechanics RealQM  be mathematically modeled in terms of two spatial charge densities, $\phi (x)$ for the proton $\psi (x)$ for the electron as functions of a Euclidean space coordinate $x$, assuming $\phi$ and $\psi$ have disjoint supports (filling space) meeting at a boundary $\Gamma$ signifying that the proton and the electron do not overlap. 

The ground state of Hydrogen is then characterised as the state of minimal total energy 

• $E(\phi ,\psi ) = PE(\phi ,\psi ) + KE(\phi ,\psi)$

where

• $PE(\phi ,\psi ) = -\int\frac{\phi^2(x)\psi^2(y)}{\vert x-y\vert}dxdy$

 is mutual potential energy, and  

• $KE(\phi ,\psi )=\int\frac{1}{2m}\vert\nabla\phi (x)\vert^2dx+\int\frac{1}{2}\vert\nabla\psi (x)\vert^2dx$

 is the sum of proton and electron compression energies under the normalisation 

• $\int \phi^2(x)dx =1$ and $\int \psi^2(x)dx =1$.

Here $m\approx 1836$ is the ratio of proton to electron mass. Eigenstates of higher energies emerge as stationary points of $E(\phi ,\psi )$. Further, $\Gamma$ is a free boundary included in the minimisation with specific boundary conditions to be decided. 

A proton-electron configuration which agrees with observations is given by a proton charge density of small radius centered at $x=0$ surrounded by an electron charge density of large radius. In the limit with the proton modeled as a constant charge distribution of vanishing radius, this gives the standard Schrödinger equation for the Hydrogen atom with Hamiltonian

• $H = -\frac{1}{2}\Delta -\frac{1}{\vert x\vert}$

in terms of the electron charge distribution $\psi (x)$ alone, with $\psi (x)\sim \exp(-\vert x\vert)$ as ground state.  

Now, a neutron is viewed to also consist of a proton and an electron, and so it is natural to ask if the above model can also describe a neutron? That would correspond to a switch of roles with now the electron at the center surrounded by a proton charge density. The compression energy would now be that of the proton resulting in a change of scale with the neutron radius about $\frac{1}{1836}$ of that of a Hydrogen atom.  

In RealQM the size of an electron, in an atom with electrons organised into shells, increases with distance to the kernel, and so electron size is variable. We may expect the same property of a proton with thus increasing size if harbouring an electron inside in the formation of a neutron. The size of a free proton  is estimated to about $10^{-15}$ meter. We compare with a Hydrogen atom of size $5\times 10^{-11}$  which with the above 1836 scaling, gives a proton size of about $10\times 10^{-15}$ when surrounding an electron in a neutron, about 10 times as big as when free.

These are speculations suggested by RealQM as a classical continuum model in terms of non-overlapping charge densities. RealQM can be seen as a form density functional theory which is different from that pioneered by Walter Kohn and Pierre Hohenberg (Nobel Prize in Physics 1998) formed by averaging in a standard multi-dimensional Schrödinger equation. 

Recall that a free neutron is unstable and decays with mean lifetime of 14 minutes into a proton, an electron and an antineutrino (but not a Hydrogen atom), while neutrons are formed in the fusion process of Hydrogen into Helium in a star like the Sun.