Review on rotation

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The rotation of a vector in a vector space can be done by either rotating the basis vector or the coordinate of the vector. Here, we always use fixed basis for rotation.

For a rigid body, its rotation can be accomplished using Euler rotation, or rotation around an axis.

Whenever a transform preserves the norm of the vector, it is a unitary transform. Rotation preserves the norm and it is a unitary transform, can it can be represented by a unitary matrix. As a unitary matrix, the eigen states are an convenient basis for the vector space.

We will start from 2-D space. Within the 2-D space, we discuss about rotation started by vector and then function. The vector function does not explicitly discussed, but it was touched when discussing on functions. In the course, the eigen state is a key concept, as it is a convenient basis. We skipped the discussion for 3-D space, the connection between 2-D and 3-D space was already discussed in previous post. At the end, we take about direct product space.


In 2-D space. A 2-D vector is rotated by a transform R, and the representation matrix of R has eigen value

\exp(\pm i \omega)

and eigenvector

\displaystyle \hat{e}_\pm = \mp \frac{ \hat{e}_x \pm i \hat{e}_y}{\sqrt{2}}

If all vector expand as a linear combination of the eigen vector, then the rotation can be done by simply multiplying the eigen value.

Now, for a 2-D function, the rotation is done by changing of coordinate. However, The functional space is also a vector space, such that

  1. a* f_1 + b* f_2 still in the space,
  2. exist of  unit and inverse of addition,
  3. the norm can be defined on a suitable domain by \int |f(x,y)|^2 dxdy

For example, the two functions \phi_1(x,y) = x, \phi_2(x,y) = y , the rotation can be done by a rotational matrix,

\displaystyle R = \begin{pmatrix} \cos(\omega) & -\sin(\omega) \\ \sin(\omega) & \cos(\omega) \end{pmatrix}

And, the product x^2, y^2, xy also from a basis. And the rotation on this new basis was induced from the original rotation.

\displaystyle R_2 = \begin{pmatrix} c^2 & s^2 & -2cs \\ s^2 & c^2 & 2cs \\ cs & -cs & c^2 - s^2 \end{pmatrix}

where c = \cos(\omega), s = \sin(\omega) . The space becomes “3-dimensional” because xy = yx, otherwise, it will becomes “4-dimensional”.

The 2-D function can also be expressed in polar coordinate, f(r, \theta) , and further decomposed into g(r) h(\theta) .


How can we find the eigen function for the angular part?

One way is using an operator that commutes with rotation, so that the eigen function of the operator is also the eigen function of the rotation. an example is the Laplacian.

The eigen function for the 2-D Lapacian is the Fourier series.

Therefore, if we can express the function into a polynomial of r^n (\exp(i n \theta)  , \exp(-i n \theta)) , the rotation of the function is simply multiplied by the rotation matrix.

The eigen function is

\displaystyle \phi_{nm}(\theta) = e^{i m \theta}, m = \pm

The D-matrix of rotation (D for Darstellung, representation in German)  \omega is

D^n_{mm'}(\omega) = \delta_{mm'} e^{i m \omega}

The delta function of m, m' indicates that a rotation does not mix the spaces. The transformation of the eigen function is

\displaystyle \phi_{nm}(\theta') = \sum_{nm} \phi_{nm'}(\theta) D^n_{m'm}(\omega)

for example,

f(x,y) = x^2 + k y^2

write in polar coordinate

\displaystyle f(r, \theta) = r^2 (\cos^2(\theta) + k \sin^2(\theta)) = \frac{r^2}{4} \sum_{nm} a_{nm} \phi_{nm}(\theta)

where a_0 = 2 + 2k, a_{2+} = a_{2-} = 1-a, a_{other} = 0.

The rotation is

\displaystyle f(r, \theta' = \theta + \omega ) = \frac{r^2}{4} \sum_{nm} a_{nm} \phi_{nm}(\theta) D^n_{mm}(\omega)  = \frac{r^2}{4} \sum_{nm} a_{nm} \phi_{nm}(\theta + \omega)

If we write the rotated function in Cartesian form,

f(x',y') = x'^2 + k y'^2 = (c^2 + k s^2)x^2 + (s^2 + k c^2)y^2 + 2(k-1) c s x y

where c = \cos(\omega), s = \sin(\omega) .


In 3-D space, the same logic still applicable.

The spherical harmonics Y_{lm} serves as the basis for eigenvalue of l(l+1), eigen spaces for difference l are orthogonal. This is an extension of the 2-D eigen function \exp(\pm n i \theta) .

A 3-D function can be expressed in spherical harmonics, and the rotation is simple multiplied with the Wigner D-matrix.


On above, we show an example of higher order rotation induced by product space. I called it the induced space (I am not sure it is the correct name or not), because the space is the same, but the order is higher.

For two particles system, the direct product space is formed by the product of the basis from two distinct space (could be identical space).

Capture.PNG

Some common direct product spaces are

  • combining two spins
  • combining two orbital angular momentum
  • two particles system

No matter induced space or direct product space, there structure are very similar. In 3-D rotation, the two spaces and the direct product space is related by the Clebsch-Gordon coefficient. While in 2-D rotation, we can see from the above discussion, the coefficient is simply 1.

Lets use 2-D space to show the “induced product” space. For order n=1, which is the primary base that contains only x, y.

For n=2, the space has x^2, y^2, xy, but the linear combination x^2 + y^2 is unchanged after rotation. Thus, the size of the space reduced 3-1 = 2.

For n = 3, the space has x^3, y^3, x^2y, xy^3 , this time, the linear combinations x^3 + xy^2 = x(x^2+y^2) behave like x and y^3 + x^2y behave like y, thus the size of the space reduce to 4 - 2 = 2.

For higher order, the total combination of x^ay^b, a+b = n is C^{n+1}_1 = n+1 , and we can find n-1 repeated combinations, thus the size of the irreducible space of order n is always 2.

For 3-D space, the size of combination of x^ay^bz^c, a + b+ c = n is C^{n+2}_2 = (n+1)(n+2)/2 . We can find n(n-1)/2 repeated combination, thus, the size of the irreducible  space of order n is always 2n+1.

Spherical Harmonics and Platonic Solids

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The Platonic solids has the most symmetric for discrete rotation in 3-D space. They are symmetry in all the faces, lines, and vertexes. The Spherical harmonics also has discrete rotational symmetry on the z-axis for all order. However, for small order, the spherical harmonics also has other symmetric axis.

For example,

\displaystyle Y_{32}(\theta, \phi) = \sqrt{\frac{105}{32\pi}} \cos\theta \sin^2\theta \exp(2i\phi)

The spherical plot and spherical contour plot for 1 + Y_{32} are

Capture2.PNG

The plot clearly shows the tetrahedron’s symmetry. In fact, the tetrahedron, hexahedron (cubic), and octahedron share the same symmetry.

The tetrahedron inscribe itself.

The hexahedron and octahedron inscribe each other.

The dodecaherdo and icosahedron also inscribe each other.


We can systematically construct the tetrahedron, hexahedron (octahedron), and dodecahedron.

The tetrahedron has 120 degree rotation symmetry around a vertex. And we can guess, Y_{33} should be a suitable candidate. Thus, we can form the tetrahedron from

f(\theta, \phi) = Y_{30}(\theta,\phi) + Y_{33}(\theta, \phi)

However, at angle \tan(\theta/2) = \sqrt{2}, \phi = \pi/3 , the value of f is less then \theta = \phi = 0 .  To restore the symmetry,

\displaystyle f_t(\theta, \phi) = Y_{30}(\theta,\phi) + \sqrt{\frac{8}{5}} Y_{33}(\theta, \phi)

The plots for f_t are show below.

Capture3.PNG

Similarly, we can from the octahedron using Y_{4m} with a coefficient.

\displaystyle f_h(\theta, \phi) = Y_{40}(\theta,\phi) + \sqrt{\frac{10}{7}} Y_{44}(\theta, \phi)

hexahedron.PNG

For dodecahedron, the use of Y_{5,5} seem to be the logical choice. However, it turns out it cannot has the correct symmetry because Y_{50} is not “even”.

\displaystyle f_d(\theta, \phi) = Y_{60}(\theta,\phi) + \sqrt{\frac{28}{11}} Y_{65}(\theta, \phi)

dodecahedron.PNG


The Mathematica code for generating the graph is

GraphicsGrid[{{
 SphericalPlot3D[1 + f(θ,φ), {θ,  0, π}, {φ, 0, 2 π}],
 ParametricPlot3D[{Cos[φ] Sin[θ], Sin[φ] Sin[θ], Cos[θ]}, {θ,  0, π}, {φ, 0, 2 π}, 
ColorFunction -> Function[{x, y, z, u, v},  Hue[Abs[1 + f(u,v)]]], 
ColorFunctionScaling -> False, Mesh -> None, PlotPoints -> 200]
}}, ImageSize -> 600]

 

The code for calculating the coefficient is

Solve[
SphericalHarmonicY[n, 0, k, 0] + a SphericalHarmonicY[n, n, k, 0] 
== SphericalHarmonicY[n, 0, 0, 0]
, a]

 

Where n is the order of the spherical harmonic, k is the angle for the next face or vertex for fixed \theta = 0.

Tetraherdon k = 2 \tan^{-1}(\sqrt{2}) .

Octaherdon k = \pi/2.

Dodecahedron k = 2 \tan^{-1}( (1+\sqrt{5})/2) .

Spherical Harmonics and Fourier Series

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Recently, I read a very interesting article on the origin of spherical harmonics. I like to summarize in here and add some personal comments.


Starting from Laplace equation

\nabla^2 \phi(\vec{r}) = 0

The Laplacian can be separated into radial and spherical part.

\nabla^2 = \nabla_r^2 + \nabla_\Omega^2

The solution is called harmonics, and it can be separated into radial part and angular part too,

\phi(\vec{r}) = R(r) \Theta(\Omega)

Since the Laplacian is coordinate-free, therefore, the solution is also coordinate free and is rotational invariant. We will come back to this point later.


A homogeneous function of degree n has property,

f(t\vec{r}) = t^n f(\vec{r})

In the case of homogeneous harmonics of degree n,

\phi_n(\vec{r}) = r^n \Theta_n(\Omega)

Here, the radial part is R_n(r) = r^n

Substitute this homogeneous harmonics into the Laplace equation, the \nabla_r^2 will produce a coefficient related to the order, and the radial part can be extracted.

0 = f(r) ( \nabla_\Omega^2 - g(n) ) \Theta(\Omega)

we have an eigenvalue problem for the angular part

 \nabla_\Omega^2 \Theta = g(n) \Theta

The eigen function for 2-D Laplacian is the Fourier Series, and that for 3-D is the Spherical Harmonics. In this sense, Fourier Series is a “polar harmonics”.


In 3-D, the angular part of the Lapacian is proportional to the angular momentum operator, -\hbar^2 \nabla_\Omega^2 = L^2 , where \hbar is the reduced Planck constant, which has the dimension of angular momentum.

L^2 Y_{lm}(\theta, \phi) = l(l+1) \hbar^2 Y_{lm}(\theta, \phi)

Here, from the previous discussion, before we solve the equation, we know that the harmonic has maximum order of l . The m is the degeneracy for same eigenvalue l(l+1)

As we mentioned before, the harmonics should be rotational invariant, such that any direction should be equal. However, when we look at the Spherical Harmonics, the poles are clearly two special points and the rotation around the “z-axis” has limited rotational symmetry with degree l. How come?

According to the article, the solution is not necessarily to be separated into \theta, \phi, such that

\displaystyle Y_{lm}(\theta,\phi) = \sqrt{\frac{2l+1}{4\pi} \frac{(l-m)!}{(l+m)!}}P_{lm}(\cos\theta) e^{im\phi}

I quote the original,

“It is not immediately obvious that we can separate variables and assume exponential functions in the φ direction. We are able to do this essentially because the lines of fixed θ are circles. We could also simply assume this form and show the construction succeeds. This organization is not forced, but separating the variables is so useful that there are no competitive options. A disadvantage of this organization is that it makes the poles into special points.”

The limited rotational symmetry with degree of l is due to the limited “band-width” that restricted by the order of the homogeneous function. The relation between the band width and the order of the harmonics can be understood that the number of “sector” or “node” on the circle/sphere is proportional to the order, thus, the “resolution” is also limited by the order and thus the “band-width”.

Since the Platonic solid is coordinate-free that they are the most symmetry. In the next post, I will show the relation between Spherical Harmonics and Platonic solid. This is related to the section 3.2 in the article,

“One would like to have an uniform discretization for the sphere, with all portions equally represented. From such an uniform discretization we could construct a platonic solid. It is known, however, that there are only a few platonic solids, and the largest number of faces is 20 (icosohedron) and largest number of vertices is 20 (dodecahedron). If we want to discretize the sphere with many points, we cannot do it uniformly. Instead we set the goal of using the fewest points to resolve the Spherical Harmonics up to some degree. Since the Spherical Harmonics themselves are “fair” and “uniform”, this gives a good representation for functions on the sphere. “


As the Fourier Series and Spherical Harmonic are closely related, they should share many properties. For instant, they are orthonormal and form a basis. This leads to the Discrete Fourier Transform and also the “Spherical Transform”,

\displaystyle f(\Omega) = \sum_{\alpha} a_\alpha \Theta_\alpha(\Omega)

where \alpha is the id of the basis. One can use the Parseval theorem,

\displaystyle \int |f(\Omega)|^2 d\Omega = \sum_{\alpha} a_\alpha^2

Also, the convolution using discrete Fourier transform can also be applied on the spherical harmonics.

Notice that, the Discrete Fourier Transform can “translate” to Continuous Fourier Transform. However, the order of the spherical harmonics is always discrete.

 

Integration formulas of spherical harmonic

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There are several important and useful formulas for the integration of spherical harmonic. We simplify the notation,

\displaystyle \int_0^{\pi} \sin(\theta)d\theta\int_0^{2\pi}d\phi  = \int d\Omega


The first one is the average of spherical harmonic.

\displaystyle \int Y_{lm} d\Omega = \sqrt{4\pi} \delta_{l0}\delta_{m0}

The 2nd one is the orthonormal  condition.

\displaystyle \int Y^{*}_{l'm'}Y_{lm} d\Omega = \delta_{l'l}\delta_{m'm}

The 3rd one is triplet integral, we use the product of spherical harmonic,

\displaystyle \int Y_{l_1m_1}Y_{l_2m_2} Y^*_{l_3m_3} d\Omega \\ = \int \sum_{lm} \sqrt{\frac{(2l_1+1)(2l_2+1)}{4\pi(2l+1)}} C_{l_10l_20}^{l0} C_{l_1m_1l_2m_2}^{lm} Y_{lm} Y^*_{l_3m_3} d\Omega \\= \sqrt{\frac{(2l_1+1)(2l_2+1)}{4\pi(2l_3+1)}} C_{l_10l_20}^{l_30} C_{l_1m_1l_2m_2}^{l_3m_3}

The 4th one is another triple integral,

\displaystyle \int Y_{l_1m_1}Y_{l_2m_2} Y_{l_3m_3} d\Omega  \\ = \int \sum_{lm} \sqrt{\frac{(2l_1+1)(2l_2+1)}{4\pi(2l+1)}} C_{l_10l_20}^{l0} C_{l_1m_1l_2m_2}^{lm} Y_{lm} Y_{l_3m_3} d\Omega  \\ = \int \sum_{lmLM} \sqrt{\frac{(2l_1+1)(2l_2+1)}{4\pi(2l+1)}} C_{l_10l_20}^{l0} C_{l_1m_1l_2m_2}^{lm} \\ \sqrt{\frac{(2l+1)(2l_3+1)}{4\pi(2L+1)}} C_{l0l_30}^{L0} C_{lml_3m_3}^{LM}Y_{LM}d\Omega

\displaystyle = \sum_{lm} \sqrt{\frac{(2l_1+1)(2l_2+1)(2l_3+1)}{4\pi}} C_{l_10l_20}^{l0} C_{l_1m_1l_2m_2}^{lm} C_{l0l_30}^{00} C_{lml_3m_3}^{00}

Notice that

C_{lmLM}^{00} = (-1)^{L+M} \sqrt{\frac{1}{2L+1}} \delta_{Ll}\delta_{-m,M}

\displaystyle \int Y_{l_1m_1}Y_{l_2m_2} Y_{l_3m_3} d\Omega = \sqrt{\frac{(2l_1+1)(2l_2+1)}{4\pi (2l_3+1)}} C_{l_10l_20}^{l_30} C_{l_1m_1l_2m_2}^{l_3,-m_3} (-1)^{m_3}

using Wigner 3-j symbol,

C_{l_1m_1l_2m_2}^{l_3m_3} = (-1)^{l_1-l_2+m_3} \sqrt{2l_3+1} \begin{pmatrix} l_1 & l_2 & l_3 \\ m_1 & m_2 & -m_3 \end{pmatrix}

\displaystyle \int Y_{l_1m_1}Y_{l_2m_2} Y_{l_3m_3} d\Omega \\= \sqrt{\frac{(2l_1+1)(2l_2+1)(2l_3+1)}{4\pi}} \begin{pmatrix} l_1 & l_2 & l_3 \\ 0 & 0 & 0 \end{pmatrix} \begin{pmatrix} l_1 & l_2 & l_3 \\ m_1 & m_2 & m_3 \end{pmatrix} 

For other integral, we can use

Y^*_{lm}(\theta, \phi) = (-1)^{m}Y_{lm}(\theta,\phi) = Y_{lm}(\theta, -\phi)

 

Product of Spherical Harmonics

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One mistake I made is that

\displaystyle Y_{LM} = \sum_{m_1 m_2} C_{j_1m_1j_2 m_2}^{LM} Y_{j_1m_1} Y_{j_2m_2}

because

\displaystyle |j_1j_2JM\rangle = \sum_{m_1m_2} C_{j_1m_1j_2 m_2}^{LM} |j_1m_1\rangle |j_2m_2\rangle

but this application is wrong.

The main reason is that, the |j_1j_2JM\rangle is “living” in a tensor product space, while |jm \rangle is living in ordinary space.

We can also see that, the norm of left side is 1, but the norm of the right side is not.


Using the Clebsch-Gordon series, we can deduce the product of spherical harmonics.

First, we need to know the relationship between the Wigner D-matrix and spherical harmonics. Using the equation

\displaystyle Y_{lm}(R(\hat{r})) = \sum_{m'} Y_{lm'}(\hat{r}) D_{m'm}^{l}(R)

We can set \hat{r} = \hat{z} and R(\hat{x}) = \hat{r}

Y_{lm}(\hat{z}) = Y_{lm}(0, 0) = \sqrt{\frac{2l+1}{4\pi}} \delta_{m0}

Thus,

\displaystyle Y_{lm}(\hat{r}) = \sqrt{\frac{2l+1}{4\pi}} D_{0m}^{l}(R)

\Rightarrow D_{0m}^{l} = \sqrt{\frac{4\pi}{2l+1}} Y_{lm}(\hat{r})

Now, recall the Clebsch-Gordon series,

\displaystyle D_{m_1N_1}^{j_1} D_{m_2 N_2}^{j_2} = \sum_{jm} \sum_{M} C_{j_1m_1j_2m_2}^{jM} C_{j_1N_1j_2N_2}^{jm} D_{Mm}^{j}

set m_1 = m_2 = M= 0

\displaystyle D_{0N_1}^{j_1} D_{0 N_2}^{j_2} = \sum_{jm} C_{j_10j_20}^{j0} C_{j_1N_1j_2N_2}^{jm} D_{0m}^{j}

rename some labels

\displaystyle Y_{l_1m_1} Y_{l_2m_2} = \sum_{lm} \sqrt{\frac{(2l_1+1)(2l_2+1)}{4\pi(2l+1)}} C_{l_10l_20}^{l0} C_{l_1m_1l_2m_2}^{lm} Y_{lm}


We can multiply both side by C_{l_1m_1l_2m_2}^{LM} and sum over m_1, m_2,  using

\displaystyle \sum_{m_1m_2} C_{l_1m_1l_2m_2}^{lm}C_{l_1m_1l_2m_2}^{LM} = \delta_{mM} \delta_{lL}

\displaystyle \sum_{m_1m_2} C_{l_1m_1l_2m_2}^{LM} Y_{l_1m_1} Y_{l_2m_2} = \sqrt{\frac{(2l_1+1)(2l_2+1)}{4\pi(2L+1)}} C_{l_10l_20}^{l0} Y_{LM}

 

 

Clebsch-Gordon Series

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One of the important identity for angular momentum theory is the Clebsch-Gordon series, that involved Wigner D-matrix.

The series is deduced from evaluate the follow quantity in two ways

\langle j_1 m_1 j_2 m_2 | U(R) |j m \rangle

If acting the rotation operator to the |jm\rangle , we insert

\displaystyle \sum_{M} |jM\rangle \langle | jM| = 1

\displaystyle \sum_{M} \langle j_1 m_1 j_2 m_2|jM\rangle \langle jM| U(R) |jm\rangle = \sum_{M} C_{j_1m_1j_2m_2}^{jM} D_{Mm}^{j}

If acting the rotation operator to the \langle j_1 m_1 j_2 m_2| , we insert

\displaystyle \sum_{N_1 N_2 } |j_1 N_1 j_2 N_2\rangle \langle  j_1 N_1 j_2 N_2| = 1

\displaystyle \sum_{N_1 N_2} \langle j_1 m_1 j_2 m_2|U(R) | j_1 N_1 j_2 N_2\rangle \langle j_1 N_1 j_2 N_2| jm\rangle

\displaystyle = \sum_{N_1N_2} C_{j_1N_1j_2N_2}^{jm} D_{m_1N_1}^{j_1} D_{m_2 N_2}^{j_2}

Thus,

\displaystyle \sum_{N_1N_2} C_{j_1N_1j_2N_2}^{jm} D_{m_1N_1}^{j_1} D_{m_2 N_2}^{j_2} = \sum_{M} C_{j_1m_1j_2m_2}^{jM} D_{Mm}^{j}

We can multiply both side by C_{j_1 N_1 j_2 N_2}^{jm} , then sum the j, m

using

\displaystyle \sum_{jm} C_{j_1 N_1 j_2 N_2}^{jm} C_{j_1N_1j_2N_2}^{jm} = 1

\displaystyle D_{m_1N_1}^{j_1} D_{m_2 N_2}^{j_2} = \sum_{jm} \sum_{M} C_{j_1m_1j_2m_2}^{jM} C_{j_1N_1j_2N_2}^{jm} D_{Mm}^{j}

 

Parseval theorem

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When a function f(x) can be expressed as a linear combination of a orthogonal basis \phi_n(x) , i.e.

\displaystyle f(x) = \sum_n a_n \phi_n(x)

\displaystyle \langle \phi_n|\phi_m \rangle = \delta_{nm}

then, the integration

\displaystyle \int |f(x)|^2 dx = \sum_n |a_n|^2 \langle \phi_n|\phi_n \rangle = \sum_n |a_n|^2

That is.

Using this theorem, many complicated integration can be calculated as a sum.

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