Recent content by paklin2

A slight correction needed. For something like ##\Delta x = y \cdot \Delta \theta##, x and y will change moving around the circumference in a single move but ##\Delta \theta## should stay constant for a single move around the circumference..

Here is a summary. I’m Interested if a certain set of matrices have any significance. To start out the unit vectors ##\vec i , \vec j, and ~\vec k ## are replaced with two dimensional matrices. ##\sigma r = \begin{pmatrix}1&0\\0&1\\\end{pmatrix}, ~\sigma z = \begin{pmatrix}1&0\\...

It appears that most likely the paths determined by Lxyz, at least those from Rlm with ##m =0##, are circles. With ##Lxyz \cdot Rlm - (l+1) Rlm = f(z) \cdot (x^2 +y^2 +z^2 – r^2)## often, ## x^2+ y^2 +z^2 - r^2## is required to be zero and this establishes that the path that Lxyz determines is a...

Here’s an argument that x, y, and z may not be a function of a single variable, ##\theta##. Often ##Lxyz \cdot Rlm - (l+1) Rlm = f(z) \cdot (x^2 +y^2 +z^2 – r^2)##. Lxyz doesn’t determine that this equation is zero by itself since ##( x^2+ y^2 +z^2 – r^2)## is determined to be zero by other...

I have to make a correction here. Circular paths are formed from ##cos(\theta)## and sin(##\theta##) in three dimensions here, but as far as I know there’s also paths formed from cos(n##\theta##) and sin(n##\theta##) here using Lxyz that are not circular but there’s still only one variable...

As far as I can determine the infinitesimals for the paths generated here by Lxyz are added to x, y, and z by the partials but are not influenced by the partials themselves. They appear to be circular paths. Except for R10 most of the Rlms if not all with l larger than 1 still generate circular...

Not sure how this got in there ## Lxyz\cdot Rlm = \begin{pmatrix} (x \frac {\partial Rlm } {\partial y}- y \frac {\partial Rlm } {\partial x})&(y \frac {\partial Rlm } {\partial z}- z \frac {\partial Rlm } {\partial y})+(z \frac {\partial Rlm } {\partial x}- x \frac {\partial Rlm } {\partial z})...

See the first post in the previous thread ‘Matrices from Spherical Harmonics with Eigenvalue l+1’ first. Originally when I came across the Lxyz operator and the Rlm matrices I had a different question. If this had to do with something like the quantum Hydrogen atom then why did it appear that...

Here’s something that seems to agree with the criticism that there are problems with SU(2). This may have to do with most Rlm’s seeming not being able to generate a path with x, y, and z as functions of a single variable ##\theta##. This contradicts a circle around an axis. One concern is that...

Thought I'd say something about the R20 matrix. Multiplying R20 times ##\sigma z## as a preference: ##R20 \cdot \sigma z = \frac {\sqrt 6} {r^2} { \sqrt \frac{15} {8 \cdot \pi}}(z x\cdot \sigma x+z y \cdot \sigma y + (z^2- \frac {r^2} {3}) \cdot \sigma z) = (X20 \cdot \sigma x + Y20 \cdot...

Thanks for the suggestions and information about SO(4),

I came across these operators like this by somehow factoring the Schroedinger Equation which also has a similar single Eigen Value as Rlm does, but may have lost the details of how I did this. This gave some credence to Rlm suggesting the Hydrogen atom and it's energy states if r is constant...

For ##Lz = (x \frac {\partial } {\partial y}- y \frac {\partial } {\partial x}),~Lx = (y \frac {\partial } {\partial z}- z \frac {\partial } {\partial y}),~Ly = (z \frac {\partial } {\partial x}- x \frac {\partial } {\partial z})## if x,y,z are replaced by ##\Delta x, \Delta y. and \Delta z## to...

I wondered if SU(2) would be more natural then vector algebra in some sense to simulate force and motion. The Rlm matrices with Lxyz seemed a good candidate for rotation. The Rlm matrices could replace normal orbits but this may only be possible only for a few of them like R10?

The xo in ##x= xo \cdot cos(\theta) +xo\cdot sin(\theta))##, ##y= xo \cdot cos(\theta) -xo\cdot sin(\theta))##, and ##z= -2 \cdot xo \cdot cos(\theta) ## is ##\frac {r} {\sqrt 6}##. The r’s cancel in the spherical harmonics in Rlm.