# Appendix H Irreducible Spherical Tensor Operators and the Wigner‐Eckart (W—E) Theorem

# Appendix H Irreducible Spherical Tensor Operators and the Wigner‐Eckart (W—E) Theorem

Appendix H

Irreducible Spherical Tensor Operators and the Wigner‐Eckart (W—E) Theorem

In our introductions to various sciences, most of us encounter problems that seem to have a set of “natural” coordinates associated with them. By choosing “natural” coordinates one can simplify the mathematical analysis. Good choices correspond to descriptions that mimic one or more symmetry elements of the problem. “Unnatural” coordinates give rise to unwieldy expressions that are unnecessarily complicated. Consequently, an entire subject is devoted to identification and exploitation of the simplest mathematical formalism to use in analyzing any problem, namely group theory. Because group theory is an important subject that transcends the limited objectives of this book, and can be a very powerful companion in the analysis of optical problems we shall not diminish it by attempting to introduce it here. However, some exposure to key results that can be explained succinctly is beneficial for application to the advanced topics considered in Chapters 6 and 7 of this book. Consequently this appendix covers the essentials of tensor analysis to explain what is known as the “irreducible” representation of physical problems.

A pure rotation *R* transforms a state vector *ψ* into *ψ*′ via a unitary transformation *U* _{R} that acts on each component in the same way that spatial coordinates transform.

In an analogous fashion, operators *V _{i}* (

*i*= 1,2,3) are said to form the Cartesian components of a vector operator

*V*, if under every rotation their expectation values transform like components of a vector. Thus a vector operator is required to have expectation values that transform according to the equation

The entries in the rotation matrix *R _{ij}* =

*∂x′*

_{i}/

*∂x*

_{j}are cosines of the angles between the rotated positive

*x′*

_{i}and the positive

*x*

_{j}‐axis prior to rotation [H.1] describing the transformation of the coordinates.

(p.284) Substitution of Eq. (H.1) into Eq. (H.2) yields

since the relation should hold for arbitrary ψ.

We now proceed to generalize Eq. (H.3) to operators of higher rank than vectors. Vectors are tensors of rank one (*k* = 1) that have three spatial components related to *x*, *y*, and *z*. Tensors of rank two (*k* = 2) can be thought of as the most general product of two vector operators *V¯* _{1m} and *V¯* _{2n}. They therefore consist of the set of all nine products of the vector components (i.e., *V* _{1m} *V* _{2n} for *m*,*n* = *x*,*y*,*z*). Irreducible spherical tensors are constructed in a basis that reflects the rotational symmetry of a sphere. So, a spherical tensor operator of arbitrary rank *k* will be defined more generally as a set of quantities [*T* ^{(k)}]_{j} which obey a transformation law similar to Eq. (H.3), except that we shall assume it is of covariant form, which transforms as the inverse of Eq. (H.3), and write it in a basis that distinguishes clockwise from counterclockwise rotations.

As a first step in determining the conventional expression for the tensor transformation law, consider extending Eq. (H.3) to the case of a second rank tensor with all Cartesian indices written out explicitly. The required transformation has the form

where the transformation coefficients are ${D}_{klij}=\frac{\partial {x}_{k}}{\partial {x}_{i}^{\prime}}\frac{\partial {x}_{l}}{\partial {x}_{j}^{\prime}}$. Now, the notation applicable to tensors of all ranks uses an index *q* (where *q*={−*k*,−*k* + 1, −*k* + 2,… + *k*}) to replace multiple Cartesian subscripts referring to individual components of the tensor. In addition, we note a distinction is introduced between *D* and *R*. We shall insist that *D* be the simplest possible representation of rotations, or in other words that it be “irreducible” in addition to producing the desired transformation for rank *k* tensors. Irreducible tensors consist of the minimum number of independent elements that span a given space, formed into linear combinations of Cartesian components that present the least mathematical complexity, and they mimic the properties and dynamics of objects and processes in the real world with which we are familiar.

Thus the tensor transformation law is a notationally simplified version of Eq. (H.4) that uses the index *q* and coefficients ${D}_{{q}^{\prime}q}^{(k)}$, from a unitary rotation based on the angular momentum operator.

(p.285) A direct comparison with vector transformation components is possible after inversion of Eq. (H.3).

Notice the summation in Eq. (H.6) is over the first subscript of *R*, just as the summation in Eq. (H.5) is over the first subscript of ${D}_{{q}^{\prime}q}^{(k)}$.

Equation (H.5) now defines tensor operator components that transform like irreducible components of vectors. This equation can be used to test whether quantities of interest have the tensor properties required to provide the simplest mathematical description of system states, energies, and dynamics. It can also be used to determine explicit components of tensors in terms of vector Cartesian components. For example we make use of Eq. (H.5) below to determine the components of a rank one tensor *T* ^{(1)}. However, it proves to be more convenient to work with commutation relations that are equivalent to the transformation rules than to use Eq. (H.5) itself.

To find the commutation relations of interest, we proceed by substituting the irreducible form for infinitesimal rotations into Eq. (H.5). The axis and magnitude of the rotation are specified by axial vector ε¯, and only terms up to first order will be considered. Borrowing expressions for the lowest order terms of the Wigner D coefficients that represent three-dimensional irreducible rotations (see, for example, Eq. (16.54) of Ref. [H.2]), we have

The transformation rule Eq. (H.5) dictates that

Substituting the first‐order expression for the rotation given in Eq. (H.7), namely

into Eq. (H.8), the left hand side becomes

(p.286) The right hand side yields

Equating left and right sides, and comparing coefficients of ε_{x},ε_{y},ε_{z}, we obtain

Now *J* _{±} = *J* _{x} ± *iJ* _{y}, so that

Hence Eqs. (H.13)–(H.15) are completely equivalent (for infinitesimal rotations) to the required transformation relations in Eq. (H.8) for irreducible tensor operators. This is sufficient to establish the equivalence between Eq. (H.8) and the commutation relations for all rotations, and make it clear that the rotation properties of any operator are determined by its commutator with the angular momentum. This is fortunate, because in practice we could not have applied Eq. (H.8) for *all* possible rotations *R*! We now turn to the problem of relating irreducible tensor components to the Cartesian components of rank zero, rank one, rank two, etc. tensors, using Eqs. (H.13)–(H.15).

(p.287)
The procedure for constructing spherical tensors is illustrated for rank one (vectors) below. Because the tensor components *T* ^{(k)} and the Cartesian vector components *V _{i}* both span the space, and obey linear transformation equations, they must be linear combinations of one another.

Substitution of Eq. (H.16) into Eqs. (H.13)–(H.15) yields nine equations in nine unknowns. We assume that the vector *V* satisfies the usual angular momentum commutation relations. Each equation can then be simplified by using the relations [*J* _{i}, *V* _{j}] = *iħε* _{ijk} *V* _{k}, where *ϵ* _{ijk} is the Levi‐Civita pseudotensor. Its value is +1 for cyclic permutations of the indices, −1 for anti‐cyclic permutations, and zero if two or more indices are equal.

These relations are completely general. But notice from Eq. (H.18) that

since *V* _{y} ≠ *V* _{x}. Hence, from Eq. (H.16) we find

Also from Eq. (H.21) we can write

and from Eq. (H.24) we find

The simplest choice is *f* = 1, for which we obtain

Based on Eqs. (H.30)–(H.32), the general form of irreducible rank one spherical tensors is

Inverse relations are provided by solving Eq. (H.33)–(H.35) for *V* _{x}, *V* _{y}, and *V* _{z}.

From Eq. (H.36)–(H.38) we can now derive an expression for the scalar product of two vectors *A¯*, *B¯* in terms of tensor components *A* _{q}, *B* _{q} adapted to spherical coordinates.

*Scalar product*

To specify individual vectors like *A¯* or *B¯* completely in spherical tensor notation, expressions for the basis vectors as well as the amplitudes are needed. Because their magnitude is unity the spherical basis vectors may be obtained directly from Eq. (H.33)–(H.35), and are denoted by

and

The coefficients of a general vector *A¯* can then be obtained by replacing all the Cartesian components introduced in the defining expression for *A¯* with their spherical tensor equivalents given by Eq. (H.36)–(H.38).

(p.290)
*Irreducible representation of V* :

The electric dipole moment operator is

Consequently, it has three irreducible tensor components given by

where

is the Racah tensor. In the same way we can write out the field in irreducible form as

Using these results, the interaction Hamiltonian becomes

(p.291)
We need matrix elements for *V* starting from a state *i* and ending in state *f*. Let the states themselves be represented by

where *j*, *j′* are the angular momenta of the two states and *m*, *m′* specify their projections on the axis of quantization. *α*, *α′* denote other quantum numbers (like the principle quantum number *n*) required to specify each state completely. Then

Using the Wigner‐Eckart (W−E) theorem (see Eq. (H.63)), this reduces to

where <*αj*ǁ*rC* ^{(1)}ǁ*α′j′*> is the so‐called reduced matrix element (independent of *m*, *m′*), and $\left(\begin{array}{ccc}j& 1& {j}^{\prime}\\ -m& 1& {m}^{\prime}\end{array}\right)$ is a 3‐*j* symbol that assures that momentum is conserved.

This particular 3‐*j* symbol is nonzero, and specifies that the interaction can take place only if

Hence Eqs. (H.53) and (H.54) constitute selection rules for electric dipole transitions excited by circularly polarized light. Transitions from *j′* = 0 to *j* = 0 are also excluded.

In writing Eq. (H.52) we used the W—E theorem to clarify as much as possible the dependence of the matrix element on the *J* and *m* values of the states involved. This theorem is readily proved starting from the transformation relation Eq. (H.8):

Evaluating both sides for the states considered above, we find

(p.292) On the left side we can make the replacements

and

These relations follow directly from the definition of ${D}_{{m}^{\prime}\mu}^{({j}^{\prime})}$ as the expansion coefficients of a rotated representation of the wavefunction. (See, for example, the defining relation and form of ${D}_{{m}^{\prime}\mu}^{({j}^{\prime})}$ given by Eqs. (16.43) and (16.50)of Ref. [H.2], respectively.)

Using Eqs. (H.56) and (H.57) in Eq. (H.55), we find

This equation has exactly the same form as an important transformation identity in which Clebsch—Gordan (C—G) coefficients replace the matrix elements on both sides of Eq. (H.58) (see, for example, Eq. (16.91) in Ref. [H.2]).

C–G coefficients furnish the unitary transformation from basis states ǀ*j* _{1} *j* _{2} *m* _{1} *m* _{2}> = ǀ*j* _{1} *m* _{1}>ǀ*j* _{2} *m* _{2}> to basis states ǀ*j* _{1} *j* _{2} *jm*>. That is,

From Eq. (H.59) we see that the ${C}_{{m}_{1}{m}_{2}}^{j}$ have the values

These coefficients have been extensively tabulated, but are often listed instead in terms of 3‐*j* symbols that have simpler permutation properties. The 3‐*j* symbols are defined by

The fact that the form of Eq. (H.58) is the same as the formula derived from the C–G series simply means that the matrix element $<{\alpha}^{\prime}{j}^{\prime}{m}^{\prime}\left|{T}_{q}^{(k)}\right|\alpha jm>$ we are interested in is proportional to the C—G coefficient. That is,

The constant of proportionality in Eq. (H.62) does not depend on *m*, *m′*, or *q* and is called the “reduced” matrix element, denoted <*α′j′*ǁ*T* ^{(k)}ǁ*αj*>. With this notation, the W—E theorem can be stated as

(p.293)
Equation (H.59) has two important implications. First, since transition probabilities depend on $|<{\alpha}^{\prime}{j}^{\prime}{m}^{\prime}|{T}_{q}^{(k)}|\alpha jm>{|}^{2}$, it is apparent from the right side of Eq. (H.59) that the relative strengths of optical transitions between different magnetic substates of the same *αj* → *α′j′* transition depend only on squared ratios of C—G coefficients. Second, since the reduced matrix element <*α′j′*ǀ*T* ^{(k)}ǁ*αj*> does not depend on *m*, *m′*, or *q*, it represents a transition multipole moment that is independent of the choice of origin. This resolves the issue that standard integral definitions of individual moments yield values of the moments that change with a shift of coordinates.

Our main interest in this book is in the tensor describing the interaction of light with matter as Hamiltonian operator *V̂* = −*μ¯* ∙ *E¯*. How can reduced matrix elements like the one in Eq. (H.52) be evaluated in the simplest, most general way? To explore this question, we first factor out and perform the radial integration for the case of an electric dipole operator. All the radially dependent quantities in the integral are well defined so one can write

In the final expression <*α′j′*ǁ*r*ǁ*αj*>_{r} is a purely radial integral of *r* between the specified states. Note, however, that the angular integral over the undetermined quantity *C* ^{(1)} must be worked out by inverting the W—E theorem. This is necessitated by the fact that <*α′j′*∥*C* ^{(1)}∥*αj*>_{θϕ} is not just the angular integral of a known combination of ${Y}_{m}^{(1)}$ or other functions. As mentioned above it is in fact independent of *q*, *m*, *m′*.

The radial dependence can be factored out of the general W—E relation (Eq. (H.63)) in the same way as above. From

we thus obtain

Since we don't know the exact form of <*α′j′*∥*C* ^{(1)}∥*αj*>_{θϕ}, we must proceed by evaluating the angular integral on the left of expressions like Eq. (H.65) in order to find the reduced matrix element. Following Sobelman [H.3], the left hand side of Eq. (H.65) is

(p.294) so that the reduced matrix element obtained by inverting Eq. (H.65) is

The 3‐*j* symbol in the expression Eq. (H.66) for the reduced matrix element dictates that Δ*j* = ±1. The selection rule with respect to magnetic quantum number *m* arises from the 3‐*j* symbol in Eq. (H.65) and dictates that Δ*m* = *m′* − *m* = 1 in the example of circularly polarized light above. For linearly polarized light, the 3‐*j* symbol in Eq. (H.65) would be

changing the magnetic quantum number selection rule to Δ*m* = *m′* − *m* = 0. Additional discussion of the relationship betweenC—G coefficients and 3‐*j* symbols can be found in Ref. [H.4] and formulas for their evaluation are on p. 60–6 of Ref. [H.3]. The choice of phase in Ref. [H.3] and Ref. [H.4] is that of Condon and Shortley [H.5] which renders the C—G coefficients real.

## References

H.1. See, for example, G. Arfken, *Mathematical Methods for Physicists*, 2nd edition, Academic Press, New York, 1970, pp. 8–11 and 121–36.

H.2. E. Merzbacher, *Quantum Mechanics*, 2nd edition, Wiley & Sons, New York, 1970.

H.3. I.I. Sobelman, *Atomic Spectra and Radiative Transitions*, Springer‐Verlag, New York, 1979, p. 78

H.4. L.I. Schiff, *Quantum Mechanics*, 3rd edition, McGraw‐Hill, New York, pp. 214–24.

H.5. E. Condon and G. Shortley, *The Theory of Atomic Spectra*, Cambridge University Press, Cambridge, 1951.