Ännu Mer Undervisningstid i Matematik!

Riksdagen har sagt ja till Regeringens förslag att ytterligare utöka den totala undervisningstiden i matematik i grundskolan med 105 tim från 1020 tim till 1125 tim, detta efter att tiden ökades med 120 tim 2013. Den totala undervisningstiden i alla ämnen är 6785 tim vilket innebär att var sjätte skoldag, eller nästan en hel dag varje vecka, skall ägnas matematik under alla grundskolans 9 år. Lagrådsremissen bakom beslutet argumenterar på följande sätt :

1. Matematik är ett av tre ämnen som krävs för behörighet till samtliga nationella program i gymnasieskolan. 
2. Grundläggande kunskaper i matematik är också en förutsättning för att klara många högskoleutbildningar.
3. För de enskilda eleverna är det av stor vikt att de får de kunskaper i matematik de kommer att behöva i yrkeslivet eller för fortsatta studier. 
4. Att de har sådana kunskaper är viktigt även för samhället i stort.
5. Mycket tyder dock på att svenska elevers matematikkunskaper försämrats under 2000-talet.
6. Som redovisas i promemorian finns det internationell forskning som stöder sambandet mellan utökad undervisningstid och kunskapsresultat.
7. Någon förändring av kursplanen och kunskapskraven i matematik med anledning av utökningen av undervisningstiden är inte avsedd.

Logiken förefaller vara att om ytterligare tid ägnas åt en kursplan/undervisning med dokumenterat dåligt resultat, så kommer resultaten att förbättras. 

Vem kan ha hittat på ett så befängt förslag? Sverker Lundin ger i Who wants to be scientific , anyway? en förklaring: Matematik (eller vetenskap) har blivit modernitetens nya religion när den gamla nu har lagt sig att dö, en religion som ingen vuxen egentligen tror på och mycket få utövar, men en religion som det blivit klädsamt och politiskt korrekt att bekänna sig till i modernitetens tecken, men då bara i "interpassiv" form med försvarslösa skolelever som mottagare av predikan. 

I detta narrspel är finns det aldrig tillräckligt med ritualer för att uppvisa sin fasta tro, och timantalet i matematik kommer således att fortsätta att öka, medan resultaten fortsätter att sjunka och det bara  blir viktigare och viktigare både för de enskilda eleverna och samhället i stort att de kunskaper i matematik som behövs i skolan också lärs ut i skolan.

De nya 105 timmarna skall företrädesvis tillföras mellanstadiet, medan de 120 som tillfördes 2013 avsåg främst lågstadiet. Detta speglar en utbredd förställning att något fundamentalt har gått snett i den tidiga matematikundervisningen, oklart dock vad, och att om bara detta tidiga misstag, oklart vad, undviks eller snabbt rättas till genom extra timmar, så kommer allt att gå så mycket bättre. Men en ensidig jakt på att undvika det första misstaget, oklart vilket det är, kommer naturligtvis medföra att det inte blir mycket tid över till förkovran i senare årskurser, men det kanske inte gör så mycket...

New Quantum Mechanics 19: 1st Excitation of He

Here are results for the first excitation of Helium ground state into a 1S2S state with excitation energy = 0.68 = 2.90 -2.22, to be compared with observed 0.72:

New Quantum Mechanics 18: Helium Ground State Revisited

Concerning the ground state and ground state energy of Helium the following illumination can be made:

Standard quantum mechanics describes the ground state of Helium as $1S2$ with a 6d wave function $\psi (x1,x2)$ depending on two 3d Euclidean space coordinates $x1$ and $x2$ of the form

• $\psi (x1,x2) =C \exp(-Z\vert x1\vert )\exp (-Z\vert x2\vert )$,       (1)

with $Z =2$ the kernel charge, and $C$ a normalising constant. This describes two identical spherically symmetric electron distributions as solution of a reduced Schrödinger equation without electronic repulsion potential, with a total energy $E =-4$, way off the observed $-2.903$. 

To handle this discrepancy between model and observation the following corrections in the computation of total energy are made, while keeping the spherically symmetric form (1) of the ground state as the solution of a reduced Schrödinger equation:  

1 . Including Coulomb repulsion energy of (1) gives  $E=-2.75$.

2. Changing the kernel attraction to $Z=2 -5/16$ claiming screening gives $E=-2.85$.

3. Changing Coulomb repulsion by inflating the wave function to depend on $\vert x1-x2\vert$ can give  at best $E=-2.903724...$ to be compared with precise observation according to Nist atomic data base $-2.903385$ thus with an relative error of $0.0001$. Here the dependence on $\vert x1-x2\vert$ of the inflated wave function upon integration with respect to $x2$ reduces to a dependence on only the modulus of $x1$. Thus the inflated non spherically symmetric wave function can be argued to anyway represent two spherically symmetric electronic distributions.

We see that a spherically symmetric ground state of the form (1) is attributed to have correct energy, by suitably modifying the computation of energy so as to give perfect fit with observation. This kind of physics has been very successful and convincing (in particular to physicists), but it may be that it should be subject to critical scientific scrutiny.

The ideal in any case is a model with a solution which ab initio in direct computation has correct energy, not a  model with a solutions which has correct energy only if the computation of energy is changed by some ad hoc trick until match.

The effect of the fix according to 3. is to introduce a correlation between the two electrons to the effect that they would tend appear on opposite sides of the kernel, thus avoiding close contact. Such an effect can be introduced by angular weighting in (1) which can reduce electron repulsion energy but at the expense of increasing kinetic energy by angular variation of wave functions with global support and then seemingly without sufficient net effect. With the local support of the wave functions meeting with a homogeneous Neumann condition (more or less vanishing kinetic energy) of the new model, such an increase of kinetic energy is not present and a good match with observation is obtained.

New Quantum Mechanics 17: The Nightmare of Multi-Dimensional Schrödinger Equation

Once Schrödinger had formulated his equation for the Hydrogen atom with one electron and with great satisfaction observed an amazing correspondence to experimental data, he faced the problem of generalising his equation to atoms with many electrons.

The basic problem was the generalisation of the Laplacian to the case of many electrons and here Schrödinger took the easy route (in the third out of Four Lectures on Wave Mechanics delivered at the Royal Institution in 1928) of a formal generalisation introducing a set of new independent space coordinates and associated Laplacian for each new electron, thus ending up with a wave function $\psi (x1,...,xN)$ for an atom with $N$ electrons depending on $N$ 3d spatial coordinates $x1$,...,$xN$.

Already Helium with a Schrödinger equation in 6 spatial dimensions then posed a severe computational problem, which Schrödinger did not attempt to solve.  With a resolution of $10^2$ for each coordinate an atom with $N$ electrons then gives a discrete problem with $10^{6N}$ unknowns, which already for Neon with $N=10$ is bigger that the total number of atoms in the universe.

The easy generalisation thus came with the severe side-effect of giving a computationally hopeless problem, and thus from scientific point meaningless model.

To handle the absurdity of the $3N$ dimensions rescue steps were then taken by Hartree and Fock to reduce the dimensionality by restricting wave functions to be linear combinations of products of one-electron wave functions $\psi_j(xj)$ with global support:

• $\psi_1(x1)\times\psi_2(x2)\times ....\times\psi_N(xN)$    

to be solved computationally by iterating over the one-electron wave functions. The dimensionality was further reduced by ad hoc postulating that only fully symmetric or anti-symmetric wave functions (in the variables $(x1,...,xN)$) would describe physics adding ad hoc a Pauli Exclusion Principle on the way to help the case. But the dimensionality was still large and to get results in correspondence with observations required ad hoc trial and error choice of one-electron wave functions in Hartree-Fock computations setting the standard.

We thus see an easy generalisation into many dimensions followed by a very troublesome rescue operation stepping back from the many dimensions. It would seem more rational to not give in to the temptation of easy generalisation, and in this sequence of posts we explore such a route.

PS In the second of the Four Lectures Schrödinger argues against an atom model in terms of charge density by comparing with the known Maxwell's equations for electromagnetics in terms of electromagnetic fields, which works so amazingly well, with the prospect of a model in terms of energies, which is not known to work.

New Quantum Mechanics 16: Relation to Hartree and Hartree-Fock

The standard computational form of the quantum mechanics of an atom with N electrons (Hartree or Hartree-Fock) seeks solutions to the standard multi-dimensional Schrödinger equation as linear combinations of wave functions $\psi (x1,x2,...,xN)$ depending on $N$ 3d space coordinates $x1$,...,$xN$ as a product:

• $\psi (x1,x2,...,xN)=\psi_1(x1)\times\psi_2(x2)\times ....\times\psi_N(x_N)$ 

where the $\psi_j$ are globally defined electronic wave functions depending on a single space coordinate $xj$.

The new model takes the form of a non-standard free boundary Schrödinger equation in a wave function $\psi (x)$ as a sum:

• $\psi (x)=\psi_1(x)+\psi_2(x)+....+\psi_N(x)$,

where the $\psi_j(x)$ are electronic wave functions with local support on a common partition of 3d space with common space coordinate $x$.

The difference between the new model and Hartree/Hartree-Fock is evident and profound.  A big trouble with electronic wave functions having global support is that they overlap and demand an exclusion principle and new physics of exchange energy.  The wave functions of the new model do not overlap and there is no need of any exclusion principle or exchange energy.

PS Standard quantum mechanics comes with new forms of energy such as exchange energy and correlation energy. Here correlation energy is simply the difference between experimental total energy and total energy computed with Hartree-Fock and thus is not a physical form of energy as suggested by the name, simply a computational /modeling error.

New Quantum Mechanics 15: Relation to "Atoms in Molecules"

Atoms in Molecules developed by Richard Bader is a charge density theory based on basins of attraction of atomic kernels with boundaries characterised by vanishing normal derivative of charge density.

This connects to the homogeneous Neumann boundary condition identifying separation between electrons of the new model under study in this sequence of posts.

Atoms in Molecules is focussed on the role of atomic kernels in molecules, while the new model primarily (so far) concerns electrons in atoms.

New Quantum Mechanics 14: $H^-$ Ion

Below are results for the $H^-$ ion with two electrons and a proton. The ground state energy comes out as -0.514, slightly below the energy -0.5 of $H$, which means that $H$ is slightly electro-negative and thus by acquiring an electron into $H^-$ may react with $H^+$ to form $H2$ (with ground state energy -1.17), as one possible route to formation of $H2$. Another route is covered in this post with two H atoms being attracted to form a covalent bond.

The two electron wave functions of $H^-$ occupy half-spherical domains (depicted in red and blue) and meet at a plane with a homogeneous Neumann condition satisfied on both sides.

New Quantum Mechanics 13: The Trouble with Standard QM

Standard quantum mechanics of atom is based on the eigen functions of the Schrödinger equation for a Hydrogen atom with one electron, named "orbitals" being the elements of the Aufbau or build of many-electron atoms in the form of s, p, d and f orbitals of increasing complexity, see below.

These "orbitals" have global support and has led to the firm conviction that all electrons must have global support and so have to be viewed to always be everywhere and nowhere at the same time (as a basic mystery of qm beyond conception of human minds). To handle this strange situation Pauli felt forced to introduce his exclusion principle, while strongly regretting to ever have come up with such an idea, even in his Nobel Lecture:

• Already in my original paper I stressed the circumstance that I was unable to give a logical reason for the exclusion principle or to deduce it from more general assumptions. 
• I had always the feeling and I still have it today, that this is a deficiency. 
• Of course in the beginning I hoped that the new quantum mechanics, with the help of which it was possible to deduce so many half-empirical formal rules in use at that time, will also rigorously deduce the exclusion principle. 
• Instead of it there was for electrons still an exclusion: not of particular states any longer, but of whole classes of states, namely the exclusion of all classes different from the antisymmetrical one. 
• The impression that the shadow of some incompleteness fell here on the bright light of success of the new quantum mechanics seems to me unavoidable. 

In my model electrons have local support and occupy different regions of space and thus have physical presence. Besides the model seems to fit with observations. It may be that this is the way it is.

The trouble with (modern) physics is largely the trouble with standard QM, the rest of the trouble being caused by Einstein's relativity theory. Here is recent evidence of the crisis of modern physics:
The LHC "nightmare scenario" has come true.

Here is a catalogue of "orbitals" believed to form the Aufbau of atoms:

And here is the Aufbau of the periodic table, which is filled with ad hoc rules (Pauli, Madelung, Hund,..) and exceptions from these rules:

 

New Quantum Mechanics 12: H2 Non Bonded

Here are results for two hydrogen atoms forming an H2 molecule at kernel distance R = 1.4 at minimal total energy of -1.17 and a non-bonded molecule for larger distance approaching full separation for R larger than 6-10 at a total energy of -1. The results fit quite well with table data listed below.

The computations were made (on an iPad) in cylindrical coordinates in rotational symmetry around molecule axis on a mesh of 2 x 400 along the axis and 100 in the radial direction. The electrons are separated by a plane perpendicular to the axis through the the molecule center, with a homogeneous Neumann boundary condition for each electron half space Schrödinger equation. The electronic potentials are computed by solving a Poisson equation in full space for each electron.

PS To capture energy approach to -1 as R becomes large, in particular the (delicate) $R^{-6}$ dependence of the van der Waal force, requires a (second order) perturbation analysis, which is beyond the scope of the basic model under study with $R^{-1}$ dependence of kernel and electronic potential energies.

%TABLE II. Born–Oppenheimer total, E

%Relativistic energies of the ground state of the hydrogen molecule

%L. Wolniewicz

%Citation: J. Chem. Phys. 99, 1851 (1993); 

for two hydrogen atoms separated by a distance R bohr

 R    energy

0.20 2.197803500 

0.30 0.619241793 

0.40 -0.120230242 

0.50 -0.526638671 

0.60 -0.769635353 

0.80 -1.020056603 

0.90 -1.083643180 

1.00 -1.124539664 

1.10 -1.150057316 

1.20 -1.164935195 

1.30 -1.172347104 

1.35 -1.173963683 

1.40 -1.174475671 

1.45 -1.174057029 

1.50 -1.172855038 

1.60 -1.168583333 

1.70 -1.162458688 

1.80 -1.155068699 

2.00 -1.138132919 

2.20 -1.120132079 

2.40 -1.102422568 

2.60 -1.085791199 

2.80 -1.070683196 

3.00 -1.057326233 

3.20 -1.045799627 

3.40 -1.036075361 

3.60 -1.028046276 

3.80 -1.021549766 

4.00 -1.016390228 

4.20 -1.012359938 

4.40 -1.009256497 

4.60 -1.006895204 

4.80 -1.005115986 

5.00 -1.003785643 

5.20 -1.002796804 

5.40 -1.002065047 

5.60 -1.001525243 

5.80 -1.001127874 

6.00 -1.000835702 

6.20 -1.000620961 

6.40 -1.000463077 

6.60 -1.000346878 

6.80 -1.000261213 

7.00 -1.000197911 

7.20 -1.000150992 

7.40 -1.000116086 

7.60 -1.000090001 

7.80 -1.000070408 

8.00 -1.000055603 

8.50 -1.000032170 

9.00 -1.000019780 

9.50 -1.000012855

10.00 -1.000008754 

11.00 -1.000004506 

12.00 -1.000002546

New Quantum Mechanics 11: Helium Mystery Resolved

The modern physics of quantum mechanics born in 1926 was a towering success for the Hydrogen atom with one electron, but already Helium with two electrons posed difficulties, which have never been resolved (to be true).

The result is that prominent physicists always pride themselves by stating that quantum mechanics cannot be understood, only be followed to the benefit of humanity, like a religion:

• I think I can safely say that nobody understands quantum mechanics. (Richard Feynman, in The Character of Physical Law (1965))

Text books and tables list the ground state of Helium as $1S^2$ with two spherically symmetric electrons (the S) with opposite spin in a first shell (the 1), named parahelium.  The energy of a $1S^2$ state according to basic quantum  theory is equal to -2.75 (Hartree), while the observation of ground state energy  is -2.903. To handle this apparent collapse of basic quantum theory, the computation of energy is changed by introducing a suitable perturbation away from spherical symmetry which delivers the wanted result of -2.903, while maintaining that the ground state still is $1S^2$.

Of course, this does not make sense, but since quantum mechanics is not "anschaulich" or  "visualisable" (as required by Schrödinger) and therefore cannot be understood by humans, this is not a big deal.  By a suitable perturbation the desired result can be reached, and we are not allowed to ask any further questions following the dictate of Dirac: Shut up and calculate.

New Quantum Mechanics resolves the situation as follows:

The ground state is predicted to be a spherically (half-)symmetric continuous electron charge distribution with each electron occupying a half-space, and the electrons meeting on at plane (free boundary) where the normal derivative for each electron charge distribution vanishes. The result of ground state energy computations according to earlier posts shows close agreement with the observed -2.903:

Notice the asymmetric electron potential and the resulting slightly asymmetric charge distribution with polar accumulation. The model shows a non-standard electron configuration, which may be the true one (if there is anything like that).