Wigner-Eckart theorem

The title of this article is ambiguous. The Wigner-Eckart theorem must not be confused with the Wigner theorem .

The Wigner-Eckart theorem (after Eugene Paul Wigner and Carl Henry Eckart ) is an aid for the calculation of the matrix elements of a tensor operator, if its symmetry properties are known.

The following applies to the defining transformation properties of a tensor operator:

${\displaystyle U^{\dagger }T_{q}^{(k)}U=\sum _{q'=-k}^{k}D_{qq'}^{(k)*}T_{q'}^{(k)}}$

in which

• ${\displaystyle U}$the unitary group transformation matrix and
• ${\displaystyle D_{qq'}^{(k)}}$is an irreducible representation of this group in the base . ${\displaystyle |\alpha ,jm\rangle }$

Theorem: The matrix element of a spherical tensor operator, expressed in the eigen-states of the angular momentum operator , satisfies the following equation:

${\displaystyle \langle \alpha ',j'm'|T_{q}^{(k)}|\alpha ,jm\rangle =\langle jm;kq|j'm'\rangle \langle \alpha 'j'\|T^{(k)}\|\alpha j\rangle }$

Here is

• ${\displaystyle T^{(k)}}$a tensor of rank  k
• j is the total angular momentum
• m is the associated magnetic quantum number
• ${\displaystyle \alpha }$all further quantum numbers of the state necessary to describe the system .

For rotational symmetry are the Clebsch-Gordan coefficients for the addition of two angular momentum and and the respective z-components or to the angular momentum with z-component . ${\displaystyle \langle jm;kq|j'm'\rangle }$ ${\displaystyle j}$ ${\displaystyle k}$ ${\displaystyle m}$ ${\displaystyle q}$ ${\displaystyle j'}$ ${\displaystyle m'}$

The factor independent of  m and  m ' as well as  q is called the reduced matrix element , identified by the 2 bars on either side of . This also has the advantage, because this matrix element, which is independent of  m and  m ' , is only calculated once, is then the same for all other matrix elements and thus enables a simple calculation of any matrix elements. ${\displaystyle T^{(k)}}$

Proof of the theorem (rotation group)

The Wigner-Eckart theorem is related to Schur's lemma . Taking advantage of this, lengthy calculations are not required for proof.

To bring the Clebsch-Gordan coefficients into play, consider the following operator, constructed for this purpose only:

${\displaystyle W=\sum _{np}T_{p}^{(k)}|\alpha jn\rangle \langle jn;kp|}$

It transforms states with two angular momentum ( and ) into states with a single angular momentum, on which tensor operators act. In the target space rotations are represented by a unitary operator , in the pre-image space by a unitary operator . The essential property of is the interchange with rotations or the invariance under rotations: ${\displaystyle j}$ ${\displaystyle k}$ ${\displaystyle U_{1}}$ ${\displaystyle U_{2}}$ ${\displaystyle W}$

${\displaystyle U_{1}W=WU_{2}\,\!}$

This is based on the similar behavior of tensor operators and angular momentum states under rotations. Specifically, the easiest way to see invariance is to use the expression

${\displaystyle \sum _{np}\sum _{n'p'}T_{p}^{(k)}|\alpha jn\rangle D_{pp'}^{(k)}D_{nn'}^{(j)}\langle jn';kp'|}$

once by summing over what results, and once by summing over what results. It is used here that the rotary dies are also unitary. ${\displaystyle np\,\!}$ ${\displaystyle U_{1}W\,\!}$ ${\displaystyle n'p'\,\!}$ ${\displaystyle WU_{2}\,\!}$

Because of the rotational invariance of , subspaces that are below irreducible are transformed into subspaces that are below irreducible. In the case of the rotation group, these subspaces are characterized by an angular momentum quantum number . According to Schur's lemma: ${\displaystyle W}$ ${\displaystyle U_{2}}$ ${\displaystyle U_{1}}$ ${\displaystyle J}$

• The parts of that mediate between different (inequivalent irreducible representations) are zero. ${\displaystyle W}$ ${\displaystyle J}$
• The parts of that mediate between equal (equivalent irreducible representations with equal representation matrices) are multiples of the one-map. ${\displaystyle W}$ ${\displaystyle J}$

The fact that the representation matrices are actually always the same for the same angular momentum is based on the use of the standard basis vectors . The proof given here is therefore only valid for the rotating group. ${\displaystyle |jm\rangle }$

If the respective multiple is denoted by a factor that is dependent on the connected subspaces, then according to Schur's lemma has the following form: ${\displaystyle \lambda \,\!}$ ${\displaystyle W}$

${\displaystyle W=\sum _{\alpha 'J}\sum _{M}\lambda _{\alpha jk\alpha 'J}|\alpha 'JM\rangle \langle JM|}$

The sum over represents the one-mapping between two irreducible subspaces. The Bra vector lacks the degeneracy index, because with angular momentum coupling (in the archetype space of ) there is no degeneracy. The indices express that the entire construction of the operator depends on them. ${\displaystyle M}$ ${\displaystyle W}$ ${\displaystyle \alpha jk}$ ${\displaystyle W}$

To complete the proof, it now forms the matrix element with the two expressions for which Ortho normality of the basis vectors exploits and identifies each with . ${\displaystyle \langle \alpha 'j'm'|W|jm;kq\rangle }$ ${\displaystyle W}$ ${\displaystyle \lambda \,\!}$ ${\displaystyle \langle \alpha 'j'\|T^{(k)}\|\alpha j\rangle }$

Individual evidence

1. ^ Albert Messiah : Quantum Mechanics. Volume 2. De Gruyter, 1985, section 13.6.3

literature

• C. Eckart: The Application of Group Theory to the Quantum Dynamics of Montomic Systems . In: Rev. Mod. Phys. 2, 1930, pp. 305-380.
• JJ Sakurai: Modern Quantum Mechanics. Addison-Wesley, 1994, pp. 239-240.
• EP Wigner: Some conclusions from Schrödinger's theory for term structures . In: Z. Physik 43, 1927, pp. 624-652.