We assume that B-matrix chains which are a product of the numerical statistical theory called Cairo technique are capable of efficiently solving the SE PDE in 1D, 2D, 3D via two natural laws or principles:
* 1-simultaneity and 2- the Pythagorean theorem,
without going through SE itself.
We also assume that our fellow Researchgate contributors can provide an equivalent or even better answer independently.
Your suggestions are greatly appreciated.
This is just an introductory answer to shed more light on the question and its answer.
The Schrödinger equation with the Bohr/Copenhagen interpretation complicates QM quantum mechanics to the maximum with the result that any physicist or mathematician who claims to understand QM is called a liar! (in N. Bohr's own words).
On the other hand, nature itself describes quantum mechanics in the simplest and fastest way, at least to comply with the least action theorem.
The question is valid:
Can we find a simple geometric solution for Schrödinger PDE via the Pythagorean theorem plus the sponaeosity assumption without needing SE or any of its derivatives?
The answer is yes, as the following analysis explains:
We assume that B-matrix chains which are a product of the numerical statistical theory called Cairo technique are capable of efficiently solving the SE PDE in 1D, 2D, 3D via two natural laws or principles:
* simultaneity and the Pythagorean theorem,
without going through SE itself
This is consistent with the geometry of the Pythagorean triangle Fig.1 below.
**The unique angle bisector of an isosceles triangle divides the opposite side into two similar right triangles
which means that the right half of the psi wavefunction is symmetric with the left half
Fig.2
📷
Fig.2 Stationary field distribution for quantum particles in a 1D infinite potential well as proposed by the geometry of Fig.1 (maximum probability)
Fig.3 Stationary field distribution for quantum particles in a 1D infinite potential well as proposed by the geometry of Fig.1 (probability next to maximum)📷
📷
Fig.4 Stationary field distribution for quantum particles in a 1D infinite potential well taken from Google search for comparison with Fig.2
📷
- ----------1/2 ----- ----- 1/2-------
*right half of psi wavefunction symmetric with left halfFig.1 Pythagoras geometry for 1D quantum wavefunction psi
**The angle bisector of the single angle of an isosceles triangle divides the opposite side into two similar right-angled triangles
***Note that Ψ=√ Ψ^2=Energy density =E
Showing results for the right half of the psi wavefunction symmetric with the left half.
To be continued.
13.60 KB
PNG
PNG
Fig 4 QM 103.png
Fig 1 QM 1033.png
Fig.2 QM103.png
Fig 1 QM 103.png
- Badges
- Science method
- Similar topics
- Methodology
- Crystallography
- X-ray Diffraction