Matrices from Spherical Harmonics with Eigenvalue l+1

[paklin2]

TL;DR Summary

I’m curious if these matrices have been discussed elsewhere and if they have any significance. They may give some insight into how the different Spherical Harmonics interact because of the differential operators.

I’m New to the forum. 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.

$\sigma r = \begin{pmatrix}1&0\\0&1\\\end{pmatrix}, ~\sigma z = \begin{pmatrix}1&0\\ 0&-1\\\end{pmatrix}, ~\sigma x = \begin{pmatrix}0&1\\ 1&0\\\end{pmatrix}, ~\sigma y = \begin{pmatrix}0&i\\ -i&0\\\end{pmatrix}$,

So $r = x \cdot \sigma x + y \cdot \sigma y + z \cdot \sigma z$ which is the same as $\begin{pmatrix}z&x+yi\\ x-yi&-z\\\end{pmatrix}$

From the following differential operators $L = \frac {1} {i} \vec r \times \nabla$ with $Lz = (x \frac {\partial } {\partial y}- y \frac {\partial } {\partial x}),~Lx = (y \frac {\partial } {\partial z}- z \frac {\partial } {\partial y}),~Ly = (z \frac {\partial } {\partial x}- x \frac {\partial } {\partial z})$ a useful operator can be defined; $Lxyz = Lx \cdot \sigma x + Ly \cdot \sigma y + Lz \cdot \sigma z$ or $Lxyz = \begin{pmatrix} (x \frac {\partial } {\partial y}- y \frac {\partial } {\partial x})&(y \frac {\partial } {\partial z}- z \frac {\partial } {\partial y})+(z \frac {\partial } {\partial x}- x \frac {\partial } {\partial z})i\\ (y \frac {\partial } {\partial z}- z \frac {\partial } {\partial y})-(z \frac {\partial } {\partial x}- x \frac {\partial } {\partial z})i&-(x \frac {\partial } {\partial y}- y \frac {\partial } {\partial x})\\ \end{pmatrix}$
If the following matrix is multiplied by the differential operator Lxyz
$Rlm = \begin{pmatrix} m \cdot Y^m_l(x,y,z) &\sqrt {( l-m) \cdot (l+1+m)}\cdot Y^{m+1}_l(x,y,z) \\ \sqrt {(l+m) \cdot (l+1-m)} Y^{m-1}_l(x,y,z) &-m \cdot Y^m_l(x,y,z) \\ \end{pmatrix}\\ +l \cdot Y^m_l(x,y,z)$
then this matrix acts like an eigenvector with Eigen value l + 1.
The $Y^m_l(x,y,z)$ are spherical Harmomics.

See en.wikipedia.org/wiki/Table_of_spherical_harmonics

Also Quantum Mechanics, Albert Messiah, Volume 1, Appendix B, Article 10
Here are some comments about Rlm;:

The vector from Lxyz, $(y \cdot \sigma x - x \cdot \sigma y )$, associated with $\frac {\partial } {\partial z}$ is perpendicular to the z axis. Similarly the other two vectors associated with $\frac {\partial } {\partial x}$ and $\frac {\partial } {\partial y}$ are perpendicular to the x vector and to the y vector. Taking the sum of the three vectors contained in Lxyz that are associated with partials but excluding the partials defines $LDelta = (y \cdot \sigma x - x \cdot \sigma y ) + (z \cdot \sigma y - y \cdot \sigma z ) + (x \cdot \sigma z - z \cdot \sigma x )$. Taking the LDelta dot product with the x, y, z vector, $\vec r$, gives $(y \cdot \sigma x - x \cdot \sigma y )z\cdot (\sigma z)^2 +(z \cdot \sigma y - y \cdot \sigma z )x\cdot (\sigma x)^2 +(x \cdot \sigma z - z \cdot \sigma x )y\cdot (\sigma y)^2$ which is zero so that LDelta is perpendicular to , $\vec r$. LDelta is also perpendicular to the x = y = z vector. LDelta$\cdot \Delta \theta$ with x, y, and z as specific functions of $\theta$ can generate R10($\theta$) that’s similar to $e^{i\theta}$. The LDelta path could define a circle around the x = y = z vector but maybe not for more than R10($\theta$) which is similar to $e^{i\theta}$
For certain Rlm in order for $Lxyz\cdot Rlm – (l+1)\cdot Rlm$ to be zero it must include the factor $(r^2- z^2 –x^2 – y^2)$ where the factor determines the difference is zero rather than zero itself determining it. A Legendre Polynomial is evident when $(x^2 + y^2)$ is replaced by $(r^2- z^2 )$.

Also significant is that the square of $(y \cdot \sigma x - x \cdot \sigma y )$ turns out to be $x^2 + y^2$. This is because the xy cross terms in $x^2 + y^2$.cancel with this math.

Multiplying the sum of squares by Lxyz gives zero suggesting $\vec r$ is a constant

 

[paklin2]

Most of the things I mentioned were things I came across my self like the Rlm matrices. I understood there were likely previous discussions about them. I learned about spherical harmonics in college.

It looked like there could be a single variable like $\theta$ that could generate a path with Lxyz based on partials multiplied what could be considered infinitesimal. For a circle going around thr x=y=z axis used $x= xo \cdot cos(\theta) +xo\cdot sin(\theta))$, $y= xo \cdot cos(\theta) -xo\cdot sin(\theta))$, and $z= -2 \cdot xo \cdot cos(\theta)$. For the R10($\theta$) matrix things like the hydrogen atom with different energy states or spin ½ are suggested?
For $R10(\theta) \cdot\sigma z$ that’s the same as the x,y,z vector, $R10(\theta) \cdot\sigma z$ can be expressed as a single variable.

With $I = \begin{pmatrix}-\frac {i} {\sqrt 3}&\frac {1} {\sqrt 3} -\frac {1} {\sqrt 3} i\\-\frac {1} {\sqrt 3} -\frac {1} {\sqrt 3}i&\frac {i} {\sqrt 3} \end{pmatrix}$ then

$\begin{pmatrix} 1&-\frac {1} {2} -\frac {1} {2} i\\-\frac {1} {2} +\frac {1} {2} \cdot& -1 \end{pmatrix} \frac {-1} {\sqrt {2\pi}}e^{I\theta}- R10(\theta) = \begin{pmatrix} 0&0\\0& 0 \end{pmatrix}$
Doing this may not be possible for l larger than 1.
Wonder if everyone can see the Latex code correctly. I attempted a preview from someone else’s post that didn’t work with the latex code.

 

[Haborix]

I think the reason you aren't getting a response is it remains unclear exactly what you are asking. You start out by asking if "these matrices" have been discussed elsewhere, but I'm unclear which matrices are "these matrices". Your posts remind me of my early drafts when I writeup calculations (and these drafts are usually understood only by me). Could you rephrase your question with more specificity?

Your Latex is rendering correctly. And if you want to use subscripts "_{subscript here}" will do it.

 

[paklin2]

Thanks for the advice. 'These matrices' are the Rlm matrices with different integer values for l and m. It did sound reasonable to me at first but I see what you're saying.

 

[paklin2]

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.

 

[vanhees71]

I do not understand what you are after, but the relation between the fundamental representation of the SU(2) and rotations SO(3). Is provided by the mapping of a usual 3D Euclidean vector $\vec{x}$ to the operator $\vec{x} \mapsto \hat{\vec{\sigma}} \cdot \vec{x}$. It is easy to show explicitly that with an $\text{SU}(2)$ representing a rotation around an axis $\vec{n}$, $\hat{D}(\varphi)=\exp(-\mathrm{i} \hat{\vec{\sigma}} \cdot \vec{n}/2)$ the transformation
$$\hat{\vec{\sigma}} \cdot \vec{x} \mapsto \hat{D}(\varphi) \hat{\vec{\sigma}} \cdot \vec{x} \hat{D}^{\dagger}(\varphi) = \hat{\vec{\sigma}} \cdot \vec{x}'$$
means a rotation $\vec{x}'=\hat{R}(\varphi) \vec{x}$ with $\hat{R} \in \mathrm{SO}(3)$.

For details (also the extension to the Lorentz group), see

R. U. Sexl and H. K. Urbantke, Relativity, Groups, Particles,
Springer, Wien (2001).

 

[paklin2]

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?

 

[paklin2]

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 give $Lz = (\Delta x \frac {\partial } {\partial y}- \Delta y \frac {\partial } {\partial x}),~Lx = (\Delta y \frac {\partial } {\partial z}- \Delta z \frac {\partial } {\partial y}),~Ly = (\Delta z \frac {\partial } {\partial x}- \Delta x \frac {\partial } {\partial z})$ and applied to $Rlm(x,y,z) = Xlm(x,y,z) \sigma x+ Ylm(x,y,z) \sigma y + Zlm(x,y,z) \sigma z$ then they appear to be trying to add the deltas to x, y, z in Rlm(x,y,z) to advance Rlm(x,y,z) along a curve.

Applying $\frac {Lz} {i} = \frac{ (x \frac {\partial } {\partial y}- y \frac {\partial } {\partial x})} {i}$ to $x+ iy$ to give $x+ iy$ demonstrates how these operators are able to generate a path in this case $e^{i\theta}$. In this case Lz ends up with a single variable $theta$. This was possible because $\frac {dx} {d \theta} = -y$ and $\frac {dy} {d \theta} = x$ for $x = cos(\theta)$ and $y = sin(\theta)$ Then $\frac {Lz (x+iy )} {i} = \frac{ (\frac {dy} {d \theta} \frac {\partial (x+iy) } {\partial y}+\frac {dx} {d \theta} \frac {\partial (x+iy)} {\partial x})} {i} = \frac{ \frac {d(x + iy)} {d \theta}} {i}$ . Both Lz and $\frac {d} {d \theta}$ give the same result which may only be for simple Rlm.

 

[vanhees71]

I'm still puzzled, what you want to do, but for sure your operators are not the usual ones. In QT one works with self-adjoint rather than anti-self-adjoint generators of groups, beause it's more natural, i.e., you have
$$\hat{\vec{L}}=\hat{\vec{x}} \times \hat{\vec{p}} = - \mathrm{i} \hbar \vec{x} \times \vec{\nabla}$$
in the position representation.

To treat spherical harmonics in Cartesian coordinates, the introduction of the complex variables $u=x + \mathrm{i} y$ and $v=x-\mathrm{i} y$ (treated as independent variables!) is convenient.

 

[paklin2]

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. I still wonder if it's still possible to represent x, y, and z with a single variable or path Rlm's with these operators.
I'm going to have to catch up on things like Lorentz groups.

 

[vanhees71]

I'm sorry, I still don't understand the question. Are you talking about the symmetries of the (non-relativistic) hydrogen atom, which is $\mathrm{SO}(4)$ for the negative-energy (bound) states, the Galilei group for $E=0$, and $\mathrm{SO}(1,3)$ for the positive-energy (scattering) states?

In that case you find a good treatment in Thirring's textbook on theoretical physics.

https://link.springer.com/book/10.1007/978-3-662-05008-8

 

[paklin2]

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

 

[paklin2]

Thought I'd say something about the R20 matrix.

Multiplying R20 times $\sigma z$ as a preference:

$R20 \cdot \sigma z = \frac {\sqrt 6} {r^2} { \sqrt \frac{15} {8 \cdot \pi}}(z x\cdot \sigma x+z y \cdot \sigma y + (z^2- \frac {r^2} {3}) \cdot \sigma z) = (X20 \cdot \sigma x + Y20 \cdot \sigma y + Z20 \cdot \sigma z)$

With $Lz = (x \frac {\partial } {\partial y}- y \frac {\partial } {\partial x}),~Lx = (y \frac {\partial } {\partial z}- z \frac {\partial } {\partial y}),~Ly = (z \frac {\partial } {\partial x}- x \frac {\partial } {\partial z})$ defining $Lxy = Lx \cdot \sigma x + Ly \cdot \sigma y$ and multiplying Lxy by $(Z20 \cdot \sigma z)$ gives a vector proportional to $(zx \cdot \sigma x+ z y \cdot \sigma y)$ for $(X20 \cdot \sigma x + Y20 \cdot \sigma y)$. This can represent the delta vector $z( \Delta x\cdot \sigma x+ \Delta y \cdot \sigma y)$ perpendicular to the z axis that along with Lyx, and Lzx generate a circle around some axis likely the x = y = z axis.

Multiplying Lxy by $(X20 \cdot \sigma x + Y20 \cdot \sigma y)$ gives a vector proportional to $-(\frac {x^2++y^2 -2z^2} {3}) \cdot \sigma z = (z^2- \frac {r^2} {3}) \cdot \sigma z$ for $Z20 \cdot \sigma z$. The new deltas could represent deltas of deltas that generate a circle and therefore have to do with the curvature of the circle being generated. The deltas seem to be $\Delta ((\Delta y-\Delta z)-( \Delta z-\Delta x)) \cdot \sigma z$ that’s in the right direction, the z direction. These deltas of deltas result in a second degree differential equation for the Legendre Polynomial

 

[paklin2]

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 the whole expression that follows a partial, say for a partial of x for example, may have to be to of the form $\frac {dx} {d \theta}$ in order for x, y, and z to be a function of $\theta$. Lxyz contains $\frac {\partial R} {\partial x} (z \cdot \sigma y – y \cdot \sigma z)$ suggesting $(z \cdot \sigma y – y \cdot \sigma z) = \frac {dx} {d \theta}$ so x would be a vector. This may not be a problem for R10 since there’s only one variable in any vector. Otherwise $dx( \theta, \phi) = \frac {\partial x( \theta, \phi)} {\partial \theta} d \theta + \frac {\partial x( \theta, \phi)} {\partial \phi} d \phi$ . This says $dx = (z \cdot \sigma y – y \cdot \sigma z) \cdot d\theta$ has to be ignored and $\frac {\partial x( \theta, \phi)} {\partial \theta} d \theta + \frac {\partial x( \theta, \phi)} {\partial \phi} d \phi$ has to be considered if possible. Does requiring the use of two independent variables, $\theta$ and $\phi$, suggest the uncertainty principle.

 

[PeterDonis]

Here’s something that seems to agree with the criticism that there are problems with SU(2).

I have no idea what you mean by "problems". SU(2) is a group--a mathematial entity. How can it have "problems"?

As for the rest of your post, it is just as unclear what question you are asking as your previous ones were. It also is not at all clear that you have the requisite background for an "A" level thread on this topic.

So I am closing this thread since it's pointless to continue it if nobody else can tell what your issue is or what question you are asking. I suggest that you take some time to think carefully about exactly what your question is, and how you can formulate it concisely so that others can understand it, before you start another thread.