Since the CG coefficient is already normalized.Thus

Since the number of is , as . Thus,

At last, the number of dimension of the coupled space or (tensor product space) is equation to , i.e.

Thus,

an adventure in Nuclear Physics!

May 16, 2018

Basic Clebsch-Gordon Leave a comment

Since the CG coefficient is already normalized.Thus

Since the number of is , as . Thus,

At last, the number of dimension of the coupled space or (tensor product space) is equation to , i.e.

Thus,

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May 16, 2018

Basic 3-j Symbol, 6-j Symbol, 9-j Symbol, Clebsch-Gordon Leave a comment

The meaning of 3-j symbol is same as Clebsch-Gordan coefficient. So, we skip in here.

I am not going to construct the 6-j symbol from 3-j symbol. In here, I just state the meaning and usage in Mathematica.

The 6-j symbol is the coupling between 3 angular momenta, .

There are 2 ways to couple these 3 angular momenta. First,

the other way is

The 6-j symbol is

We can see that there are 4 vector-sum must satisfy.

If we draw a line to connect these 4 vector-sum, we have:

In Mathematica, there is a build in function

The 9-j symbol is the coupling between 4 angular momenta, .

The 9-j symbol can be used in coupling 2 nucleons, .

The 9-j symbol is

We can see, each row and column must satisfy the vector-sum.

Unfortunately, there is no build in function in Mathematica. The formula for 9-j symbol is

Where sum all possible value, which can be calculate using the 6 couplings inside the 3 6-j symbols.To check your result, the coupling between and to from a state is

October 17, 2017

atomic, Basic, Math Clebsch-Gordon, Eigen System, Hartree-Fock, helium, hydrogen, Laguerre Leave a comment

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

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

where the single-particle Hamiltonian and mutual interaction are

In the last step, we use atomic unit, such that . And the energy is in unit of Hartree, .

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

I like the left the , because in the integration , the can be cancelled. Also, the is a compact index of the orbital.

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

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

Using the well-known expansion,

The angular integral

where the integral .

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

The Clebsch-Gordon coefficient imposed a restriction on .

The radial part,

The algebraic calculation of the integral is complicated, but after the restriction of 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.

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

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

The angular part

Thus, the mutual interaction energy is

The radial part

We can easy to see that . 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

The Hartree matrix is

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

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

After 13th trial,

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

In which ,

So the energies for

From He to He++.

From He+ to He++, .

From He to He+, is

The experimental 1 electron ionization energy for Helium atom is

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

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

From the , 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 , 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,

The mutual energy is further reduced to 1.05415 Hartree = 28.6848 eV. The . If 4s orbital included, the . We can expect, if more orbital in included, the will approach to .

August 15, 2017

Basic, review angular momentum, Clebsch-Gordon, Eigen System, Euler, Fourier transform, Laplacian, rotation, spherical harmonic, Wigner-D Leave a comment

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

and eigenvector

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

- still in the space,
- exist of unit and inverse of addition,
- the norm can be defined on a suitable domain by

For example, the two functions , the rotation can be done by a rotational matrix,

And, the product also from a basis. And the rotation on this new basis was induced from the original rotation.

where . The space becomes “3-dimensional” because , otherwise, it will becomes “4-dimensional”.

The 2-D function can also be expressed in polar coordinate, , and further decomposed into .

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 , the rotation of the function is simply multiplied by the rotation matrix.

The eigen function is

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

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

for example,

write in polar coordinate

where .

The rotation is

If we write the rotated function in Cartesian form,

where .

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

The spherical harmonics serves as the basis for eigenvalue of , eigen spaces for difference are orthogonal. This is an extension of the 2-D eigen function .

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 , which is the primary base that contains only .

For , the space has , but the linear combination is unchanged after rotation. Thus, the size of the space reduced .

For , the space has , this time, the linear combinations behave like and behave like , thus the size of the space reduce to .

For higher order, the total combination of is , and we can find repeated combinations, thus the size of the irreducible space of order is always 2.

For 3-D space, the size of combination of is . We can find repeated combination, thus, the size of the irreducible space of order is always .

August 8, 2017

Basic Clebsch-Gordon, rotation, spherical harmonic, Wigner-D Leave a comment

One mistake I made is that

because

but this application is wrong.

The main reason is that, the is “living” in a tensor product space, while 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

We can set and

Thus,

Now, recall the Clebsch-Gordon series,

set

rename some labels

We can multiply both side by and sum over , using

August 8, 2017

Basic Clebsch-Gordon, rotation, Wigner-D Leave a comment

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

If acting the rotation operator to the , we insert

If acting the rotation operator to the , we insert

Thus,

We can multiply both side by , then sum the

using

June 22, 2017

Basic Clebsch-Gordon, Eigen System, Wigner-Eckart Leave a comment

The mathematical form of the theorem is, given a tensor operator of rank , , The expectation value on the eigen-state of total angular momentum is,

where, 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 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 of rank-k, using the theorem, we have,

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

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

The expectation value with same state ,

using Wigner-Eckart theorem, the right side becomes,

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

,

Using the Replacement theorem,

Thus, we have,

as the state is arbitrary,

this is same as the classical vector projection.