ROTATIONS AND ANGULAR MOMENTUM 161 The relation between symmetry and degeneracy can be summarized as follows. If the

Hamiltonian is invariant under a symmetry operationU, and if| iis an eigenstate ofH with eigenvalue E, then | ⁰i=U| i is either a di↵erent eigenstate, though degenerate with the first, or | ⁰i =| i, apart from a phase. Further, the Hamiltonian eigenstates are not necessarily invariant under the same symmetry operation which leaves H in-variant, such as when| ⁰i 6=| i; in this case, the state has a lower symmetry than the Hamiltonian itself, and, especially in the context of many-body physics, such state is said to have broken symmetry. Again, the H atom provides an example: H is rotationally invariant, but onlys-states are spherically symmetric.

8.5.6 Tensor Operators and the Wigner-Eckart theorem

Refs.: G§36; S§3.10; Baym Ch. 17.

8.5.6.1 Motivation

We have already established that a scalar observable, S, is invariant under rotations.

Since angular momentum states, {|jmi}, form a very useful basis, and have definite transformation properties under rotations, one expects that general selection rules for matrix elements of S can be derived.

Indeed, since [D(R), S] = 0, one has

[S, J²] = [S, J_z] = [S, J_±] = 0, (8.5.72) which imply

hj⁰m⁰|S|jmi= 0, ifj6=j⁰ orm6=m⁰ (8.5.73) This selection rule can be interpreted as due to geometrical constraints, in the sense that a spherically symmetric operator carries no angular momentum, so that when it acts upon a state |jmi, the magnitudeJ² (meaning j) is preserved, so thatS|jmi only overlaps with |j⁰m⁰i if j⁰ =j. By the same token, it does not alter the orientation (m) of the state it acts upon, so the outcome has no superposition with |jm⁰i, if m⁰ 6= m.

On the other hand, the specific nature ofS should a↵ect other quantum numbers, apart from (j, m).

The existence of selection rules like these provide important tools to extract infor-mation about the interactions at play in a given physical system, without their detailed knowledge. A systematic way of achieving this goal is to set up a classification of ob-servables into scalars, vectors, and tensors, and exploit their transformation properties under rotations. As we will see, although selection rules can be drawn for the cartesian components of vector operators, a unified framework of tensor operators is readily es-tablished in terms of theirspherical components,Tq^(k), wherekis the rank of the tensor, andqlabels the component; see below. In terms of this representation, somegeometrical selection rules will emerge in a simple way, with the aid of the Wigner-Eckart theorem.

8.5.6.2 Definitions

We have seen that the cartesian components of a vector operator, V, have a simple transformation rule under rotation:

V_i⁰=D^†(R)V_iD(R) =X

j

R_ijV_j, (8.5.74)

whereRis the matrix that performs the corresponding rotation of usual vectors in three-dimensional space. Likewise, we have seen that Eq. (8.5.74) leads to specific commutation relations with the cartesian components of the angular momentum operator,

[J_i, V_j] =i~✏_ijkV_k, (8.5.75) which can also be used as adefinitionof vector operators. Further, one should also note that the first equality in Eq. (8.5.74) also applies to any function of the vector operator, V:

f⁰(V) =D^†(R)f(V)D(R). (8.5.76) In classical Physics, a cartesian tensor of rank k is an object with 3^k components, whose transformation rules under rotations is an immediate generalization of those for vectors,

T_ijl...= X

i⁰,j⁰,l⁰,...

R_ii0R_jj0R_ll0T_i0j⁰l⁰...; (8.5.77) where the R_ii0 are the elements of the cartesian transformation matrix, and k is the number of indices: scalars correspond tok= 0, vectors tok= 1, and so forth. The fact that one has to perform products ofktransformation matrices makes it hard to extract general information from these rotations.

O exemplo mais simples de um tensor cartesiano de ordem 2 ´e uma di´adica, formada a partir de dois vetoresU eV, da seguinte forma:

T_ij ⌘U_iV_j. (8.5.78)

O leitor deve verificar que, de fato, as nove componentesTij se transformam sob rota¸c˜oes de acordo com a Eq. (8.5.77).

Further, we may foresee that cartesian components are not universally convenient to fully explore their behaviour under rotations. Indeed, cartesian tensors such as (8.5.77) arereducible; that is, they can be decomposed in objects which, as expressed in (8.5.77), follow di↵erent rotation rules. Por exemplo, a di´adica (8.5.78) pode ser escrita da seguinte forma: O primeiro termo do lado direito ´e T, o tra¸co do tensor, que, sendo um escalar, ´e invariante sob rota¸c˜oes. O segundo termo pode ser escrito como um produto vetorial,

8.5. ROTATIONS AND ANGULAR MOMENTUM 163 A_k ⌘✏_ijk(U⇥V)_k, de modo que se transforma como um vetor. O terceiro termo pode ser escrito como um tensor sim´etrico de tra¸co nulo, que tem apenas cinco componentes independentes (dos trˆes elementos da diagonal, apenas dois s˜ao independentes). Logo, estes trˆes objetos se transformam de acordo com as representa¸c˜oes irredut´ıveis`= 0, 1 e 2, respectivamente; suas respectivas multiplicidades (2`+ 1) somam 3² = 9, que ´e o n´umero de componentes do tensor.

A combination of cartesian components in some form of spherical coordinates should be preferred instead. In order to pursue along this line, we start by recalling that the

`= 1 spherical harmonics can be written in terms of cartesian coordinates as Y₁⁰= r= 1, so that the tensor spherical coordinatesof the vector rare defined as

r_q⌘

The transformation analogous to Eq. (8.5.74) is therefore the one that takes nˆ ! nˆ⁰ = Rˆn, or, to be more precise,Y₁^q(ˆn)!Y₁^q(Rn), as given byˆ where we have used ` instead of 1, in order to highlight (once again) that all spherical harmonics transform according to the `-th irreducible representation of the rotation group, instead of according to the rotation matrix, as in the second equality of Eq.

(8.5.74). We can then write

r⁰_q=X

q⁰

D_qq⁽¹⁾0^⇤(R)r_q0, (8.5.85) where we used the dummy indices q, q⁰ = `, `+ 1, . . . ,`, instead ofm, m⁰.

The above procedure can be extended along two directions. First, we replace the cartesian coordinates of the position vector by a generic vector operator,Vin Eq. (8.5.83):

r_±1 !V_±1 ⌘ ⌥V_x±iV_y

p2 and r₀ !V₀ =V_z. (8.5.86)

Therefore, while the cartesian components of a vector operator transform according to the three-dimensional rotation matrix, its spherical tensor components transform accord-ing to the `= 1 irreducible representation of the rotation group, which, consistently, is also three-dimensional.

The second extension is to define higher order spherical tensor operators, made to transform under rotations like the spherical harmonics, Eq. (8.5.47). Indeed, for` = 2 we have

so that the corresponding tensor operators are T_q^(k)=

The transformation of these tensor operator components is therefore [T_q^(k)]⁰ =D^†(R)T_q^(k)D(R) =X

q⁰

T_q^(k)₀ hkq⁰|D^†(R)|kqi. (8.5.94) Since they transform like thek-th representation of the rotation group, they are referred to asirreducible tensor operators.

If one now considers infinitesimal rotations, sayD(R)'1 (i"/~)J·u, whereˆ "⌧1 is the angle of rotation aroundˆu, the following commutation relations emerge:

hJ_z, T_q^(k)i These relations can be considered asdefinitionsof irreducible spherical tensor operators;

that is, any set of operators satisfying these relations, even if they cannot be written in terms ofY_k^q(V), are irreducible tensor operators. For instance, (Ux+iUy)(Vx+iVy) is the q = 2 component of a spherical tensor operator of rank k = 2, which, unlike (V_x+iV_y)², cannot be written as Y₂^q(V).

8.5. ROTATIONS AND ANGULAR MOMENTUM 165