The Real Essence of Quantum Mechanics

Essential Real Elements of Schrödinger's Life as a Scientist.

1. Theory 

Quantum Mechanics QM grew out a need to explain observations that (i) an atom has a stable ground state without interaction with the environment and (ii) an atom can interact with light to exhibit an absorption/emission line spectrum. Next step was to explain molecules formed by atoms. There was no need to explain light since that was already done by Maxwell’s wave equations.

Since light is known to consist of electromagnetic waves of different frequencies $\nu$ and an atom is seen to interact with light, it is natural to seek an atomic wave equation for a function $\Psi (x,t)$ depending on a space variable $x$ and a time variable $t$ of the form of a harmonic oscillator (in non-dimensional form): 

• $i\frac{\partial\Psi}{\partial t} = H\Psi$   (S)

where $H$ is a Hamiltonian operator with a set of real-valued eigenfunctions $\psi_j(x)$ with eigenvalues $E_j$ satisfying $H\psi_j=E_j\psi$ where $E_1<E_2<E_3...$, forming the following representation:

• $\Psi (x,t) =\sum_{j\ge 1}\exp(-iE_jt)c_j\psi_j(x)$, 

with certain coefficients $c_j$. It is natural to associate $\vert\Psi (x,t)\vert^2$ with electronic charge density and $E_j$ with atomic energy. 

The charge density of the pure eigenstates $\exp(-iE_jt)\psi_j (x)$ including the ground state with $j=1$ is independent of time and so naturally can be seen as non-radiating states. 

Consider now a superposition of two eigenstates such as 

• $\exp(-iE_t)\psi_1(x)+\exp(-iE_2t)\psi_2(x)$
• $=\exp(-iE_1t)(\psi_1(x) + \exp( -i(E_2-E_1)t)\psi_2(x))$

for which the charge density is varying in time with the "beat frequency" $\Delta E=E_2-E_1$ as the difference of atomic energy between two eigenstates.  We thus see that superposition of two eigenstates generates a time varying charge density with frequency $\Delta E$ as difference in atomic energies. 

We know that an electric charge oscillating in space generates radiation/electromagnetic waves and it is natural to expect the same from oscillation in time with the frequency of the radiation set by the frequency of the oscillation. 

We can thus naturally connect the above superposition to radiation of frequency $\Delta E$ in interaction with light of the same frequency thus with  $\nu =\Delta E$, or $h\nu =\Delta E$ with $h$ Planck's constant defining space and time dimensions. 

Now, a prediction of atomic spectrum can thus be made from the eigenvalues of $H$ which can be compared with observation. For the Hydrogen atom with one electron Schrödinger formed by the Hamiltonian in non-dimensional form: 

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

with $\Delta$ the Laplacian, which gave very close agreement with observations. Schrödinger very happily concluded that he had created a mathematical model of the Hydrogen atom in a wave function representing charge density, and he was rocketed to fame. Notice that in this wave model there is no need to speak about energy quanta $h\nu$, only frequencies which can be observed, as macroscopic spectral lines.

What then about atoms with more than one electron? The standard procedure is to make a formal extension into multi-dimensional configuration space with a probabilistic non-physical interpretation of the wave function named Copenhagen Interpretation CI made into a canon by Bohr/Hesienberg/Born but never accepted by Schrödinger arguing that the CI interpretation of the wave function as a probability to find an electron as particle at a particular spot was void of meaning.

A different generalisation in physical terms is presented as Real Quantum Mechanics RealQM.  

Recall that Planck introduced energy quanta $h\nu$ to derive his law of black body radiation, which was then picked up by Einstein to (heuristically) explain the photoelectric effect, which lacking anything better gave him the Nobel Prize in Physics in 1921. 

RealQM and Computational Black Body Radiation show that energy quanta are not needed to explain these phenomena, and so loose their role and can be removed from the discussion, which brings a relief since nobody knows what an energy quanta is. In particular, the idea of explaining light as a stream of energy quanta or “photons” lacks physical basis.

CI comes with many problems which have never been resolved (see shocking review). RealQM offers a new start in the spirit of Schrödinger. 

The beauty of (S) for the Hydrogen atom is that it has clear physical meaning as an electronic cloud subject to Coulomb attraction from the kernel with spectrum in agreement with observation. Observe that electron cloud density itself is not observable, only the atomic spectrum as this is what reflects interaction with the environment, recalling that observation/measurement requires interaction.

Compare with the state of affairs as expressed by John Bell:

• Nobody knows what quantum mechanics says exactly about any situation, for nobody knows where the boundary really is between wavy quantum systems and the world of particular events.

But the boundary is clear as concerns atomic spectra.  Maybe RealQM opens to a resolution of the basic open question: The Measurement Problem. 

2. Theory vs Observation 

Let is now confront the eigenstates of (S) with observation of line spectrum for Hydrogen.

We observe a frequency of $2.469\times 10^{15}$ Hz corresponding to the ground state vs 1st excited state as lowest frequency in the Lyman (ultra-violet) series.   

We compute from (S) the energy level eigenvalues $-\frac{1}{2n^2}$ for $n=1,2,3,..$ with smallest $\Delta E = 3/8$ in Hartree or $10.2$ electronVolts eV. 

From the equation $\Delta E =h\nu$ with $\Delta E$ computed and $\nu$ observed, we can now compute Planck's constant $h$ to find the value given in physics books $h=4.136\times 10^{-15}$ eV.

We see that Planck's constant can be seen as a constant determined to make the model (S) fit with observation of the Hydrogen spectrum, thus as a form of model calibration (setting the relation between kinetic and potential energy) in (S). The wavelength of the lowest frequency in the Lyman series is $121.56701x10^{-9}$ meter which gives a connection to dimensional reality. 

Note the idea of energy quanta or photons $h\nu$ with some kind of physical realisation connects to the idea of phlogiston ("fire of the Earth") as carrier of energy in chemical reactions. The phlogiston theory was found to lack physical reality and so was abandoned before the end of the 18th century, while energy quanta has survived. Energy is a measure of the state of a system but is not a physical substance. 

(S) as a model of atoms and molecules does not need energy quanta, just continuum physics, which can help to demystify QM. It is the application of QM to light as a stream of photons which has brought the main mystery. It is time to let photons meet the same fate as phlogistons. 

Interaction between matter (atoms) and light can be modeled by QM for atoms and Maxwell's equations for light, and there seems to be no need to extend QM to light with all its complications. Yet this has become the objective of foundational quantum mechanics occupying the minds of philosophers of quantum mechanics or explorers of quantum computing.

3. Formality without physics  

Note that in the standard formulation of Schrödinger's equation in dimensional form the Laplacian $\Delta$ is multiplied with the factor $\frac{{\bar h}^2}{2m}$ with $\bar h =\frac{h}{2\pi}$ Planck's reduced constant and $m$ the mass of the electron. 

The appearance of the mass of electron here is strange, since it plays no role in the electro-magnetics of the Hydrogen atom captured by Schrödingers equation. It comes from a formal similarity to the kinetic energy $\frac{p^2}{2m}$ with $p=mv$ and $v$ velocity of classical mechanics, formally replacing $p$ by ${\bar h}^2\frac{\partial}{\partial x}$ without physics rationale. 

The energy associated with the Laplacian $\Delta$ is given by 

• $\int\frac{{\bar h}^2}{2m}\vert\nabla\psi\vert^2dx$, 

which motivated by the above formality is referred to as "kinetic energy". But this is a misnomer since kinetic refers to motion and here nothing is moving. Better would to refer this energy to a form of "elastic energy" or "compression energy" since it measures $\vert\nabla\psi\vert$.  

4. Physical size of Hydrogen atom ground state 

If we change the non-dimensional spatial coordinates $x$ in (1) into physical coordinates $\bar x=a_0x$, where $a_0=5.3\times 10^{-11}$ meter is the Bohr radius, then the Hamiltonian $H$ takes the following standard form in physical dimensions: 

• $\bar H = -\frac{{\bar h}^2}{2m}\bar\Delta - \frac{e^2}{4\pi\epsilon_0}\frac{1}{\vert\bar x\vert}$,

where $m$ here is (reduced) electron mass, $e$ electron charge and $\epsilon_0$ dielectric constant with $a_0=\frac{4\pi\epsilon_0h^2}{me^2}$. The Bohr radius gives the size of the electron cloud of the Hydrogen ground state in the range of 0.05 nanometers.   

5. Electron Mass?

The appearance of the electron mass in the coefficient $\frac{{\bar h}^2}{2m}$ of the Laplacian is strange since the electron mass $m$ is not part of the quantum physics of the Hydrogen atom building on electrostatic Coulomb attraction on the electron cloud balanced by the "compression force" from the Laplacian term. As said above the presence of the mass comes from a formal similarity to the kinetic energy $\frac{p^2}{2m}$ of classical mechanics. The value assigned to $m$ is $0.511$ MeV  based Einstein's formula $m=\frac{E}{c^2}$ translating energy to mass, to be compared with $10.2$ eV corresponding to the lowest frequency in the Lyman spectral sequence. The presence of the electron mass in the standard formulation of Schrödinger's equations (apparently) lacks rationale and the assigned large value of millions of eV appears to be ad hoc. 

In the 2019 redefinition the unit of mass 1 kg $\approx 8.98\times 10^{16}$ Joule as the energy of a collection of photons with frequencies summing to $1.356\times 10^{50}$, that is mass is defined in terms of energy, as a tribute to $E=mc^2$ showing Einstein's influence on modern physics. 

But mass is according to Newton's 2nd Law $m=\frac{F}{a}$ or $a=\frac{F}{m}$ a measure of resistance to motion with $F$ force and $a$ acceleration. This is inertial mass which is equal to gravitational mass. You discover the mass of your body by weighing it on a scale without any connection to energy. 

The $E=mc^2$ equivalence of energy is maybe Einstein's biggest mistake, and that is huge!