Atomic Emission Spectrum: StdQM vs RealQM

In textbook Standard Quantum Mechanics StdQM the emission spectrum of an atom (spontaneous or stimulated) of an atom consists of light of certain frequencies $f=\frac{E}{h}$ where $E=E_1-E_2$ is the difference in energies $E_1$ and $E_2$  between two different electronic quantum states given by two wave functions $\Psi_1$ and $\Psi_2$ forming a superposition $\Psi =\Psi_1+\Psi_2$ carrying the beat frequency $f$. Here $h$ is Planck's constant connecting frequency and energy by $E=hf$. 

The functions $\Psi_1$ and $\Psi_2$ carry no real physical meaning, while the emission of light is observable as physical reality, which creates a gap: 

• How can components without physical meaning form meaningful physics?    

This is the basic question of StdQM as theory about real physics. We know that an oscillating dipole  creates an electromagnetic wave/light of the frequency of the oscillation and so we expect to find the origin of the emission spectrum in oscillation of electronic charge density between states with different energy. 

For example, oscillation of the electron of the H atom between the ground state and the first excited state would give the first line in the H spectrum as radiation from a real oscillation of charge density. 

But this is not what StdQM delivers since the wave functions lack connection to real charge density and instead have meaning as probability densities, which cannot radiate.  This means that in StdQM the frequency of emission is connected to a beat frequency of two states in superposition and the oscillating dipole origin is lost. 

On the other hand, RealQM as directly based on non-overlapping charge densities carries the oscillating dipole origin of the emission spectrum with more clear connection to real physics than that of superposition in StdQM. 

The Unfortunate Start of Quantum Mechanics in 1926

Attempts to theoretically explain the observed emission spectrum of the Hydrogen atom were initiated as soon as the spectrum was observed starting with Ångström in 1853, followed by Balmer 1885 who discovered an algebraic formula 3 years later generalised by Rydberg into  

• $\frac{1}{\lambda}=R_H (\frac{1}{n_1^2}-\frac{1}{n_2^2})$    (R)

where $\lambda$ is wave length, $R_H$ is Rydberg's constant, and $1\le n_1<n_2$ are natural numbers.

The challenge was to find a mathematical model of the H atom which reproduced (R). Bohr in the 1910s came up with an ad hoc model in terms of classical mechanics, but the real breakthrough came in 1926 when the 38 year old Austrian physicist Erwin Schrödinger formulated an eigenvalue problem for a partial differential equation which he could solve analytically and so find to exactly agree with (R). This model  was coined Schrödinger's Equation SE and was formulated in terms of a wave function $\Psi (x)$ with $\Psi^2(x)$ representing electronic charge density and $x$ a spatial  Euclidean 3d coordinate. The success was complete and rocketed Schrödinger to fame.

At the same time the young 24 year old Werner Heisenberg from a different school of physics had developed another mathematical model as a new form of algebraic model named matrix mechanics with focus on what could be measured (the spectrum) rather than on underlying physics like Schrödinger. It turned out that the two models could be identified. But it was Schrödinger who insisted on a physically meaningful model, not only formality fitted to observation. 

Anyway, Heisenberg supported by his mentor Max Born took over the scene by developing a SE for systems with many electrons by a purely formal mathematical generalisation by adding a new 3d coordinate for each new electron. 

So was the foundation of modern physics as Standard Quantum Mechanics StdQM as a Schrödinger Equation SE in terms of a (complex-valued) wave function $\Psi (x)$ depending on a spatial coordinate $x$ which ranges over a configuration space with $3N$ dimensions (plus a time coordinate $t$), by Born given the following meaning to be named the Copenhagen Interpretation CI:

•  $\vert\Psi (x)\vert^2$ is a probability density of configurations $x\in\Re^{3N}$.

But a probability density does not represent any actuality of physical nature, only a possibility of physical nature. Since reality consists of actualities and not of possibilities, many physicists including Schrödinger and Einstein, did not find CI convincing. Later other interpretations were tried to give the wave function over configuration space physical meaning (Bohm, Many Worlds,...), but on the whole were less convincing.

The result is that modern physics still today is based on a mathematical model in the form of SE in wave function over configuration space, for which the physical meaning is lacking. This means that the message to students of modern physics from the highest authorities of theoretical physics including many Nobel Laureates today is something like:

• Do not worry/ask about physical meaning of solutions to SE. There is no answer.
• Accept that predictions about physics from solving SE always agree with experimental observation. 

Of course this is not a healthy situation and the result is a crisis of modern physics deepening with each Nobel Prize to StdQM. 

In any case, textbooks present StdQM and then in particular teaches about superposition as basic element of atom physics not present in classical physics, and recall Schrödinger's Cat which can both alive and dead at the same time, as a wave function $\Psi =\Psi_a + \Psi_d$ with $\Psi_a$ representing a live cat and $\Psi_d$ a dead with the + sign still to give a physical meaning.

The student will be told that the spectrum of an atom arises from superpositions of eigenstates with different eigenvalues which allows the superposition to express a beat frequency as a difference of eigenvalues. But the physics of the beat frequency will be unclear because the physics of the eigenstates are. 

The physical origin of an emission line in the spectrum must be a physical dipole generated by oscillating electron charge densities, but the connection to beat frequency in superposition lacks physics because  eigenstates lack physics.  

RealQM is an alternative to StdQM based on non-overlapping electron densities with direct physical presence which directly connects to a dipole origin of the spectrum. In short, RealQM is a reformulation of StdQM into a model with direct physical interpretation as non-overlapping charge densities. The clear physicality in the case $N=1$ is kept when going to $N>1$.

The basic problem with StdQM is lack of physics for $N>1$. RealQM comes with physics for $N>1$.

A restart from SE with $N=1$ appears to be necessary, and RealQM delivers an option. 

Comment by chatGPT:

Your post makes a passionate case for re-examining the foundations of quantum mechanics, particularly the move from Schrödinger’s single-electron wave equation to the high-dimensional configuration-space formalism used in standard quantum mechanics (StdQM). It’s true that the interpretation of the wave function — whether it *represents physical reality or only a probability distribution — has been debated since the early days of the theory, and this debate continues in the physics community. csc.kth.se+1

However, it’s important to separate formal mathematical success from interpretational issues. Standard quantum mechanics predicts experimental outcomes with extremely high precision across countless systems, from atomic spectra to quantum field theory and particle physics. Its use of a wave function on configuration space and probabilistic interpretation (as in the Born rule) may seem abstract, but these features are not just philosophical; they match observation. math.columbia.edu

Interpretational discomfort — like Schrödinger’s own unease with the Copenhagen Interpretation or Einstein’s critiques — doesn’t mean the theory is wrong. Many physicists acknowledge that interpretations (Copenhagen, Many-Worlds, Bohmian mechanics, etc.) differ yet agree on observable predictions. math.columbia.edu

If your RealQM approach can reproduce all known experimental results and make new, testable predictions, then the physics community would certainly be interested. The real challenge for any alternative formulation is not just providing a more intuitive picture, but showing it agrees with all precise experimental data where standard quantum mechanics excels.

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.

Computing Spectra of Nuclei

Quantum Electro Dynamics QED forms a low-energy version of the Standard Model SM within quantum electro-magnetics. QED theory agrees to high precision with  precise measurements of the (anomalous) magnetic moment of the electron using a Penning trap including a single electron. This device measures resonances of the electron in a way similar to measuring the spectrum of an atom, or resonance frequencies of a mechanical system, by subjecting the system to input of varying frequency and recording peaks in system output for certain frequencies showing resonance between input to and output from the system. 

The common understanding is that QED describes electrons of atoms and molecules, but not atomic nuclei composed of protons and neutrons asking for an extension of SM to Quantum Chromo Dynamics QCD including the strong force and the weak force of different nature than the Coulomb force of QED. 

The following question was asked 100 years ago during rapid development of quantum mechanics in 1920s with prospects to be all-encompassing: 

• Is it possible that QED can describe not only the electrons of atoms around nuclei with protons, but also also nuclei of atoms as systems of protons and electrons? 

The answer was negative: It is impossible for a nucleus to include an electron, because the nucleus is so small and squeezing an electron into that size would require energies of 100s of MeV, while the total energy of a nucleus per nucleon is around 8 MeV. 

RealNucleus takes up that old idea again from the new perspective of RealQM where charge densities of electrons and protons can meet with continuity, which allows electrons to be localised to the same extent as protons, thus allows electrons to hide inside a nucleus. RealNucleus thus offers a model of a nucleus as a system of non-overlapping charge densities of positive and negative sign interacting by Coulomb potentials. The model shows stability/existence of nuclei with in basic case $Z$ electrons forming a kernel surrounded by a shell system pf $2Z$ protons, with full quantum resolution of both electrons and protons, signifies by negative ground state energies in fair agreement with observation. 

The computational complexity of RealNucleus scales linearly with $Z$ which allows computation of full spectrum even for large $Z$, which is unthinkable with QCD. Results for RealNucleus under way...

Why Atomic Emission at Beat Frequencies Only?

An atom can emit radiation of frequency $\nu =E_2-E_1$ (with Planck's constant $h$ normalized to unity and allowing to replace energy by frequency) and $E_2>E_1$ are two frequencies as eigenvalues $E$ of a Hamiltonian $H$ with corresponding eigenfunction $\psi (x)$ depending on a space coordinate $x$ satisfying $H\psi =E\psi$ and corresponding wave function $\Psi (x,t)=\exp(iEt)\psi (x)$ satisfying Schrödingers wave equation

• $i\frac{\partial\Psi}{\partial t}+H\Psi =0$

and $t$ is a time variable.

Why is the emission spectrum generated by differences $E_2-E_1$ of frequencies of the Hamiltonian as "beat frequencies" and not the frequencies $E_2$ and $E_1$ themselves? Why does an atom interact/resonate with an electromagnetic field of beat frequency $E_2-E_1$, but not $E_2$ or $E_1$?

In particular, why is the ground state of smallest frequency stable by refusing electromagnetic resonance?  

This was the question confronting Bohr 1913 when trying to build a model of the atom in terms of classical mechanics terms. Bohr's answer was that "for some reason" only certain "electron orbits" with certain frequencies "are allowed" and that "for some reason" these electron orbits cannot resonate with an electromagnetic field, and then suggested that observed resonances at beat frequencies came from "electrons jumping between energy levels".  This was not convincing and prepared the revolution into quantum mechanics in 1926.

Real Quantum Mechanics realQM gives the following answer: The charge density $\vert\Psi (t,x)\vert^2=\psi^2(x)$ of a wave function $\Psi (x,t)=\exp(iEt)\psi (x)$ with $\psi (x)$ satisfying $H\psi =E\psi$, does not vary with time and as such does not radiate.

On the other hand the difference $\Psi =\Psi_2-\Psi_1$ between two wave functions $\Psi_1(x,t)=\exp(iE_1t)\psi_1(x)$ and $\Psi_2(x,t)=\exp(iE_2t)\psi_2(x)$ with $H\psi_1=E_1$ and
$H\psi_2=E_2\psi_2$, is a solution to Schrödinger's equation and can be written

• $\Psi (x,t)=\exp(iE_1t)(\exp(i(E_2-E_1)t)\psi_2(x)-\psi_1(x))$

with corresponding charge density

• $\vert\Psi (t,x)\vert^2 = \vert\exp(i(E_2-E_1)t)\psi_2(x)-\psi_1(x)\vert^2$

with a visible time variation in space scaling with $(E_2-E_1)$ and associated radiation of frequency $E_2-E_1$ as a beat frequency. 

Superposition of two eigenstates thus may radiate because the corresponding charge density varies in space with time, while pure eigenstates have charge densities which do not vary with time and thus do not radiate.

In realQM electrons are thought of as "clouds of charge" of density $\vert\Psi\vert^2$ with physical presence, which is not changing with time in pure eigenstates and thus does not radiate, while superpositions of eigenstates do vary with time and thus may radiate, because a charge oscillating at a certain frequency generates a electric field oscillating at the same frequency.

In standard quantum mechanics stdQM $\vert\Psi\vert^2$ is instead interpreted as probability of configuration of electrons as particles, which lacks physical meaning and as such does not appear to  allow an explanation of the non-radiation/resonance of pure eigenstates and radiation/resonance at beat frequencies. In stdQM electrons are nowhere and everywhere at the same time, and it is declared that speaking of electron (or charge) motion is nonsensical and then atom radiation remains as inexplicable as to Bohr in 1913.

So the revolution of classical mechanics into quantum mechanics driven by Bohr's question and unsuccessful answer, does not seem to present any real answer. Or does it?

PS I have already written about The Radiating Atom in a sequence of posts 1-11 with in particular 3: Resolution of Schrödinger's Enigma connecting to this post.