Hartree method for Helium ground state

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After long preparation, I am ready to do this problem.

The two electron in the helium ground state occupy same spacial orbital but difference spin. Thus, the total wavefunction is

\displaystyle \Psi(x,y) = \frac{1}{\sqrt{2}}(\uparrow \downarrow - \downarrow \uparrow) \psi(x) \psi(y)

Since the Coulomb potential is spin-independent, the Hartree-Fock method reduce to Hartree method. The Hartree operator is

F(x) = H(x) + \langle \psi(y)|G(x,y) |\psi(y) \rangle

where the single-particle Hamiltonian and mutual interaction are

\displaystyle H(x) = -\frac{\hbar^2}{2m} \nabla^2 - \frac{Ze^2}{4\pi\epsilon_0 x} = -\frac{1}{2}\nabla^2 - \frac{Z}{x}

\displaystyle G(x,y) = \frac{e^2}{4\pi\epsilon_0|x-y|} = \frac{1}{|x-y|}

In the last step, we use atomic unit, such that \hbar = 1, m=1, e^2 = 4\pi\epsilon_0. And the energy is in unit of Hartree, 1 \textrm{H} = 27.2114 \textrm{eV}.

We are going to use Hydrogen-like orbital as a basis set.

\displaystyle b_i(r) = R_{nl}(r)Y_{lm}(\Omega) \\= \sqrt{\frac{(n-l-1)!Z}{n^2(n+l)!}}e^{-\frac{Z}{n}r} \left( \frac{2Z}{n}r \right)^{l+1} L_{n-l-1}^{2l+1}\left( \frac{2Z}{n} r \right) \frac{1}{r} Y_{lm}(\Omega)

I like the left the 1/r, because in the integration \int b^2 r^2 dr, the r^2 can be cancelled. Also, the i = nlm is a compact index of the orbital.

Using basis set expansion, we need to calculate the matrix elements of

\displaystyle H_{ij}=\langle b_i(x) |H(x)|b_j(x)\rangle = -\delta \frac{Z^2}{2n^2}

\displaystyle G_{ij}^{hk} = \langle b_i(x) b_h(y) |G(x,y) |b_j(x) b_k(y) \rangle

Now, we will concentrate on evaluate the mutual interaction integral.

Using the well-known expansion,

\displaystyle G(x,y) = \frac{1}{|x-y|}=\frac{1}{r_{12}} = \sum_{l=0}^{\infty} \sum_{m=-l}^{l} \frac{4\pi}{2l+1} \frac{r_<^l}{r_>^{l+1}} Y_{lm}^{*}(\Omega_1)Y_{lm}(\Omega_2)

The angular integral

\displaystyle \langle Y_i(x) Y_h(y)| Y_{lm}^{*}(x) Y_{lm}(y) | Y_j(x) Y_k(y) \rangle \\ = \big( \int Y_i^{*}(x) Y_{lm}^{*}(x) Y_j(x) dx \big) \big( \int Y_h^{*}(y) Y_{lm}(y) Y_k(y) dy \big)

where the integral \int dx = \int_{0}^{\pi} \int_{0}^{2\pi} \sin(\theta_x) d\theta_x d\phi_x .

From this post, the triplet integral of spherical harmonic is easy to compute.

\displaystyle \int Y_h^{*}(y) Y_{lm}(y) Y_k(y) dy = \sqrt{\frac{(2l+1)(2l_k+1)}{4\pi (2l_h+1)}} C_{l0l_k0}^{l_h0} C_{lm l_km_k}^{l_hm_h}

The Clebsch-Gordon coefficient imposed a restriction on l,m.

The radial part,

\displaystyle \langle R_i(x) R_h(y)| \frac{r_<^l}{r_>^{l+1}} | R_j(x) R_k(y) \rangle \\ = \int_0^{\infty} \int_{0}^{\infty} R_i(x) R_h(y) \frac{r_<^l}{r_>^{l+1}} R_j(x) R_k(y) y^2 x^2 dy dx \\ = \int_0^{\infty} R_i(x) R_j(x) \\ \left( \int_{0}^{x} R_h(y) R_k(y) \frac{y^l}{x^{l+1}} y^2dy  + \int_{x}^{\infty} R_h(x)R_k(x) \frac{x^l}{y^{l+1}}  y^2 dy   \right) x^2 dx

The algebraic calculation of the integral is complicated, but after the restriction of l from the Clebsch-Gordon coefficient, only few terms need to be calculated.

The general consideration is done. now, we use the first 2 even states as a basis set.

\displaystyle b_{1s}(r) = R_{10}(r)Y_{00}(\Omega) = 2Z^{3/2}e^{-Zr}Y_{00}(\Omega)

\displaystyle b_{2s}(r) = R_{20}(r)Y_{00}(\Omega) = \frac{1}{\sqrt{8}}Z^{3/2}(2-Zr)e^{-Zr/2}Y_{00}(\Omega)

These are both s-state orbital. Thus, the Clebsch-Gordon coefficient

\displaystyle C_{lm l_k m_k}^{l_h m_h} = C_{lm00}^{00}

The radial sum only has 1 term. And the mutual interaction becomes

\displaystyle G(x,y) = \frac{1}{|x-y|}=\frac{1}{r_{12}} = 4\pi \frac{1}{r_>} Y_{00}^{*}(\Omega_1)Y_{00}(\Omega_2)

The angular part

\displaystyle \langle Y_i(x) Y_h(y)| Y_{lm}^{*}(x) Y_{lm}(y) | Y_j(x) Y_k(y) \rangle = \frac{1}{4\pi}

Thus, the mutual interaction energy is

G_{ij}^{hk} = \displaystyle \langle b_i(x) b_h(y) |G(x,y) |b_j(x) b_k(y) \rangle = \langle R_i(x) R_h(y)| \frac{1}{r_>} | R_j(x) R_k(y) \rangle

The radial part

G_{ij}^{hk} = \displaystyle \langle R_i(x) R_h(y)| \frac{1}{r_>} | R_j(x) R_k(y) \rangle \\ \begin{pmatrix} G_{11}^{hk} & G_{12}^{hk} \\ G_{21}^{hk} & G_{22}^{hk} \end{pmatrix} = \begin{pmatrix} \begin{pmatrix} G_{11}^{11} & G_{11}^{12} \\ G_{11}^{21} & G_{11}^{22} \end{pmatrix} & \begin{pmatrix} G_{12}^{11} & G_{12}^{12} \\ G_{12}^{21} & G_{12}^{22} \end{pmatrix} \\ \begin{pmatrix} G_{21}^{11} & G_{21}^{12} \\ G_{21}^{21} & G_{21}^{22} \end{pmatrix} & \begin{pmatrix} G_{22}^{11} & G_{22}^{12} \\ G_{22}^{21} & G_{22}^{22} \end{pmatrix} \end{pmatrix} \\= \begin{pmatrix} \begin{pmatrix} 1.25 & 0.17871 \\ 0.17871 & 0.419753 \end{pmatrix} & \begin{pmatrix} 0.17871 & 0.0438957 \\ 0.0439857 & 0.0171633 \end{pmatrix} \\ \begin{pmatrix} 0.17871 & 0.0438957 \\ 0.0438957 & 0.0171633 \end{pmatrix} & \begin{pmatrix} 0.419753 & 0.0171633 \\ 0.0171633 & 0.300781 \end{pmatrix} \end{pmatrix}

We can easy to see that G_{ij}^{hk} = G_{ji}^{hk} = G_{ij}^{kh} = G_{hk}^{ij} = G_{ji}^{kh} . Thus, if we flatten the matrix of matrix, it is Hermitian, or symmetric.

Now, we can start doing the Hartree method.

The general solution of the wave function is

\psi(x) = a_1 b_{1s}(x) + a_2 b_{2s}(x)

The Hartree matrix is

F_{ij} = H_{ij} + \sum_{h,k} a_h a_k G_{ij}^{hk}

The first trial wave function are the Hydrogen-like orbital,

\psi^{(0)}(x) = b_{1s}(r)

F_{ij}^{(0)} = \begin{pmatrix} -2 & 0 \\ 0 & -0.5 \end{pmatrix}  + \begin{pmatrix} 1.25 & 0.17871 \\ 0.17817 & 0.419753 \end{pmatrix}

Solve for eigen system, we have the energy after 1st trial,

\epsilon^{(1)} = -0.794702 , (a_1^{(1)}, a_2^{(1)}) = (-0.970112, 0.242659)

After 13th trial,

\epsilon^{(13)} = -0.880049 , (a_1^{(13)}, a_2^{(13)}) = (-0.981015, 0.193931)

F_{ij}^{(13)} = \begin{pmatrix} -2 & 0 \\ 0 & -0.5 \end{pmatrix}  + \begin{pmatrix} 1.15078 & 0.155932 \\ 0.155932 & 0.408748 \end{pmatrix}

Thus, the mixing of the 2s state is only 3.7%.

Since the eigen energy contains the 1-body energy and 2-body energy. So, the total energy for 2 electrons is

E_2 = 2 * \epsilon^{(13)} - G = -2.82364 \textrm{H} = -76.835 \textrm{eV}

In which ,

G = \langle \psi(x) \psi(y) |G(x,y) |\psi(x) \psi(y) \rangle = 1.06354 \textrm{H} = 28.9403 \textrm{eV}

So the energies for

From He to He++.  E_2 = -2.82364 \textrm{H} = -76.835 \textrm{eV}
From He+ to He++, E_1^+ = -Z^2/2 = 2 \textrm{H} = -54.422 \textrm{eV} .
From He to He+, is E_1 = E_2 - E_1^+ = -0.823635 \textrm{H} =  -22.4123 \textrm{eV}

The experimental 1 electron ionization energy for Helium atom is

E_1(exp) = -0.90357 \textrm{H} = -24.587 \textrm{eV}
E_1^+(exp) = -1.99982 \textrm{H} = -54.418 \textrm{eV}
E_2(exp) = -2.90339 \textrm{H} = -79.005 \textrm{eV}

The difference with experimental value is 2.175 eV. The following plot shows the Coulomb potential, the screening due to the existence of the other electron, the resultant mean field, the energy, and r \psi(x)


Usually, the Hartree method will under estimate the energy, because it neglected the correlation, for example, pairing and spin dependence. In our calculation, the E_2 energy is under estimated.

From the (a_1^{(13)}, a_2^{(13)}) = (-0.981015, 0.193931) , we can see, the mutual interaction between 1s and 2s state is attractive. While the interaction between 1s-1s and 2s-2s states are repulsive. The repulsive can be easily understood. But I am not sure how to explain the attractive between 1s-2s state.

Since the mass correction and the fine structure correction is in order of 10^{-3} \textrm{eV} , so the missing 0.2 eV should be due to something else, for example, the incomplete basis set.

If the basis set only contain the 1s orbit, the mutual interaction is 1.25 Hartree = 34.014 eV. Thus, the mixing reduce the interaction by 5.07 eV, just for 3.7% mixing

I included the 3s state,

\epsilon^{(13)} = -0.888475 , (a_1^{(13)}, a_2^{(13)}, a_3^{(13)}) = (0.981096, -0.181995, -0.06579)

The mutual energy is further reduced to 1.05415 Hartree = 28.6848 eV. The E_2 = -77.038 \textrm{eV} . If 4s orbital included, the E_2 = -77.1058 \textrm{eV} . We can expect, if more orbital in included, the E_2 will approach to E_2(exp).



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).


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.

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}


\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}


\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}


\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


\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}


Wigner-Eckart theorem

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The mathematical form of the theorem is, given a tensor operator of rank k, T^{(k)}, The expectation value on the eigen-state \left|j,m\right> of total angular momentum J is,

\left<j m\right|T_q^{(k)} \left|j' m'\right> = \left<j' m' k q| j m \right> \left<j||T^{(k)}||j'\right>

where, \left<j||T^{(k)}||j\right> is reduced matrix element. The power of the theorem is that, once the reduced matrix element is calculated for the system for a particular (may be the simplest) case, all other matrix element can be calculated.

The theorem works only in spherical symmetry. The state are eigen-state of total angular momentum. We can imagine, when the system rotated, there is something unchanged (which is the reduced matrix element). The quantum numbers m, m' define some particular direction of the state, and these “direction” will cause an additional factor, which is the Clebsch-Gordan coefficient.

Another application is the Replacement theorem.

If any 2 spherical tensors A^{(k)}, B^{(k)} of rank-k, using the theorem, we have,

\displaystyle \left<j m|A^{(k)}|j' m' \right> = \frac{\left<j||A^{(k)}||j'\right>}{\left<j||B^{(k)}||j'\right>} \left<j m|B^{(k)}|j' m' \right>

This can prove the Projection theorem, which is about rank-1 tensor.

L , J are orbital and total angular momentum respectively. The projection of L on  J is

L\cdot J = L_z J_z - L_+ J_- - L_-J_+

The expectation value with same state \left|j m\right> ,

\left< L\cdot J\right> = \left< L_z J_z\right> - \left< L_+ J_-\right> - \left<L_- J_+\right>

using Wigner-Eckart theorem, the right side becomes,

\left< L \cdot J \right> = c_j \left<j||L||j\right>

where the coefficient c_j only depends on j as the dot-product is a scalar, which is isotropic. similarly,

\left< J \cdot J \right> = c_j \left<j||J||j\right> ,

Using the Replacement theorem,

\displaystyle \left< L \right> = \frac{\left<j||L||j\right>}{\left<j||J||j\right>} \left<J \right>

Thus, we have,

\displaystyle \left< L \right> = \frac{\left< L\cdot J \right>}{\left<J\cdot J\right>} \left<J \right>

as the state is arbitrary,

\displaystyle L = \frac{L\cdot J}{J\cdot J} J

this is same as the classical vector projection.



Angular distribution of emission that carry angular momentum

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Before decay, the nucleus is in state with total angular momentum J and symmetry axis quantization M :


Say, the emitted radiation (can be EM wave or particle ) carries angular momentum  l and axis quantization m, its wavefunction is:


then the daughter nucleus has angular momentum j and m_j, the wave function is

\Psi_{j m_j}

their relation is:

\Phi_{JM} = \sum_{m, m_j}{\phi_{lm} \Psi_{j m_j} \left< l m j m_j | JM \right>}

where the $Latex \left< l m j m_j | JM \right> $ is Clebsch-Gordan coefficient.

The wave function of the emitted radiation from a central interaction takes the form:

\phi_{lm} = A_0 u_{nl}(r) Y_l^m(\theta,\phi)

The angular distribution is:

\int |\phi_{lm}|^2 dx^3 =A_0^2 \int |u_{nl}|^2 r^2 dr |Y_l^m|^2

for a fixed distance detector, the radial part is a constant. Moreover, not every spherical harmonic contribute the same weight, there is weighting factor due to Clebsch-Gordan coefficient.   Thus, the angular distribution is proportional to:

W(\theta) \propto \sum_{m_j=M-m}{|Y_l^m|^2 |\left<l m j m_j |JM\right>|^2 }

For example, JM=00, The possible (l, j) are (0,0), (1,1), (2,2) and so on, the m=-m_j . The C-G coefficient are,

\langle 0000 | 00 \rangle = 1

\langle lml-m | 00 \rangle = \frac{1}{\sqrt{2l+1}}



Y_0^0 = \frac{1}{4\pi}

\displaystyle \sum_{m}{\left|Y_l^m\right|^2  \left|\langle l m l -m |0 0 \rangle \right|^2 } =  \sum_m|Y_l^m|^2 \frac{1}{2l+1}= constant

Thus, the angular distribution is isotropic.

Clebsch – Gordan Coefficient II

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As last post discussed, finding to CG coefficient is not as straight forward as text book said by recursion.
However, there are another way around, which is by diagonalization of J^2

first we use the identity:

J^2 = J_1^2+J_2^2 + 2 J_{1z} J_{2_z} + J_{1+} J_{2-} + J_{1-} J_{2+}

when we “matrix-lize” the operator. we have 2 choice of basis. one is \left| j_1,m_1;j_2;m_2 \right> , which give you non-diagonal matrix by the J_{\pm} terms. another one is \left|j,m\right>, which give you a diagonal matrix.

Thus, we have 2 matrixs, and we can diagonalized the non-diagonal. and we have the Unitary transform P, from the 2-j basis to j basis, and that is our CG coefficient.

oh, don’t forget the normalized the Unitary matrix.

i found this one is much easy to compute.

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