## tisdag 27 januari 2015

### Physical Quantum Mechanics 7: Towards Many-Electron Atoms

The extension of the second order Schrödinger equation considered in previous posts to an atom with $N$ electrons may tentatively take the form:

Find $N$ (normalized) wave functions $\psi_1(x,t),\psi_2(x,t),...,\psi_N(x,t)$ depending on a 3d space coordinate $x$ and time coordinate $t$ with non-overlapping spatial supports $\Omega_1(t),\Omega_2(t),...\Omega_N(t)$ filling 3d space, such that the combined wave function $\psi (x,t)$ defined by $\psi (x,t)=\psi_j(x,t)$  for $x\in\Omega_j(t)$ for $j=1,...,N,$ satisfies
• $\ddot\psi (x,t)+H^2\psi(x,t) =0$ for all $(x,t)$,                       (1)
where the (normalized) Hamiltonian $H=H(x)$ is given by
• $H(x) = -\frac{1}{2}\Delta - \frac{N}{\vert x\vert}+\sum_{k\neq j}\int\frac{\psi_k^2(y,t)}{2\vert x-y\vert}dy$, for $x\in\Omega_j(t)$.
The combined wave function $\psi (x,t)$ is thus assumed to be continuous, more precisely so smooth that $H^2\psi (x,t)$ is defined, and the potential acting in $\Omega_j(t)$ is given as the attractive kernel potential together with the repulsive kernel potential resulting from the combined electronic charge distributions $\psi_k^2$ for $k\neq j$.

The time-independent ground state $\psi (x)$ satisfying $H\psi =E\psi$ with minimal energy $E$ is then determined as the minimizer of the total energy functional
• $\frac{1}{2}\int\vert\nabla\psi\vert^2 dx-\int\frac{N\psi^2}{\vert x\vert}dx+\sum_{k\neq j}\int_{\Omega_j}\int_{\Omega_k}\frac{\psi_j^2(x_j)\psi_k^2(x_k)}{2\vert x_j-x_k\vert}dx_jdx_k$.
This is a free-boundary electron (or charge) density formulation keeping the individuality of the electrons, which can be viewed as a "smoothed $N$-particle problem" of interacting non-overlapping "electron clouds" under Laplacian smoothing. The model (1) connects to the study in Quantum Contradictions showing a surprisingly good agreement with observations.

For example, the ground state of Helium in this model consists of two disjoint (contacting) half-spherical electron clouds with energy in close agreement with observation, see Quantum Contradictions 12, and PS2 below.

This is to be compared with the postulated ground state according to the standard Schrödinger equation considered to be two overlaying spherical electron configurations named as $1s2$, which however has wrong energy and thus is not the true ground state, and thus asks for perturbative correction.

PS1 In the series Quantum Contradictions 1-29 we considered a model closely related to (1) with each one-electron wave function $\psi_j$ a smooth function defined in 3d space satisfying a one-electron wave equation with the Laplacian smoothing acting on the sum of the wave functions,  and not on each individual wave function, with the effect of reducing contributions to the smoothing energy from angular variation. In this case the one-electron wave functions have overlapping support, while being largely separated by repulsion and loosing individuality under overlapping Laplacian smoothing.

PS2  Computing an approximation of the ground-state energy $E$ of Helium using the two normalized separated half-spherical wave functions
• $\psi_1(x)=\sqrt{\frac{2\alpha^3}{\pi}}\exp(-\alpha\vert x\vert )$ for $x=(x_1,x_2,x_3)$ with $x_3\ge 0$
• $\psi_2(x)=\sqrt{\frac{2\alpha^3}{\pi}}\exp(-\alpha\vert x\vert )$ for $x$ with $x_3\le 0$,
we find with $\alpha =27/16$ the following value:
• $E=-2.902\pm 0.002$,
to be compared with the observed $-2.903$. The standard 1s2 ground state of two overlaying fully spherical wave functions gives a best value of -2.85 asking for so called perturbation correction, which effectively introduces electron separation. We have thus found evidence that the ground state of Helium is not the standard 1s2 state with two overlaying spherical wave functions, without or with perturbation correction, but instead consists of two non-overlapping half-spherical wave functions. If this conclusion indeed shows to be  correct, then electronic wave functions will have to be recomputed for all atoms. Mind-boggling!

PS3 For Li+ we get E = -7.32 (-7.28), for Be2+ we get = -13.72 (-13.65) and for B3+ we get -22.12 (-22.03) with measured value in parenthesis....more to come...

### Physical Quantum Mechanics 6: Interpretation of Ground State

Ground state wave function of Hydrogen (surface plot of 2d section) as minimizer of potential energy under Laplacian regularization.

The ground state wave function $\psi (x)$ of a Hydrogen atom is the minimizer of the (normalized) energy cost functional
• $E(\psi ) =\int \frac{1}{2}\vert\nabla\psi\vert^2dx -\int\frac{\psi^2(x)}{\vert x\vert}dx$,
under the normalization
• $\int\psi^2dx =1$.                                  (1)
The ground state wave function thus emerges as the minimizer of potential energy
• $PE(\psi ) = -\int\frac{\psi^2(x)}{\vert x\vert}dx$
under Laplacian regularization expressed by the cost functional
• $RE(\psi )=\int\frac{1}{2}\vert\nabla\psi\vert^2dx$.
From the form of the potential energy $PE(\psi )$ we understand that the physical meaning of $\psi^2(x)$ must be charge distribution with (1) setting the total charge. This is a key element of the new formulation of Schrödinger's equation as a real-valued second order equation , which we are exploring in this sequence of posts.

Minimization of energy distributes the electronic charge around the proton kernel with grading monitored by the Laplacian regularization. There is nothing left to chance in this process. All Hydrogen ground states look the same, as well as excited states emerging as stationary points of $E(\psi )$.

We compare with the standard formulation as a complex-valued first order equation, in which $\vert\psi\vert^2$ is not interpreted as charge distribution but as a probability distribution of particle position.

The key point is that charge distribution has a direct real physical meaning in connection to a potential in a classical continuum mechanical sense, while a probability distribution of particle position has no direct real physical meaning, and thus (probably) is meaningless.

## måndag 26 januari 2015

### Physical Quantum Mechanics 5: Does the Scientific Method Need Revision?

Niels Bohr brainwashed a whole generation of theorists into thinking that the job of interpreting quantum theory was done 50 years ago. (1969 Nobel Laureate Murray Gell-Mann)

Sabine Hossenfelder asks Does The Scientific Method Need Revision? motivated by the following observations and reflections:
• Theoretical physics has problems.
• It is our lacking understanding of space, time, matter, and their quantum behavior that prevents us from better using what nature has given us.
• And it is this frustration that lead people inside and outside the community to argue we’re doing something wrong, that the social dynamics in the field is troubled, that we’ve lost our path, that we are not making progress because we keep working on unscientific theories.
• Somewhere along the line many physicists have come to believe that it must be possible to formulate a theory without observational input, based on pure logic and some sense of aesthetics.
• They must believe their brains have a mystical connection to the universe and pure power of thought will tell them the laws of nature. But the only logical requirement to choose axioms for a theory is that the axioms not be in conflict with each other.
• You can thus never arrive at a theory that describes our universe without taking into account observations, period. The attempt to reduce axioms too much just leads to a whole “multiverse” of predictions, most of which don’t describe anything we will ever see.
• See, in practice the origin of the problem is senior researchers not teaching their students that physics is all about describing nature. Instead, the students are taught by example that you can publish and live from outright bizarre speculations as long as you wrap them into enough math.
• I cringe every time a string theorist starts talking about beauty and elegance. Whatever made them think that the human sense for beauty has any relevance for the fundamental laws of
nature?
• There isn’t any one scientific method. The only thing that matters is that you honestly assess the use of a theory to describe nature. If it’s useful, keep it. If not, try something else. This method doesn’t have to be changed, it has to be more consistently applied. You can’t assess the use of a scientific theory without comparing it to observation.
• Theories might have other uses than describing nature...but if they don’t describe nature don’t call  them science.
I agree: Modern physics is not pursued according to a scientific method. How can that be, when classical physics represents the highest incarnation of the scientific method? Why is modern physics irrational and without contact to observation, when physics if anything should be rationalization of observations?

The answer is simple:  The diversion or Fall away from the Paradise of rational classical physics, happened in the 1920s with Born's interpretation of Schrödinger's wave function $\psi$ in statistical terms with $\vert\psi\vert^2$ as a probability distribution of particle position, against the scientific principles of Schrödinger,  which in the merciless hands of Bohr crushed Schrödinger and along with that the scientific method.

To insist like Bohr that the ground state of a Hydrogen atom is the result of a microscopic game of roulette with a large variability, instead of a fully determinstic interplay of forces without variability making the ground state the same for all Hydrogen atoms,  is to make physics into a crazy casino or zoo of an inifinite variety of Hydrogen atoms all different.

In the present series of posts on Physical Quantum Mechanics I return to Schrödinger's original second order wave equation, which can be given a deterministic interpretation without statistics, and thus may open a possibility of returning to the Paradise of rationality of classical physics.

Of course Lubos Motl, as a represenative of the generation of physicists brainwashed by Bohr, opposes to everything Sabine says.

### Physical Quantum Mechanics 4: Interpretation of the Wave Function

We are exploring a second order alternative formulation of Schrödinger's wave equation as the basic model of quantum mechanics, which for a Hydrogen atom takes the form
• $h^2\ddot\psi +H^2\psi =0$,        (1)
for all $(x,t)$, where $\psi =\psi (x,t)$ is a real-valued function of a 3d space coordinate $x$ and time coordinate $t$, $\dot\psi =\frac{\partial\psi}{\partial t}$, and $H=-\frac{h^2}{2m}\Delta +V$ is the standard Hamiltonian with $\Delta$ the Laplacian differential operator, $h$ Planck's constant, $m$ the mass of the electron, and $V=V(x)=-\frac{e^2}{4\pi\epsilon_0\vert x\vert}$ is the kernel potential with $\epsilon_0$ the dielectric constant of vacuum and $e$ the charge of the electron. We view (1) as a generalized harmonic oscillator.

We introduce the eigenfunctions $\psi_j$ and corresponding real eigenvalues $E_j$ of the Hamiltonian $H$ satisfying
• $H\psi_j = E_j\psi$ for $j=1,2,3...$ with $E_1\leq E_2\leq E_3,...$
We then redefine $H$ into $H-E_1$ and $E_j$ into $E_j-E_1$, so that $H\psi_1 =0$ and $E_1=0$ and we consider $\psi_1=\psi_1(x)$ to be the ground state.

The solution $\psi (x,t)$ of (1) can then be expressed as a real-valued linear combination of eigen-modes
• $\exp(i\frac{E_j}{h}t)\psi_j(x)$ for $j=1,2,3,...$
What may here be the physical meaning of the wave function $\psi (x,t)$ as a scalar real-valued function?

We seek guidance comparing with the equation for a vibrating thin 2d elastic plate with plane stress-free ground configuration, which may take the form
• $\ddot\phi +\Delta^2\phi =0$,      (2)
where $\phi (x,t)$ is the transversal displacement of the plate at a position $x$ in the 2d plane of the pale at time $t$.

We are thus led to interpret the wave function $\psi (x,t)$ of (1) as a "transversal displacement" of a 3d "elastic body" at a position $x$ in 3d and time $t$, with the "transversal displacement" acting so to speak into a "virtual 4th dimension" as a measure of change away from a "stress-free ground configuration".  We may then view (1) to express force balance according to Newton's 2nd law with $h^2\ddot\psi$ rate of change of momentum $h^2\dot\psi$ and $H^2\psi$ a corresponding force.

We seek further guidance in the following conserved quantities of (1) as different forms of energy:
• $OE= \frac{1}{2}\int (H\psi )^2 + h^2\dot\psi^2)dx$,                                               (3)
• $AE= \frac{1}{2}\int (\psi H\psi+h^2\dot\psi H^{-1}\dot\psi )dx$                           (4)
• $Q=\frac{1}{2}\int (\psi^2+h^2(H^{-1}\dot\psi )^2dx$                                            (5)
as results of multiplication (modulo the ground state) of (1) by  $\dot\psi$,  $H^{-1}\dot\psi$ and $H^{-2}\dot\psi$,  respectively, and integrating in space.

It is here natural to view $OE$ as total oscillator energy, $AE$ as total atomic energy, and it may also be natural to view $Q$ as total charge, viewing thus charge as a form of energy.  We are thus led to define
• $\frac{1}{2}(H\psi )^2 + h^2\dot\psi^2) =$ local oscillator energy
• $\frac{1}{2}(\psi H\psi +h^2\dot\psi H^{-1}\dot\psi ) =$ local atomic energy
• $\frac{1}{2}(\psi^2+h^2(H^{-1}\dot\psi )^2) =$  local charge.
We thus view the wave function $\psi (x,t)$ to represent (scalar) displacement away from a static ground state $\psi_1$ satisfying $H\psi_1=0$, and we view $\psi^2$ to describe charge distribution.

Note that
• $\int \psi H\psi dx =\frac{h^2}{2m}\int\nabla\psi\vert^2dx +\int V(x)\psi^2(x)dx$,   (6)
with the appearance of $\psi^2$ in the kernel potential directly connection to an interpretation as charge. In particular, the ground state $\psi_1(x)$ emerges as the minimizer of (6) with minimal
potential energy under Laplacian space regularization.

This is radically different from the standard interpretation with $\vert\psi\vert^2$ as a probability distribution of particle position with $\psi$ complex-valued.

Let us thus compare the two interpretations of (i) $\psi^2$ as charge distribution and (ii) $\vert\psi\vert^2$ as probability distribution of particle position:

1. A function value $\vert\psi\vert^2(x,t)$ is a non-negative number and as such cannot represent a 3d position coordinate, while $\psi^2 (x,t)$ may naturally directly represent a scalar quantity like charge (or mass).

2. The scalar function $\psi^2(x,t)$ generates a 3d charge distribution through the dependence on $x$ and thus has a 3d quality, which gets expressed in radiation from oscillating charges.

3. The only way to connect the scalar $\vert\psi\vert^2$ to 3d space is to interpret
it as a probability distribution in space, but a probability is not a direct physical quantity.

The conclusion is that connecting the scalar $\vert\psi\vert^2$ to 3d particle position, as in standard quantum mechanics, is both irrational and unneccessary.

We will next extend to radiation summarizing previous posts on the radiating atom.

Then we will extend to atoms/ions with more than one electron. With the wave function $\psi (x,t)$
connecting to charge distribution $\psi^2(x,t)$, we will not be misl(led) to introduce a multi-dimensional wave function $\psi (x1,x2,...,xN,t)$ depending on $N$ 3d space coordinates $x1,x2,...xN$ as in the standard formulation (which makes the Schrödinger equation uncomputable for several electrons) because such a function does not connect to a physical many-electron charge distribution, only to a probability distribution of many-particle positions without direct physical interpretation. Instead we will be led to a system of one-electron Schrödinger equations with direct physical meaning.

## söndag 25 januari 2015

### Physical Quantum Mechanics 3: Back to Continuum Mechanics

We are exploring a second order alternative formulation of Schrödinger's wave equation as the basic model of quantum mechanics, which for a Hydrogen atom takes the form
• $\ddot\psi +H^2\psi =0$,        (1)
for all $(x,t)$, where $\psi =\psi (x,t)$ is a real-valued function of a 3d space coordinate $x$ and time coordinate $t$, $\dot\psi =\frac{\partial\psi}{\partial t}$, and $H=-\frac{h^2}{2m}\Delta +V$ is the standard Hamiltonian with $\Delta$ the Laplacian differential operator with respect to $x$, $h$ Planck's constant, $m$ the mass of the electron, and $V=V(x)=-\frac{1}{\vert x\vert}$ is the (normalized) kernel potential.

We observe that (1) upon multiplication by $m$ takes the form
• $m\ddot\psi +(-\frac{h^2}{2}\Delta +mV)^2\psi =0$    (2)
with the electron mass $m$ appearing as if $m\dot\psi$ represents momentum and (2) expresses force balance according to Newton's 2nd law.  We observe that the scaling of the Laplacian conforms with an interpretation as space regularization.

We further observe that in the limit with $h=0$, (2) decouples into a set of ordinary differential equations indexed by $x$:
• $\ddot\psi (x,t)+\frac{1}{\vert x\vert^2}\psi (x,t) =0$,
which reflects Newton's 2nd law with a gravitational force scaling with $\frac{1}{\vert x\vert^2}$.

In the case $V=0$, (2) reduces to (with normalization)
• $m\ddot\psi +\Delta^2\psi =0$,
which can be viewed as a model of a vibrating elastic solid.

We thus find that it is possible to interprete the atomic model (2) in classical continuum mechanical terms. Of particular interest is then the conserved quantities of (2), and of course the physical meaning of the wave function, which is not the standard one with $\vert\psi\vert^2$ a particle position probability, to which we return in the next post.

### Physical Quantum Mechanics 2: Schrödinger's Original Wave Equation

Schrödinger first posed a second order Schrödinger equation as the basic wave equation of quantum, in the fourth of his ground-breaking articles in 1926.

The experience behind a second-order time-dependent Schrödinger equation of the form
• $\ddot\psi +H^2\psi =0$,    (1)
with $H$ a Hamiltonian and the dot signifying differentiation with respect to time, is as follows: Observed frequencies $\nu$ of atomic light emission show to be proportional to differences $E_j-E_k > 0$ of energy levels $E_1\lt E_2\lt E_3...$ as eigenvalues with corresponding eigen-states $\psi_j=\psi_j(x)$ depending on a space coordinate $x$, satisfying the time-independent equation:
• $H\psi_j =E_j\psi_j$.       (2)
We can here assume that $E_1=0$ by replacing $H$ by $H-E_1$ and what is observed is thus a linear relation between observed frequencies $\nu_j$ and eigenvalues $E_j$ of a mathematical atomic model:
• $E_j = h\nu_j$                    (3)
where $h$ is a constant named Planck's constant (which we here normalize to 1).

We now observe that (1) is a natural extension of the real-valued eigenvalue problem (2) to dynamic time-dependency in accordance with (3), as a real-valued wave equation which can be given a physical interpretation in classical continuum mechanical terms with solutions as real-valued linear combinations of $\exp(i\nu_jt)\psi_j(x)$.

We compare with Schrödinger's equation in standard first-order complex form
• $i\dot\psi + H\psi =0$     (4)
which also respects (3), but does not lend itself to physical interpretation.

We thus have two possible Schrödinger equations with solutions $\psi$ named as wave functions, the second order real-valued form (1) and the standard first order complex-valued form (4), both which fit with the experience of (3).

Here (1) has a physical interpretation in classical continuum mechanical terms, while (4) lacks such a physical interpretation and has only been given a statistical unphysical interpretation.

It seems pretty clear that (1) is to prefer before (4) as a the basic mathematical model of quantum mechanics, and this is the possibility I want to explore.  In particular the whole mess of interpreting the wave function in statistical terms can probably be avoided this way, and much now wasted effort saved.

## fredag 23 januari 2015

### Physical Quantum Mechanics (Based on Second Order Schrödinger Equation) 1

String vibration as deterministic physics.

The founding pillars of modern physics are (i) quantum mechanics of small scale atomistic physics and (ii) relativity theory of large scale physics. Unfortunately (i) and (ii) have shown to be incompatible, which gives modern physics a shaky foundation loaded with mysteries. In particular, quantum mechanics is viewed to be fundamentally different from classical continuum mechanics, and so beyond human comprehension.

Quantum mechanics describes the atomistic world in terms of wave functions $\psi$ satisfying Schrödinger's equation, which for the basic case of the Hydrogen atom takes the (normalized) form
• $i\dot\psi \pm H\psi =0$,        (1)
where $\psi =\psi (x,t)$ is a complex-valued function of a 3d space coordinate $x$ and time coordinate $t$, $\dot\psi =\frac{\partial\psi}{\partial t}$, and $H=-\frac{1}{2}\Delta +V$ is a Hamiltonian with $\Delta$ the Laplacian differential operator and $V=V(x)=-\frac{1}{\vert x\vert}$ is the kernel potential. The mystery of the wave function $\psi (x,t)$ and the equation (1), is that $\psi$ has no direct physical meaning, only an indirect unphysical meaning with $\vert\psi (x,t)\vert^2$ viewed as a probability distribution of particle position.

Schrödinger obtained (1) in 1926 starting from a second-order wave equation
• $\ddot\phi + H^2\phi =0$,      (2)
in terms of a real-valued wave function $\phi (x,t)$, by a formal decomposition
• $\frac{\partial^2}{\partial t^2} + H^2 = -(i\frac{\partial}{\partial t}+H)(i\frac{\partial}{\partial t}-H)$,
thus viewing formally the complex-valued first order equation (1) as the "square root" of the real-valued second order equation (2).

This decomposition is analogous to the decomposition of the second order wave equation
• $\frac{\partial^2\phi}{\partial t^2}-\frac{\partial^2\phi}{\partial x^2}=0$,   (3)
in a 1d space coordinate $x$, into the first order equations
• $\frac{\partial\psi}{\partial t}\pm\frac{\partial\psi}{\partial x}=0$.           (4)
We want to compare the physics expressed by (1) and (2), and then start comparing the physics of (3) and (4).

We know that the function $\phi (x,t)$ in (3) can be interpreted as the transversal displacement of a vibrating elastic string at $(x,t)$ with (3) expressing a balance of inertial and elastic forces according to Newton's law.

We know that (4) expresses constancy along characteristics $x\pm t=constant$ describing convection or translation of a quantity with speed 1 in the positive or negative $x$-direction.

We know that the second order wave equation (3) admits waves traveling in both positive and negative $x$-direction, while each of the two equations (4) admits waves traveling in only one direction. We conclude that the physics described by (3) and (4) is different: the elastic string of (3) is not present in (4) and the physics of the translation in (4) is unknown or unspecified.

We now understand that also (1) and (2) may describe different physics, or no physics.

Our conclusion is that the second order real-valued form (2), which is close to (3), may describe physics similar to that of a vibrating string as a form of vibrating electron with again (2) expressing force balance, while the physics of the first order complex-valued conventional form (1) has remained a mystery since 1926.

The meaning of the wave function $\phi$ of (2) is "displacement in space" with $\phi^2+(H^{-1}\dot\phi )^2$ representing charge carried as a concrete physical phenomenon. We compare with the accepted meaning of the wave function $\psi$ in (1) with $\vert\psi\vert^2$ a probability distribution of particle position, which is not carried as a physical phenomenon, only as a phantasm in the mind of a physicist.

The difference between (1) and (2) thus appears to be most essential, if external physical reality is maintained to be what makes physics different from mathematics and philosophy, which do not require an external world to exist.

The unfortunate result is that insisting to take (1) as the basic equation of quantum mechanics while lacking direct physical meaning, has led generations of physicists following Max Born to attribute a non-physical meaning to the wave function $\psi (x,t)$  as a probability distribution $\vert\psi (x,t)\vert^2$ of particle position.

The result is a collapse of determinism and casuality and thus scientific rationality, which could have been avoided if instead, along with Schrödinger's original thoughts, (2) had been chosen as the basic equation of quantum mechanics. This is the line of thought I would like to explore further with the hope of finding a deterministic rational physical quantum mechanics as a form of classical continuum mechanics, which can replace probabilistic irrational unphysical quantum mechanics as atomistic physics. It is then encouraging to note that the present highest form of modern physics of string theory, connects to (2) rather than (1).

Atomic vibration as deterministic physics.

• There is nothing more deterministic and with less free will than the ground state of a Hydrogen atom.  (Nietzsche)
• The assumption of an absolute determinism is the essential foundation of every scientific enquiry. (Planck)
PS Compare with Physicists debate whether quantum math is as real as atoms discussing the difference between ontic (what is) and epistemic (what we know) aspects of quantum mechanics.
The most clever among us like Motl Lubos insists that the wave function is neither ontic nor epistemic:
• However, Nature around us doesn't work in either way. Just like the electron in Nature is neither a classical particle nor a classical wave, the wave function is neither "ontic" nor "epistemic". The world is simply described neither by classical physics evolving a point in the phase space; nor by classical statistical physics.
This is so clever: By removing what physics is not, what remains must be what physics is! Right?

## tisdag 20 januari 2015

### Funeral of Schrödinger's Cat in Sweden

Swedish physics professors Karl Erik Eriksson and Bengt Gustavsson performed a symbolic academic funeral of (i) Schrödinger's cat along with (ii) multiversa and (iii) probabilistic dice interpretations of quantum mechanis, in a worthy ceremony at the Alma-Löv art museum in Värmland in the heart of Sweden on November 20, 2014.

I fully agree with these professors of physics that the modern physics of (i)-(iii) is dead and that the funeral thus puts an end to three tragic episodes of physics, as the Queen of sciences and tremendous success, see post of Jan 16.

From the ashes a new form of quantum mechanics may emerge, maybe in the form of a physical quantum mechanics based on a second order real-valued Schrödinger equation without cat, dice and parallel worlds, as discussed in recent posts.

Recall that the reason to introduce the dice leading to the cat and parallel worlds, was that the standard first order complex-valued form of Schrödinger's equation, does not describe any physics.

### New Physical Quantum Mechanics

Schrödinger (left, laughing) and Heisenberg (right, also laughing) together with a (taller) representative of Swedish Kingdom quantum mechanics (middle, serious).

In the recent sequence of posts on The Radiating Atom 1-11, I have been led to a formulation of Schrödinger's equation as the basic equation of quantum mechanics, as a scalar second order wave equation in terms of a real-valued wave function $\psi (x,t)$ of space-time $(x,t)$ of the form (for a one-electron atom/ion to start with):
• $\ddot\psi (x,t) + H^2\psi (x,t) = 0$,  for all $(x,t)$,              (1)
where the dot denotes differentation with respect to time and thus $\ddot\psi =\frac{\partial^2\psi}{\partial t^2}$, and $H$ is a Hamiltonian. We have observed that (1) is closely related to the standard formulation of Schrödinger's equation in complex form (normalizing to $h=1$)
• $i\dot\psi \pm H\psi =0$              (2)
which appears as a form of "square-root of (1)". Or the other way around, (1) appears as the "square of (2)".

We have argued that (1) lends itself better to physical interpretation and extension to radiation than (2), and we recalled that (1) was the original starting point for Schrödinger in 1926.

We have observed that solutions of (1) satisfy conservation of
• total charge = $\frac{1}{2}\int (\psi^2+(H^{-1}\dot\psi )^2dx$
• total atomic energy = $\frac{1}{2}\int (\psi H\psi+\dot\psi H^{-1}\dot\psi )^2dx$
• total oscillator energy = $\frac{1}{2}\int (H\psi )^2+\dot\psi^2)dx$,
as results of multiplication of (1) by $H^{-2}\dot\psi$, $H^{-1}\dot\psi$ and $\dot\psi$, respectively, and integrating in space.

We have seen that (1) naturally extends to radiation and forcing in the form:
• $\ddot\psi + H^2\psi -\gamma\dddot\psi = f$,
with $f=f(x,t)$ scalar forcing and and $\gamma\ge 0$ a small radiation coefficient, and we have observed the following basic energy balance in the case of near resonant forcing:
• total outgoing radiation = $\int \gamma\ddot\psi^2dxdt\approx\int f^2dxdt$ = total incoming radiation.
We have argued that (1) can be interpreted as a force balance with the wave function $\psi$ as a form of "scalar virtual displacement" connecting to classical mechanics with the Hamiltonian involving "internal elastic forces" connected to the presence of the Laplacian and to elastic spring forces from kernel potential, and $\gamma\dddot\psi$ connecting to the Abraham-Lorentz radiation recoil force. We have then noted that the wave function $\psi$ as "scalar virtual displacement" is given a physical realization as 3d local charge displacement.

The wave function of $\psi$ of (1) can thus be given a deterministic physical meaning, as an alternative to the standard interpretation of the wave function of (2) as probabilistic particle position.
Physical conservation of total charge in (1) will then replace unphysical conservation of total probability in (2).

We thus have compared with (2), which is viewed to be an ad hoc model without physical interpretation; if (1) has a physical meaning, it does not follow that (2) as a "square-root of (1)" must have a physical meaning.

We expect that (1) to extend in a natural way to the case of several atoms as a system of one-electron equations expressing force balance of a collection of wave functions depending on $(x,t)$, which is computable. We compare with the standard extension of (2) into a complex equation for a wave function depending on $3N$ spatial coordinates for $N$ electrons, which leads to an uncomputable model.

In short, there is evidence that (1) may offer a better foundation of quantum mechanics than the standard (2), in accordance with the original thoughts of Schrödinger, which unfortunately became muddled by the later declared success of (2) in the Copenhagen Interpretation by Bohr and Heisenberg.

In short: (1) appears to be deterministic, physical and computable, while (2) appears to be probabilistic, unphysical and uncomputable. Further study will show if the expectations for (1) can be met.

## fredag 16 januari 2015

### Perimeter Institute: Very Deep Crisis of Modern Physics: Ultimate Catastrophe

• Our donors believe that discoveries made in theoretical physics are really important for humanity.
• This is certainly true if you look at the past:
• Without theoretical physics we wouldn't have modern technology and a vast array of sciences.
• We know that the world is made of atoms. Who found that out? Physicists!
• We know that atoms stick together to molecules. Who discovered that? Physicists!
• The Higgs boson, a billion times smaller that the atom, predicted 50 years earlier on the basis of quantum theory and relativity, was found in a 10 billion Euro LHC machine ...
• The second great discovery was made by the Planck satellite as a pattern of density variations from the Big Bang...very precise...although a few unknowns like dark matter and energy...
• The level of agreement between theory and data is extraordinary and not matched in any other area of science.
• Yet, what we are describing is the whole universe itself, which turns out to be very simple on very large scales and on very small scales.
• The level of precision and reach of physics is beyond any other area of science.
• That is the good news about physics: It is the most successful field in science and the most conceptually simple.
Turok continues with some philosophical reflections:
• What is is going on?
• Why is it that human beings have this capacity to see into the heart of nature way beyond the world of everyday experience?
• We don't know.
Turok then (humorously, intertwingled with laughter) informs the young minds eager to learn:
• Theoretical physics is at a crossroads.
• We have entered a very deep crisis...grand unified models, supersymmetric models, superstring models, loop quantum models, models, models...
• Nature turns out to more simple than all these models.
• Theorists are in state of confusion. What is going on?
• Theories have failed, because they did not really introduce new concepts.
• The number of parameters in supersymmetric models is 120, and explodes into thousands...
• String theory seems to predict $10^{1000}$ different possible laws of physics:
• The multiverse: The ultimate catastrophe. Crazy situation.
• Theorists utterly confused...seem to have no predictions at all.
• Very deep crisis. The vacuum is not empty...it has energy...
• Theory has kind of selfdestructed.
• The theories we know are missing some very important principle.
• You are entering a field which requires a complete re-invention.
• What type theory will replace quantum theory and relativity?
• You will need to develop your own ideas...
What would the result be if Turek's message was directed to the donors of Perimeter Institute and not just to innocent students? Is the idea of presenting modern physics, the most successful field of all of science, as an ultimate catastrophe, to stimulate donors to give more money to both theorists and supercollider experimentalists, because it is so important to humanity?

Note that this post connects to an earlier post on the same theme. The reason I am returning is to find motivation in my own search for new models, in which endevour I may have as good chance of finding something as any other confused physicist.

## onsdag 14 januari 2015

### The Radiating Atom 11: Connection to String Theory

Michio Kaku: In string theory, all particles are vibrations on a tiny rubber band; physics is the harmonies on the string; chemistry is the melodies we play on vibrating strings; the universe is a symphony of strings, and the 'Mind of God' is cosmic music resonating in 11-dimensional hyperspace.

We have been led to a alternative formulation of Schrödinger's equation as a second order wave equation in terms of a real-valued wave function $\psi =\psi (x,t)$ depending on a space coordinate $x$ and time $t$ (here for a one electron atom or ion):
• $\frac{\partial^2\psi}{\partial t^2} + H^2\psi =0$,          (1)
where $H = -\frac{1}{2}\Delta + V(x)$ is a Hamiltonian with $\Delta$ the Laplacian and $V(x)$ a potential (with $V(x)=-\frac{1}{\vert x\vert}$ in the basic case of the Hydrogen atom and normalizing to atomic units).

In the model case of one space dimension and $V(x) = 0$,  Schrödinger's equation (1) takes the form
• $\frac{\partial^2\psi}{\partial t^2} + \frac{1}{4}\frac{\partial^4\psi}{\partial x^4} =0$,
which can be interpreted as a model of a vibrating thin elastic beam. We compare with the basic model of a vibrating elastic string:
• $\frac{\partial^2\psi}{\partial t^2} - \frac{\partial^2\psi}{\partial x^2} =0$,
This is also the basic model of string theory (cf. (3.31) here or here) as modern fundamental physics supposedly describing a subatomic world on Planck scales of $10^{-35}$ m.

We thus find a second order in time wave equation to be a basic model of physics on all scales, from macroscopic, over atomic to extremely subatomic scales.

We know that a macroscopic wave equation expresses a force balance, a balance of
• inertial forces proportional to $\frac{\partial^2\psi}{\partial t^2}$ with $\psi (x,t)$ interpreted as a displacement at position $x$ at time $t$,
• elastic forces proportional to $\frac{\partial^2\psi}{\partial x^2}$ or $\frac{\partial^4\psi}{\partial x^4}$.
It is natural to interpret also an atomic and subatomic wave equation as a force balance. This opens to interpret Schrödinger's equation in the form (1) as a force balance. In the previous post we saw that this naturally opens to extension to atomic radiative absorption and emission under forcing.  We compare with the standard complex first oder form of Schrödinger's equation $i\frac{\partial\psi}{\partial t}+H\psi =0$,  for which a physical interpretation is missing.

We know that a wave equation can be viewed to express in mathematical terms stationarity of an action integral as an integral in space and time of a Lagrangian $L(\psi )$, which for an elastic string takes the form
• $L(\psi )=\frac{1}{2}(\frac{\partial^2\psi}{\partial t^2})^2-\frac{1}{2}(\frac{\partial^2\psi}{\partial x^2})^2$.
Force balance can thus in mathematical terms be viewed to express stationarity of an action integral, but unfortunately physicists have become so impressed by this mathematical equivalence as to elevate stationarity of action to be the basic principle of physics, and then so before force balance. But stationarity of an action integral is not physics, only mathematics, because there is no physical process computing action integrals and finding stationarity, while force balance is the essence of physics, as expressed by Newton's law in terms of intertial force.

Unfortunately, confusion of physics with mathematics has led physicists to search for Lagrangians irrespective of possible lack of physical meaning, rather than seeking wave equations expressing force balance. The formulation of Schrödinger's equation as the second order wave equation (1) is a step in the other direction towards physical meaning, understanding that mathematics is not always physics.

We are then led to interpret the wave function $\psi (x,t)$ in (1) as a displacement in space at  position $x$ and time $t$, and thus $\frac{\partial\psi}{\partial t}=\dot\psi$ as a displacement velocity. From the force balance (1) then follows conservation of the following physical entities
1. total charge = $\frac{1}{2}\int (\psi^2+(H^{-1}\dot\psi )^2dx$
2. total atomic energy = $\frac{1}{2}\int (\psi H\psi+\dot\psi H^{-1}\dot\psi )^2dx$
3. total oscillator energy = $\frac{1}{2}\int (H\psi )^2+\dot\psi^2)dx$,
from multiplication of (1) by $H^{-2}\dot\psi$, $H^{-1}\dot\psi$ and $\dot\psi$, respectively, and integrating in space. Note that 3. naturally connects to radiation scaling with $\nu^4$ with $\nu$ frequency, with $f\dot\psi$ representing work by forcing $f$ acting on displacement velocity $\dot\psi$, with (1) extended to forcing and radiation as in the previous post.

We are thus led interpret charge as a form of energy, and the wave function $\psi$ as a measure of charge displacement, like the displacement of an elastic string, or rather a 3d elastic body.

A true physicist would probably say that (1) is no good since since it is not Lorentz invariant and thus not relativistically correct. But is this true? Well, the speed of light is $c = 3\times 10^{18}$ m/s and, a typical frequency may be $10^{15}$ Hz, the radius of an atom typically smaller than $3\times 10^{10}$ m and so the speed $v=\vert\dot\psi\vert$ will satisfy $\frac{v}{c}\le 10^{-3}$ and thus relativistic effects appear to be very small, if any.

PS Note that in this setting there is no reason to interprete $\psi^2(x,t)$ as a probability of the presence at $(x,t)$ of the electron as particle, since $\psi^2(x,t)$ is not conserved, but instead it is $\psi^2 +(H^{-1}\dot\psi )^2$ as charge, which is conserved. The scientific gamble of viewing atom physics as microscopic roulette physics with inevitable major losses, as in the textbook version of quantum mechanics, can thus possibly be avoided, and much be gained.

## måndag 12 januari 2015

### The Radiating Atom 10: Restart from Schrödinger IV 1926

Schrödinger in Quantization and Proper Values IV, 1926:
Meantime, there is no doubt a certain crudness in the use of a complex wave function. If it were unavoidable in principle, and not merely a faciliation of the calculation, this would mean that there are in principle two wave functions, which must be used together in oder to obtain information on the state of the system. This somewhat unacceptable inference admits, I believe, of the very much more congenial interpretation that the state of the system is given by a real function and its time derivative. Our inability to give more accurate information about this is intimately connected with the fact that we have before us only the substitute, extraordinarily convenient for the calculation, to be sure, for a real wave equation of probably fourth order, which, however, I have not succeeded in forming in the non-conservative case.

In the present series of posts on the radiating atom, I have restarted from the last of Schrödinger's four legendary 1926 articles formulating Schrödinger's equation, for Hydrogen to start with:
• $i\dot\Psi + H\Psi =0$         (1)
where $\Psi (x,t)=\psi +i\phi$ is a complex-valued electronic wave function of a space coordinate $x=(x_1,x_2,x_3)$ and time $t$, with real-valued real and imaginary parts $\psi (x,t)$ and $\phi (x,t)$, $H$ is the Hamiltonian defined by
• $H =-\frac{h^2}{2m}\Delta + V$,
where $\Delta$ is the Laplacian with respect to $x$, $V(x)=-\frac{1}{\vert x\vert}$ is the kernel potential, $m$ the electron mass, $h$ Planck's constant and the dot signifies differentiation with respect to time $t$.

Before ending up with (1) Schrödinger considered the following second-order equation in terms of a real-valued wave function $\psi (x,t)$, which can be the real or imaginary part of $\Psi (x,t)$:
• $\ddot\psi +H^2\psi = 0$      (2)
which formally follows by writing (1) as the first order system
• $\dot\psi - H\phi =0$         (3)
• $\dot\phi + H\psi =0$          (4)
and then eliminating $\phi$ by differentating (3) with respect to time and replacing $H\dot\phi$ by
$-H^2\psi$ after application of $H$ to (4).

Schrödinger thus considered both (1) and (2), but then decided to choose the complex first order form (1), while regretting that the real-valued second-order form (2) in principle was to prefer, because of its very much more congenial interpretation

What Schrödinger referred to was that (2) could be given a physical interpretation as force balance pretty much as in classical mechanics, while the physical meaning of (1) was mysterious to Schrödinger and has so remained to all physicists into our time:

The accepted wisdom, presented in all books, is that (1) arises from classical Hamiltonian mechanics by formally representing momentum by the differential operator $-ih\nabla$ acting in space and energy by the differential operator $ih\frac{\partial}{\partial t}$ in time, which is however ad hoc and without physical reason as expressed by Schrödinger himself and acknowledged by all physicists into our time.

The lack of physical interpretation of (1) means that modern physics as the foundation and model of modern science rests on a quantum foam of mystery, which is the opposite of scientific enligthenment.

Schrödinger stated that his choice of (1) before (2) came from a perceived difficulty of generalizing (2) to a non-conservative case including radiation. But maybe Schrödinger dismissed (2) too quickly.

To check this out, let us consider the following natural generalization of (2) to include radiation as a direct generalization of the classical mechanical or electromagnetic wave equation with (small) radiative damping under near-resonant forcing considered in Mathematical Physics of Black Body Radiation (and Computational Black Body Radiation):
• $\ddot\psi +H^2\psi -\gamma\dddot\psi = f$      (5)
where $f(x,t)$ is external forcing,  and $\gamma =\gamma (\psi )$ is a (small) positive radiation damping coefficient. The equation (5) has the physical meaning of force balance with
• $\ddot\psi +H^2\psi$ out-of-balance force of an electronic resonator
• $-\gamma\dddot\psi$ the Abraham-Lorentz radiation recoil force
• $f$ component of an electrical field force.
Let us now subject the model (5) to a basic study:  First we observe that if $f=0$ and  $\gamma =0$,  then conservation of total charge expressed as
• $\frac{d}{dt}\int\rho (x,t)dx =0$,                   (6)
with $\rho =\psi^2+(H^{-1}\dot\psi )^2$ the charge intensity, is obtained by multiplying (5) with $H^{-2}\dot\psi$ and integrating in space.

Next, letting $\{\Psi_1,\Psi_2,\Psi_3....\}$ be an orthonormal basis of eigenfunctions $\Psi_k=\Psi_k(x)$ of the Hamiltonian $H$ satisfying $H\Psi_k =E_k\Psi_k$ with corresponding sequence of eigenvalues $E_1\le E_2\le E_3 ...$ , we spectrally decompose $\psi (x,t) =\sum\psi_k(t)\Psi_k(x)$ and $f(x,t)=\sum f_k(t)\Psi_k(x)$  and obtain after multiplication of (5) by $\Psi_k$ and integrating in space, for $k=1,2,..$ and for all $t$ :
• $\ddot\psi_k(t) +E_k^2\psi_k(t) -\gamma\dddot\psi_k(t) = f_k(t)$   (7)
which is a set of harmonic oscillators with damping under forcing, each one which can be analyzed as in Mathematical Physics of Black Body Radiation.

Let us now consider the basic case $\psi (x,t) = \psi_1(t)\Psi_1(x) +\psi_2(t)\Psi_2(x)$ with $\Psi_1$ the ground state eigenfunction with smallest eigenvalue $E_1$ and $\Psi_2$ an eigenfunction of the next eigenvalue $E_2 > E_1$ with non-vanishing $f_2(t)$ (assuming $f_k=0$ for $k>2$). By a shift of the Hamiltonian by $E_1$, we may assume that $E_1=0$ and then also that $f_1=0$. We thus have the system
• $\ddot\psi_1(t) -\gamma\dddot\psi_1(t) = 0$,  thus $\psi_1(t)=\psi_1=constant$,
• $\ddot\psi_2(t) +E_2^2\psi_2(t) -\gamma\dddot\psi_2(t) = f_2(t)$,
and conclude under an assumption of near-resonant forcing $f_2(t)\sim \cos(\nu t)$ with $\nu\approx E_2/h$ and small damping as in Mathematical Physics of Black Body Radiation:
• $\int\gamma\ddot\psi_2^2dt \approx \int f_2^2(t)dt$   (8)
or in terms of the wave function $\psi$ and the forcing $f$
• $\int\gamma\ddot\psi^2dxdt \approx\int f^2dxdt$       (9)
which expresses that in periodic equilibrium state:
We now recall the basic energy balance of (5) obtained by multiplying (5) by $\dot\psi$ and integrating in space:
• $\dot A(t) +R(t) = W(t)$,
• $A(t)=\frac{1}{2}(\int\dot\psi^2dx+\int (H\psi )^2dx)$ = internal oscillator energy
• $R(t)=\int\gamma\ddot\psi_2^2dt$ = outgoing radiation (per unit of time)
• $W(t) = \int f(x,t)\dot\psi dx$ = work by incoming radiation (per unit of time),
with in equilibrium periodic state, $\dot A(t)=0$ and $R(t)=W(t)$ effectively expressing (9) or (10).

Of particular concern is now the charge conservation in (5). We note that the internal oscillator energy $A(t)$ may increase under forcing with $W(t)>R(t)$, or decrease if $W(t) < R(t)$,
reflecting a change of balance of the spectral weights $\psi_k(t)$. The question is then if such a change of internal oscillator energy may take place under conservation of total charge, and we
are then led to compare the work $f_k\dot\psi_k$ connected to energy and $f_kE_k^{-2}\dot\psi_k$ connected to charge with the corresponding coefficient $E_k^{-2}$ for $k>1$.

Now, in typical cases, $E_k\approx 10^{15}$ and thus $E_k^{-2}\approx 10^{-30}$, which signifies that in the model (5) energy may change under almost perfect charge conservation.

Note that (8) can be expressed as
• $\gamma \nu_2^4\int\psi_2^2dt\approx \int f_2^2dt$,
thus connecting the amplitude of the excited component $\psi_2\Psi_2$ to the forcing $f_2^2$, which itself may be of the form $\gamma\nu_2^4$ with a possibly different $\gamma$. The radiation balance (10) can thus be viewed to express radiative equilbrium of a collection of atoms under mutual radiative absorption/emission.

We sum up the virtues of (5) as a semi-classical continuum wave model of a radiating atom subject to forcing,
1. (5) lends itself to physical interpretation as force balance in a classical sense with the Laplacian representing some form of elastic energy, and the value of wave function $\psi (x,t)$ at position $(x,t)$  representing the "displacement" of the electron at $(x,t)$ from a ground state.
2. The Abraham-Lorentz recoil force is small compared to forcing and oscillator imbalance,  because $\gamma$ is very small, which means that self-interaction is avoided and the forcing $f$ can be viewed to be independent of the wave function $\psi$.
3. (5) lends itself to mathematical analysis as energy balance under charge conservation.
4. (5) has a natural extension to a model for a many-electron atom as a system of one-electron equations, which is computable and thus potentially useful.
5. (5) coincides in the case $f=0$ and $\gamma =0$ with the standard model (1) and thus with experiments.
6. (5) admits the ground state to be independent of time as a stable solution without radiation and forcing.
7. (5) fits with observed radiation of frequency $\nu =(E_k-E_1)/h$ under near-resonant forcing.
8. Outgoing and incoming radiation can be shifted in (10), which allows (5) to model both absorption of radiation and stimulated or spontaneous emission of radiation.
Our conclusion is that maybe (5) is the basic model of quantum mechanics asking for thorough analysis and waiting for extensive practical use, rather than (1), corresponding to a restart from the original idea of Schrödinger as the true father of quantum mechanics.

It appears that the advantage of (5) allowing natural extension to radiation and forcing, was (paradoxically so) by Schrödinger perceived instead as a disadvantage making him prefer (1). Too bad that Schrödinger is not around anymore, so that he could have clarified the reason for his choice.

The above virtues 1-8 of (5) may be compared to the following acknowledged deficiencies/difficulties of (1):
1. The physical meaning of (1) as a strange ad hoc "square-root" of (5) is unknown.
2. Extension to radiation of (1) is typically accomplished through a time-dependent potential representing forcing, which does not include the Abraham-Lorentz recoil force and thus appears to miss essential physics.
3. The attribution of kinetic energy to $\vert\nabla\psi\vert^2$, resulting form formally replacing classical momentum by the differential operator $-ih\nabla$, is irrational from physics point of view.
4. The generalization of (1) to include radiation under forcing is commonly viewed to require extensions to QED which is further away from classical mechanics, and thus loaded with difficulties.
5. Extension of (1) to many electrons introduces a multi-dimensional wave function, which makes (1) uncomputable and thus useless.
PS Note that the wave equation $\ddot\psi +H^2\psi -\gamma\dddot\psi = f$ is a scalar equation in a real-valued function $\psi (x,t)$ with scalar forcing $f(x,t)$, which may be any component of the electrical field, with non-zero $f_k(t)$ in near-resonant interaction with an eigenmode $\psi_k(t)\Psi_k (x)$. It is thus the multiplicity of eigenvalues with in particular 3 independent $2p_{x1}$, $2p_{x2}$ and $2p_{x3}$ eigenstates oriented in the coordinate directions $x =(x1,x2,x3), which in the basic case of resonant radiation connects the scalar wave equation to the vector$E=(E_{x1},E_{x2},E_{x3})$of the electrical field, see The Radiating Atom 9. Interpreting the scalar wave function$\psi$as an (oscillating) virtual "displacement" connects to a corresponding (oscillating) real physical displacement of charge in 3d space, as the connection between the scalar$\psi (x,t)\$ and the vectors of charge displacement and related electrical field.

## lördag 10 januari 2015

### Wolodarski: DN = Charlie Hebdo: Politisk Korrekthet

Den politiskt korrekte chefredaktören Peter Wolodarski på det politiskt korrekta huvudsstadsbladet Dagens Nyheter kommenterar terrordådet mot Charlie Hebdo med:
• Vi är en tidning. Vi har sorg i dag.
• Någon kanske frågar: Var det nödvändigt av Charlie Hebdo att ständigt utmana och provocera, ja, stundtals slå vilt åt alla möjliga håll? Svaret är givet: det är varje medborgares rätt att få göra just det, även om det svider och till och med råkar vara olämpligt. Demokratins ramar måste vara så pass vida att de rymmer också grova provokationer och förolämpningar.
• Häri ligger en avgörande skillnad mot de samhällsskick där fundamentalism och repression härskar.
• Det måste vara slut på naivitet. Det måste vara slut på undfallenhet mot dem som hotar och sprider hat.
Wolodarski uttrycker solidarisk enhet: Dagens Nyheter = Charlie Hebdo.

Men vad betyder egentligen DN = CH? Vi vet att en matematisk ekvation A = B antingen uttrycker
• (1) identitet dvs att A = A, där B alltså är identisk med A, vilket är tautologiskt självklart och därför inte har något egentligt informationsinnehåll,
eller
• (2) likhet gäller för vissa aspekter av A och B, men inte all som ju täcks av (1), vilket har ett informationsinnehåll, så långt dessa aspekter är specificerade.
Sålunda uttrycker 1 + 1 = 2 likhet i antal men inte identitet,  eftersom 1 + 1 uttrycker addition som process medan 2 endast är slutresultatet av processen. Skillnaden uttrycks av att notationen inte är den samma.

Wolodarski sammanfattar med:
• DN publicerar i dag bilder på de mördade i attacken. Vi gör det för att hedra offren och uttrycka solidaritet med deras anhöriga.
• Men vi gör det också för att de som angriper det fria ordet även angriper allt det som vi själva står för.
Lars Bern jämför Wolodarskis här framförda försvar av det fria ordet med sin erfarenhet av suppression av det fria ordet som DN länge utövat i klimatfrågan i form av politisk korrekthet, och som jag också fick känna av i samband med att min röst tystades av KTH i ett mediadrev beskrivet som KTH-gate.

DN uttrycker således solidaritet med det fria ordets uttryck i form av islamsatir, som nu angripits, vilket är politiskt korrekt och därför inte är så modigt.

Men DN undertrycker samtidigt det fria ord som på vetenskaplig grund ifrågasätter den politiskt korrekta koldioxidalarmism som anses utgöra mänsklighetens största hot, långt större än hotet från extrem islamism, vilket inte heller kräver något mod, bara dumhet.

Nu är enhetstanken grunden för det politisk korrekta, som innebär att vi alla är en och delar samma uppfattningar i vårt gemensamma (om än odefinierade) innanförskap. Utom de förstås som står utanför, men de är ju så avskyvärda att de inte kan räknas som hedervärda medborgare.

Men satir kan vara både politiskt korrekt och inkorrekt och kan uppfattas olika: Vad skulle DN säga om Charlie Hebdos islamsatirer fick en framträdande plats på SDs hemsida?  Jämför med dessa synpunkter.

Och hur skall vi tolka Wolodarskis Det måste vara... ultimatum: Det måste vara slut på naivitet. Det måste vara slut på undfallenhet mot dem som hotar och sprider hat.

Frågan kvarstår: Vad menas med att DN = CH? Varför är politisk korrekthet ledstjärnan för DN?