Clebsch–Gordan coefficients

The Clebsch - Gordan coefficients find their use in the coupling of quantum angular momenta. It is to the expansion coefficients, with which it goes from the base of the individual pulses in the rotation based on the total angular momentum. They are used to calculate the spin -orbit coupling as well as used in the isospin formalism.

They are named after Alfred Clebsch (1833-1872) and Paul Gordan ( 1837-1912 ).

Angular momentum coupling

See also the section "addition of angular momenta " in the article angular momentum operator.

It is of two rotational pulses, and from, each having quantum numbers, and (z- component), respectively, and. In this and the following values ​​assume: and and angular momentum exchange among themselves (see quantum mechanical commutator ). This means that the individual angular momenta independently of one another can measure sharp. Each of these has its own angular momentum eigenspace spanned by the eigenvectors respectively. In the basis of these eigenvectors has a simple diagonal form; The same applies to (more specifically, for the square and the z- component of the particular operator ).

Now couple the individual angular momenta and a total angular momentum ( addition of the individual components). This total angular momentum has the quantum numbers and that can take the following values ​​:

Since the total angular momentum consists of two angular momenta and, it can be represented in the product space of the individual eigenstates:

Where the tensor referred.

However, these are not the eigenvectors of the total angular momentum, so that it has no diagonal shape in this base.

Eigenbasis of the total angular momentum operator

The eigenvectors of are, and uniquely determined by the quantum numbers. With respect to the new basis of eigenvectors of the total angular momentum has a simple diagonal form again. The following applies:

The Clebsch -Gordan coefficients indicate completed the transition of product base in the eigenbasis of ( unitary transformation ):

Here are the Clebsch -Gordan coefficients.

Properties of the Clebsch - Gordan coefficients

  • The Clebsch - Gordan coefficients are equal to zero if one of the two conditions and is not satisfied:
  • The Clebsch - Gordan coefficients are real by convention:
  • The following Clebsch -Gordan coefficient to be positive by convention:
  • The Clebsch -Gordan coefficient is equal in magnitude to the Clebsch -Gordan coefficient according to
  • The Clebsch - Gordan coefficients satisfy the orthogonality relation
  • The Clebsch - Gordan coefficients satisfy the orthogonality relation

Determination of Clebsch - Gordan coefficients

The eigenstate with and is immediately in the product base specify (just a Clebsch -Gordan coefficient equal to 1, all others zero):

By applying the dump operator gives the states up, so to all the states with.

This condition is obtained from the requirement of orthogonality to the Convention and that the Clebsch -Gordan coefficient is positive.

With the lowering operator all states can be generated with again. This process is repeated iteratively until.

SU ( N) Clebsch - Gordan coefficients

The angular momentum algebra corresponds to the mathematical sense of the algebra su ( 2), the Lie algebra of the special unitary group. In quantum mechanics, it is not only states couple that carry angular momentum quantum numbers or su ( 2) quantum numbers, but also states with su ( N) quantum numbers. This happens, for example, in quantum chromodynamics. To calculate the Clebsch - Gordan occurring coefficients algorithms are now known.

Generalization: Ausreduzierung a product representation

One can understand the theory of Clebsch - Gordan coefficients as a special case of the representation theory of groups. And that is that the two ( or more) products of functions spanned " product presentation " ia is reducible. You can therefore " ausreduziert " after the irreducible representations, where the integer " multiplicities " with which this can occur in the general case, assume in the rotation group, enter the value 1.

In this case, at any rate the above-mentioned products of the form and the associated irreducible representation is defined by features of the mold.

So abstract, with the irreducible representations of the rotation group

The occurring in this Ausreduzierung complex expansion coefficients are the Clebsch -Gordan coefficients.

A simple example

Besides the above- treated nuclear functions, the following example is instructive, in which it comes to the simplest two-spin problem: two particles thus are considered with the spin. Which results in the four functions wherein the first factor, the second refers to the one on the other particles. The specified conditions are illustrated by arrows below.

Ausreduktion this product also results in a total of four " irreducible " states. These are equipped with a so-called singlet state,

And three so-called triplet states with, namely

The Clebsch - Gordan coefficients in this case correspond to the values ​​or that occur in this representation.

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


Which of the two states, a singlet or triplet energy dominates depends on the details of the interaction from: If the dominating mechanism is the attraction of the electrons by the nucleus, such as hömöopolarer binding dominates the singlet state and the resulting molecule or. the solids are non-magnetic or diamagnetic. If, however, dominates the mutual Coulomb'abstoßung of electrons is obtained paramagnetic molecules or ferromagnetic solids.

The first part of the article implicitly dominant quantum mechanical angular momentum in-depth physics ( " angular momentum gymnastics" ) is obtained with the standard interpretation that is considered first, not two, but only a single particle and and sets. This results in a variety of applications in nuclear and particle physics.