Clebsch-Gordan coefficient

The Clebsch-Gordan coefficients are used in the coupling of quantum mechanical angular momentum . These are expansion coefficients with which one goes from the basis of the individual angular momentum to the basis of the total angular momentum. They are used to calculate the spin-orbit coupling and in the isospin formalism.

They were named after Alfred Clebsch (1833–1872) and Paul Gordan (1837–1912). Instead of Clebsch-Gordan coefficients, one can also use the related 3j symbols according to Eugene Wigner .

Angular momentum coupling

• → see also the section on addition of angular momentum in the article angular momentum operator

One assumes two angular momenta and , which each have the quantum numbers and (z-component), or and . Here, take and following values: and : and the angular momentum exchange with each other (s. Quantum mechanical commutator ). This means that the individual angular momentum can be measured sharply independently of one another. Each of these angular momenta has its own eigenspace, which is spanned by the eigenvectors or . In the basis of these eigenvectors , the square of and a component of this operator has a diagonal shape. The same applies in an analogous manner to . ${\displaystyle {\vec {J}}_{1}}$ ${\displaystyle {\vec {J}}_{2}}$ ${\displaystyle j_{1}}$ ${\displaystyle m_{1}}$ ${\displaystyle j_{2}}$ ${\displaystyle m_{2}}$ ${\displaystyle m_{1}}$ ${\displaystyle m_{2}}$ ${\displaystyle m_{1}\in \{-j_{1},-j_{1}+1,\dots ,j_{1}\}}$ ${\displaystyle m_{2}\in \{-j_{2},-j_{2}+1,\dots ,j_{2}\}}$ ${\displaystyle [{\vec {J}}_{1},{\vec {J}}_{2}]=0}$ ${\displaystyle \left|j_{1},m_{1}\right\rangle }$ ${\displaystyle \left|j_{2},m_{2}\right\rangle }$ ${\displaystyle \left|j_{1},m_{1}\right\rangle }$ ${\displaystyle {\vec {J}}_{1}}$ ${\displaystyle {\vec {J}}_{2}}$

The individual angular momentum and now couple to a total angular momentum . Ie the individual components add vectorially. The eigen-states of the total angular momentum have the quantum numbers and . They can have the following values: ${\displaystyle {\vec {J}}_{1}}$ ${\displaystyle {\vec {J}}_{2}}$ ${\displaystyle {\vec {J}}={\vec {J}}_{1}+{\vec {J}}_{2}}$ ${\displaystyle J}$ ${\displaystyle M}$

${\displaystyle |j_{1}-j_{2}|\leq J\leq |j_{1}+j_{2}|}$and . ${\displaystyle M=[-J,-J+1,\dots ,J]}$

Since the total angular momentum consists of both angular momentum and , the states of the total angular momentum can be represented in the product space of the individual eigenstates: ${\displaystyle {\vec {J}}}$ ${\displaystyle {\vec {J}}_{1}}$ ${\displaystyle {\vec {J}}_{2}}$

${\displaystyle \left|j_{1},m_{1};j_{2},m_{2}\right\rangle =\left|j_{1},m_{1}\right\rangle \otimes |j_{2},m_{2}\rangle ,}$

where denotes the tensor product . ${\displaystyle \otimes }$

However, these states are generally not eigenvectors of the total angular momentum , so that it does not have a diagonal shape in this basis. ${\displaystyle {\vec {J}}}$

Eigen basis of the total angular momentum operator

The eigenvectors of will by the quantum numbers , , and clearly defined. With regard to the new basis of eigenvectors, the total angular momentum again has a simple diagonal shape. The following applies: ${\displaystyle {\vec {J}}}$ ${\displaystyle J}$ ${\displaystyle M}$ ${\displaystyle j_{1}}$ ${\displaystyle j_{2}}$ ${\displaystyle {\vec {J}}}$

${\displaystyle {\vec {J}}^{2}\left|J,M,j_{1},j_{2}\right\rangle =J(J+1)\hbar ^{2}\left|J,M,j_{1},j_{2}\right\rangle }$

${\displaystyle J_{z}\left|J,M,j_{1},j_{2}\right\rangle =M\hbar \left|J,M,j_{1},j_{2}\right\rangle }$

The Clebsch-Gordan coefficients now indicate the transition from the product basis to the eigen basis ( unitary transformation ): ${\displaystyle \left|j_{1},m_{1};j_{2},m_{2}\right\rangle }$ ${\displaystyle \left|J,M,j_{1},j_{2}\right\rangle }$

${\displaystyle \left|J,M,j_{1},j_{2}\right\rangle =\sum _{m_{1},m_{2}}\left|j_{1},m_{1};j_{2},m_{2}\right\rangle \langle \ j_{1},m_{1};j_{2},m_{2}|J,M,j_{1},j_{2}\rangle .}$

Where are the Clebsch-Gordan coefficients. ${\displaystyle \langle \ j_{1},m_{1};j_{2},m_{2}|J,M,j_{1},j_{2}\rangle }$

Properties of the Clebsch-Gordan coefficients

• The Clebsch-Gordan coefficients are equal to zero if either of the two conditions is met or not: ${\displaystyle |j_{1}-j_{2}|\leq J\leq j_{1}+j_{2}}$ ${\displaystyle M=m_{1}+m_{2}}$

${\displaystyle \langle j_{1},m_{1};j_{2},m_{2}|J,M,j_{1},j_{2}\rangle \neq 0\quad \Rightarrow \quad |j_{1}-j_{2}|\leq J\leq j_{1}+j_{2}\ \ \wedge \ \ M=m_{1}+m_{2}}$  ("Selection Rules").

• By convention, the Clebsch-Gordan coefficients are real:

${\displaystyle \langle j_{1},m_{1};j_{2},m_{2}|J,M,j_{1},j_{2}\rangle \in \mathbb {R} .}$

• The following Clebsch-Gordan coefficient is positive by convention: ${\displaystyle M=J}$

${\displaystyle \langle j_{1},j_{1};j_{2},J-j_{1}|J,J,j_{1},j_{2}\rangle >0.}$

• The amount of the Clebsch-Gordan coefficient zu is equal to the Clebsch-Gordan coefficient zu according to ${\displaystyle M}$ ${\displaystyle -M}$

${\displaystyle \langle j_{1},m_{1};j_{2},m_{2}|J,M,j_{1},j_{2}\rangle =(-1)^{j_{1}+j_{2}-J}\langle j_{1},-m_{1};j_{2},-m_{2}|J,-M,j_{1},j_{2}\rangle .}$

• The Clebsch-Gordan coefficients satisfy the orthogonality relation

${\displaystyle \sum _{m_{1},m_{2}}\langle j_{1},m_{1};j_{2},m_{2}|J,M,j_{1},j_{2}\rangle \langle j_{1},m_{1};j_{2},m_{2}|J',M',j_{1},j_{2}\rangle =\delta _{JJ'}\delta _{MM'}.}$

• The Clebsch-Gordan coefficients satisfy the orthogonality relation

${\displaystyle \sum _{J,M}\langle j_{1},m_{1};j_{2},m_{2}|J,M,j_{1},j_{2}\rangle \langle j_{1},m_{1}';j_{2},m_{2}'|J,M,j_{1},j_{2}\rangle =\delta _{m_{1}m_{1}'}\delta _{m_{2}m_{2}'}.}$

Determination of the Clebsch-Gordan coefficients

The eigenstate with and can be specified immediately in the product base (only one Clebsch-Gordan coefficient equal to 1, all others zero): ${\displaystyle J=j_{1}+j_{2}}$ ${\displaystyle M=J}$

${\displaystyle |j_{1}+j_{2},j_{1}+j_{2},j_{1},j_{2}\rangle =|j_{1},j_{1};j_{2},j_{2}\rangle }$

By using the relegation operator one obtains the states up to , i.e. all states with . ${\displaystyle J_{-}=J_{1\,-}+J_{2\,-}}$ ${\displaystyle |j_{1}+j_{2},j_{1}+j_{2}-1,j_{1},j_{2}\rangle }$ ${\displaystyle |j_{1}+j_{2},-j_{1}-j_{2},j_{1},j_{2}\rangle }$ ${\displaystyle J=j_{1}+j_{2}}$ ${\displaystyle M=-J,\dots ,J=-j_{1}-j_{2},\dots ,j_{1}+j_{2}}$

The state is obtained from the requirement of orthogonality and the convention that the Clebsch-Gordan coefficient is positive. ${\displaystyle |j_{1}+j_{2}-1,j_{1}+j_{2}-1,j_{1},j_{2}\rangle }$ ${\displaystyle |j_{1}+j_{2},j_{1}+j_{2}-1,j_{1},j_{2}\rangle }$ ${\displaystyle M=J}$

With the relegation operator all states can be generated again. This process is now repeated iteratively until . ${\displaystyle J=j_{1}+j_{2}-1}$ ${\displaystyle M=-j_{1}-j_{2}+1,\dots ,j_{1}+j_{2}-1}$ ${\displaystyle J=|j_{1}-j_{2}|}$

SU (N) -Clebsch-Gordan coefficients

The commutator relations of the angular momentum operators show that every angular momentum defined in this way forms an algebra that is isomorphic in the mathematical sense to that of the Lie algebra of the special unitary group SU (2).

In quantum mechanics, however, not only states with angular momentum quantum numbers or su (2) quantum numbers can be coupled, but also states with su (N) quantum numbers. This happens e.g. B. in quantum chromodynamics . Algorithms are also known to calculate the Clebsch-Gordan coefficients that occur in this process.

Generalization: Reduction of a product representation

The theory of the Clebsch-Gordan coefficients can be understood as a special case from the representation theory of groups. In fact, the "product representation" spanned by two (or more) products of the functions i. a. is reducible. It can therefore be “reduced” according to the irreducible representations , whereby the integer “multiples” with which these can occur in the general case only assume the value 1 for the rotating group. ${\displaystyle u_{\alpha _{1}}^{\gamma _{1}}\cdot u_{\alpha _{2}}^{\gamma _{2}}\,\,(\cdot \dots )}$ ${\displaystyle {\hat {\gamma }}_{1}\otimes {\hat {\gamma }}_{2}\,\,(\otimes \dots )}$ ${\displaystyle {\hat {J}}}$

In the present case, the products mentioned are of the form and the associated irreducible representation is spanned by functions of the form . ${\displaystyle u_{m_{l}}^{l}\cdot v_{m_{s}}^{s}}$ ${\displaystyle w_{M_{J}}^{J}}$

So abstract, with the irreducible representations of the rotating group

${\displaystyle {\hat {\gamma }}_{1}\otimes {\hat {\gamma }}_{2}\ \ {\stackrel {\text{ausred.}}{=}}\ \ {\hat {J}},}$where z. B. corresponds to the size l and is analogous to s . ${\displaystyle {\hat {\gamma }}_{1}}$ ${\displaystyle {\hat {\gamma }}_{2}}$

The complex-valued expansion coefficients that occur during this reduction are the Clebsch-Gordan coefficients.

A simple example

In addition to the atomic functions treated above, the following example is instructive, which deals with the simplest two-spin problem: So two particles are considered with the spin . This results in the four functions where the first factor relates to one particle, the second to the other particle. The states indicated are illustrated in the following by arrow symbols. ${\displaystyle 1/2}$ ${\displaystyle \{|s=1/2,m_{s}=\pm 1/2\rangle \}\otimes \{|s=1/2,m_{s}=\pm 1/2\rangle \},}$

Reduction of this product also results in a total of four “irreducible” states. These are a so-called singlet state with , ${\displaystyle J=0}$

${\displaystyle |J=0,\,M_{J}=0\rangle \ {\hat {=}}(1/{\sqrt {2}})\ (\uparrow \downarrow -\downarrow \uparrow )}$

as well as three so-called triplet states with , namely ${\displaystyle J=1}$

${\displaystyle |J=1,M_{J}=+1\rangle \ {\hat {=}}\uparrow \uparrow ,}$

${\displaystyle |J=1,\,M_{J}=0\rangle \ {\hat {=}}(1/{\sqrt {2}})\ (\uparrow \downarrow +\downarrow \uparrow )}$ and

${\displaystyle |J=1,M_{J}=-1\rangle \ {\hat {=}}\downarrow \downarrow .}$

In this case, the Clebsch-Gordan coefficients correspond to the values or which occur in this representation. ${\displaystyle \pm 1/{\sqrt {2}}}$ ${\displaystyle 1}$

In the absence of magnetic fields, the three triplet states have one and the same energy.

Applications

Which of the two states, singlet or triplet, dominates energetically depends on the details of the interaction: If the dominant mechanism is the attraction of the electrons by the nucleus, e.g. B. with homeopolar bond , the singlet state dominates and the resulting molecule or solid are non-magnetic or diamagnetic . If, on the other hand, the mutual Coulomb repulsion of the electrons dominates, paramagnetic molecules or ferromagnetic solids are obtained.

The quantum-mechanically deepened angular momentum physics implicitly dominating in the first part of the article ("angular momentum gymnastics") is obtained with the standard interpretation that, first of all, not two, but only one particle is considered and and then set. This results in a variety of applications in nuclear and particle physics . ${\displaystyle j_{1}\to l}$ ${\displaystyle j_{2}\to s}$

Web links

• Table with examples of specific values ​​for and (Particle Data Group) ${\displaystyle j_{1}}$ ${\displaystyle j_{2}}$ (PDF, 732 kB)
• Web interface for listing the SU (N) -Clebsch-Gordan coefficients

literature

• Wachter, Hoeber: Theoretical Physics . Springer publishing house. ISBN 3-540-21457-7

Individual evidence

1. ^ A. Alex, M. Kalus, A. Huckleberry, and J. von Delft: A numerical algorithm for the explicit calculation of SU (N) and SL (N, C) Clebsch-Gordan coefficients . In: J. Math. Phys . 82, February 2011, p. 023507. doi : 10.1063 / 1.3521562 . Retrieved April 13, 2011.
2. ↑ See all standard textbooks on group representation theory; especially those with main applications in physics.
3. ^ A. Lindner: Basic course theoretical physics , Wiesbaden, Vieweg & Teubner, 3rd edition (2012), ISBN 978-3-8348-1895-9