Tensor operators

1. Definition of a rotated operator

Let's write the rotation operator as U(R). This operator 
rotates a ket |ψ> to the ket |ψ'>.

If A is any operator, the rotated operator A' must be 
unchanged with respect to its expectations before and 
after rotation. That is:
<ψ'|A'|ψ'>  = <ψ|A|ψ>. 

We have then:
<ψ|A|ψ> = <ψ|U+(R)A'U(R)ψ>. Therefore:
A = U+(R)A'U(R) or A' = U(R)AU+. 
Hence:

The rotated operator of the operator A is A' = U(R)AU+, 
where U(R) is the rotation operator.

2. Scalar Operators

A scalar operator is invariant under rotations. That is:
K = U(R)KU+ under any rotation operator U(R).

This definition implies that U(R) and K commute: [U(R),K] = 0. 

The scalar operator commutes with any rotation operator U(R); 
in particular with the rotation operator R(θ) = exp {- i θ J/ħ}
or its infinitesimal rotation operator: R(dθ) = 1 - idθJ/ħ; 
hence, it commutes with the angular momentum J. 
We have then a new definition:

A scalar operator K commutes with the 
angular momentum J if 
 
   [J, K] = 0 

3. Vector Operators

In three-dimensional Euclidean space, a vector is 
defined as a set of three numbers. Under rotation, 
these numbers are transformed according certain rules. 
Similarly, in quantum mechanics, we define a vector operator 
as a vector of operators (that is, a collection of 
three operators) with certain transformation 
properties under rotations.

For its definition,a vector operator "V" must have 
its expectation rotated according the definition of 
the rotation of any ordinary vector. That is: 
If |ψ'> is the rotated state of 
the original one |ψ>, we have:

R <ψ|V|ψ> = <ψ'|V|ψ'>

In components form:
V'i = <ψ'|Vi|ψ'> = Σ (j) Rij <ψ|Vj

Using |ψ'> = U(R) |ψ>,, the definition becomes:

R <ψ|V|ψ> = <ψU(R)+|V|U(R)ψ>. So 

R V = U(R)+|V|U(R), or R-1 V = U(R)VU+(R)
(the adjoint of the rotation R is equal to its inverse R-1)

The definition of a vector operator becomes:

R V = U+(R)VU(R)


For an infinitesimal rotation about the z-axis 
by an angle ε, we have:




V'i = (V'x , V'y , V'z) = Rz(ε) (Vx , Vy , Vz) 
= (Vx - εVy, Vy + εVx, Vz)

Using the expression of the rotation operator 
U(R) = 1 - iεJz/ħ, to find:
U+(R)Vi)U(R) = (1 + iεJz/ħ)Vi)(1 - iεJz/ħ) = 
Vi + (iε/ħ)[Jz, Vi]

Therefore:
i[Jz, Vx] =  - ħ Vy
i[Jz, Vy] =  - ħ Vx

The cyclic equivalents give the related commutation 
relations of the components of any vector operator V:
[Vi, Jj] = iħεijkVk

[Vi, Jj] = iħεijkVk


εijk is the Levi-Civita element:
εijk = 
+1 for even permutations: 123,231,312
-1 for odd permutations: 132,213,321
0 otherwise

4. Tensor Operators

A tensor operator  is a matrix of operators. That is each element 
of the matrix is an operator. For example, if we have:
A = (A1, A2, A3), and 
B = (B1, B2, B3); we 
can construct a 3x3 tensor Tij = AiBj. 
This is the case of a rank-2 tensor.

Before rotation, we have Vi and Wj, then 
a tensor Tij. After rotation we have T'ij. 
We are going to express the matrix elements of T', that is 
T'ij function of Tij.

We have already V'i = Σ(k) Rik Vk
W'j = Σ(l) Rjl Wl
Then:
T' = V' ⊗ W' and T'ij = V'i ⊗ W'j = Σ(k) Rik Vk Σ(l) Rjl Wl = 
 Σ(k) Σ(l)  Rik Rjl VkWl

 ViWj = Tij, then:
T'ij =  Σ(k)Σ(l) Rik Rjl Tkl

Therefore:

T'ij = U+(R)TijU(R) = Σ(k)Σ(l) Rik Rjl Tkl


Since the suffixes i, j, ... refer to Cartesian axes. Tensors 
written this way are called Cartesian tensors. The 
number of suffixes in "T" is the rank of the Cartesian 
tensor (Tij has the rank 2). The rank n 
tensor has 3n components. A rank 3 Cartesian tensor 
is transformed as:
T'ijk =  Σ(l) Σ(m) Σ(n) Ril Rjm Rkn Tlmn,
having 27 components.

Now we will represent a Cartesian tensor in spherical coordinates 
and talk about spherical tensor

4.1.Spherical Vector

The angular momentum eigenkets |l,m> = Ylm(θ,φ), 
called spherical harmonics can be expressed for l = 1:
|1,0> = Y10(θ,φ) = [3/4π]1 (z/r)
|1,-1> = Y1-1(θ,φ) = +[3/4π]1(x - i y)/21/2 r
|1,1> = Y11(θ,φ) = - [3/4π]1(x + i y)/21/2 r
x, y , and z are considered position operators. 
r = (x, y, z) is a vector operator that will be transformed 
under rotation U(R) to the vector operator r' by rotating 
the eigenstates |l,m>

The eigenstates operators |l,m> will be considered as 
eigenstates operators |j,m>.

|j=1,m> → U(R(θ))|j=1,m> = exp {-iθJ/ħ} |j=1,m> =
Σ(m') <j=1,m'|exp {-iθJ/ħ}|j=1,m> |j=1 m'> 

with:
 D(j)m'm(Rθ)) = <j,m'|exp {-iθJ/ħ)}|j,m> 
 
We have:
|j=1,m> → U(R(θ))|j=1,m>Σ(m') D(j=1)m'm(Rθ) |j=1 m'> 

 Therefore:
 
 If (Vx,Vy,Vz) are the operators components of 
 a vector operator V in Cartesian coordinates, 
 in spherical coordinates, they become:
V11, V-11, V01, with:

 V-11 = +(Vx - iVy)/[2]1/2
 V+11 = -(Vx + iVy)/[2]1/2
 V01 =  Vz
 
 To work with the same notation T, we write:
  T-11 = +(Vx - iVy)/[2]1/2
 T+11 = -(Vx + iVy)/[2]1/2
 T01 =  Vz
 
 These components are denoted Tq1. 


The definition : 
V'i = Σ (j) Rij |Vj
becomes:
T'q = Σ(q') Rqq' Tq', or
T'q1 = Σ(q') Rqq' Tq'1

More generally, a rank k vector operator having 2k+1 
(q varies from = k to +k) components is written as 
Tqk. If k = 2, we have a 
spherical tensor.

4.2.Spherical Tensor

We ca generalize the latter result to define a spherical
 tensor of rank k as a set of 2k + 1 operators: 
 Tqk; q = -k, ...+k, which 
under rotation R, they are transformed with the  
matrix of 2j+1 elements D(k)mm' = <j,m|exp{-iθJ/ħ|jm>} . 
That is:
U(R)TqkU+(R) = Σ(q) D(k)q'q'Tq'k

Let's recall:


D(k)mm' = <j,m|exp{-iθJ/ħ|jm>} 
U(R)TqkU+(R) = Σ(q) D(k)q'q'Tq'k


Using the expression of the infinitesimal rotation 
operator U(R(dθ)) = 1 - i εJ/ħ, we find the 
commutation relations:

[J+, Tqk] = +ħ [(k-q)(k+q+1)]1/2 Tq+1k
[J-, Tqk] = -ħ [(k+q)(k-q+1)]1/2 Tq-1k
[Jz, Tqk] = ħ q Tqk