Spherical Tensor Operators in NMR

Muhammad Sabieh Anwar

March 5, 2004

1. Quick Review of Vector Operators: Considering an infinitesimal rotation through δabout the axis Ju = J.u, the components of a vector operator V, transform Vi → V ′

i

according to the following transformation rules:

ViD(R)−−−→ V ′

i = D(R)ViD†(R) (1a)

= Vi − iδ

h[Vi, Ju] (1b)

MoreoverD†(R)ViD(R) =

∑

j

Rij Vj (1c)

alsoD(R) = 1− exp (−i

δ

hJu) (1d)

The equality in (1c) holds because of the following concepts:

(a) The expectation value of a component 〈α|Vi|α〉 remains unchanged in the rotated frame(with respect to the transformed operator), i.e., 〈α′|V ′

i |α′〉 = 〈α|Vi|α〉 where |α〉 −→D(R)|α〉 = |α′〉 and V ′

i is the operator transform. The expectation value in the twobases must be the same, because we are assuming that space is isotropic, and all physicalobservables and physical laws must remain invariant under the rotation of the entiresystem, including the measuring apparatus. In such a scenario, V ′

i |α′〉 = Vi|α〉.(b) The expectation value 〈α|Vi|α〉 is unchanged with respect to transformed operator and

kets, but with respect to the kets and operators in the un-rotated frame, it trans-forms like the components of a cartesian vector, i.e., 〈α|Vi|α〉 −→

∑j Rij〈α|Vj |α〉 =

〈α′|Vi|α′〉 = 〈α|D†(R)ViD(R)|α〉. This last equation is not to be confused with theequality 〈α′|V ′

i |α′〉 = 〈α|Vi|α〉, in which both the operator and the kets are transformed.

(c) Since the matrix with elements Rij is orthogonal, i.e.,∑

k RikRjk = δij , (1c) can alsobe written as:

D(R)ViD†(R) =

∑

j

VjRji (1e)

The above equation will be used as our starting point for defining tensor operators.

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The defining transformation rule for vector operators can also be written in a more compactform:

[Vi, Jj ] = ihεijkVk (1f)

We have observed that scalar and vector operators are defined through the behaviour underrotations. Likewise we can also define tensor operators by characterizing their behaviourunder transformations — scalar and vector operators being particular cases of these tensoroperators (ranks 0 and 1 respectively).

2. Elaboration of point (b) Rotations in Quantum Mechanics present some confusion be-cause they involve a rotation of both the variables as well as the states. Another confusionarises because of the possibility of active and passive rotations. We shall be concerned withpassive rotations, which involve a rotation of the co-ordinate axes – affording what is calleda basis transformation. The active conceptualism is totally equivalent, however it pays tofocus on just either of the two! Consider a state represented by a ket |ψ〉 in the Hilbertspace. This state will be depend on a dynamic variable, like the position co-ordinate −→r . Acomplete description of the state is thus |ψ(−→r )〉. Now we rotate the co-ordinate axes suchthat the position vector transforms as −→r 7→ −→r ′. We assume that this rotation in the physicalspace is represented by a rotation matrix D , such that D−→r = −→r ′. In the new basis, thestate will also have been modified. We define the transformation of the state |ψ〉 7→ |ψ′〉,such that:

|ψ′(−→r ′)〉 = |ψ(−→r )〉 (2a)

(2a) would strictly be true upto some global phase e iφ. It expresses our intuition that allobservables and spectra of eigenvalues remain unchanged when both the variable and thestate are rotated: the same idea as has been expressed in [1b] as 〈α′|V ′

i |α′〉=〈α|Vi|α〉. From(2a), we continue:

|ψ′(−→r ′)〉 = |ψ(D−1−→r ′)〉 (2b)

As −→r ′ is just an arbitrary variable, we can replace it by −→r :

|ψ′(−→r )〉 = |ψ(D−1−→r )〉 = D |ψ(−→r )〉 (2c)

D is unitary because the norms of D |ψ−→r 〉 and |ψ−→r 〉 are the same. Moreover from the lastequality in (2c), note that we are using the same rotation matrix D for rotations of both theco-ordinates and the states. If we are considering rotations in the real, physical space, allelements of the matrix representing D are real and hence its representation is a symmetricmatrix. In general Hilbert space, the matrix representation of an arbitrary rotation R isdenoted as D(R).

3. Cartesian Tensor Operator Motivated by the definition of a vector operator presented in(1e) , we can define a tensor operator T with components {Ts} which transforms accordingto the equation:

Ti −→ T ′i = D(R)TiD†(R) =

∑s

TsDs,i(R) (3)

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Given two vector operators V and W, we can form a cartesian tensor with 9 componentsViWj where i, j = 1, 2, 3. This cartesian tensor, however is not irreducible and it is notconvenient to work with it. The component ViWj can be reduced into a scalar, a tensorproduct (or antisymmetric tensor of rank 1) and a traceless, symmetric tensor of rank 2 asexpressed below:

ViWj =V.W

3δij +

(ViWj − WiVj)2

+( ViWj + WiVj

2− V.W

3δij

)(4)

The number of components corresponding to these tensors of ranks 0, 1 and 2 are 1, 3 and 5respectively, which add up to 9. These numbers correspond well to 2l +1 components of thespherical harmonic functions Y m

l of rank l where l = 0, 1 and 2. In fact the irreducible tensoroperators transform just like the spherical harmonic functions and hence they are most oftencalled the spherical tensor operators.

4. Transformation of Spherical Harmonics under Rotations The spherical harmonicfunction parametrically depends on the angles θ and φ in the spatial co-ordinate basis andis given by the wavefunction representation in space as:

Y ml (θ, φ) = 〈n|l,m〉 (5)

where n is a unit vector that points in the θ,φ direction. Now consider the rotation |n〉 7−→|n′〉 = D(R)|n〉. Taking the inner product with the bra 〈l,m|, we arrive at the relationship:

〈l,m|n′〉 = 〈l, m|D(R)|n〉〈n′|l, m〉 = 〈n|D(R)†|l, m〉

=⇒ Y ml (n′) = 〈n|D(R)†|l, m〉

=∑

m′〈n|l, m′〉〈l, m′|D(R)|l, m〉 (6a)

where in (6a), we have suppressed the sum over all possible values of l, because of theinvariance of the E (l) sub-space. Finally (6a) can written in the more amenable form as:

Y ml (n′) =

∑

m′Y m′

l (n)D lm′,m(R) (6b)

5. Definition of Spherical Tensor Operators Being motivated by the transformation rela-tions of tensors (3) and spherical harmonics (5), we can define a tensor operator of rank kas an operator whose 2k +1 components transform according to the relations given below:

D†(R)T kq D(R) =

k∑

q′=−k

D kq,q′(R)T k

q′ (7a)

If we write the transpose of (7a), we obtain the following definition of a tensor operator ofrank k:

D(R)T kq D†(R) =

k∑

q′=−k

T kq′D

kq′,q(R) (7b)

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Note the similarity of (7b) with (1e). The tensor operators T kq transform in a similar fashion

to the spherical harmonic wavefunctions Y ml with k = l and q = m. Some authors prefer to

use the notation T qk , whereas I shall use the former. A more convenient operational definition

of a spherical tensor operator, analogous to the relation given in (1f) is given by the followingcommutation rleations that must be satisfied by a tensor operator:

[Jz, Tkq ] = hq T k

q (8a)and

[J±, T kq ] = h

√k(k + 1)−m(m± 1)T k

q±1 (8b)

6. Spherical Tensor Operator Reduction of a Rank 1 Tensor (Vector) A vector Ahas components {Ax, Ay, Az}. These components do not form an irreducible representation.However, we can find the following so-called standard components of A that transform likethe components of a rank 1 tensor, T 0,±1

1 :

T 10 = Az (9a)

T 11 = − 1√

2(Ax + iAy) (9b)

T 1−1 =

1√2(Ax − iAy) (9c)

It can be verified that this spherical tensor operator of rank 1 satisfies the commutationrelations given in (8). A simple recipe for the formation of tensor operators is by using thethe general forms of the spherical tensor operators, Y m

l (θ, φ) = Y ml (n) that are generally

presented in standard tables. The unit vector n can be replaced by an arbitrary vector V,with components {Vx, Vy, Vz}. The individual components of the unit vector are replacedaccording to the relations (n)z = z/r 7→ Vz, (n)x = x/r 7→ Vx and (n)y = y/r 7→ Vz. Weconsider the example of the formation of the components of a rank 1 tensor operator fromthe spherical harmonics Y 0,±1

1 :

Y 01 =

√34π

cos θ =

√34π

z

r=⇒ T 1

0 = aVz (10a)

Y ±11 = ∓

√38π

e±iφsin θ = ∓√

38π

x± iy

r=⇒ T ±1

0 = ∓b (Vx ± Vy) (10b)

In deriving (10), we have also made use of the substitutions commonplace in the sphericalcoordinate system, i.e., x = r cos φ sin θ, y = r sin φ sin θ and z = r cos θ. The coefficientsa and b above will be determined by the normalization of the tensor operators. We need tore-normalize because we have replaced a unit vector n by an arbitrary vector V.

7. A straightforward but useful point to remember is that the spherical tensor operators areirreducible in standard E (α, J) basis, where α represents the eigenvalues of the operator H,such that {H, J2, Jz} form a C.S.C.O.

8. Construction of Higher Rank Tensors We can also extend the same technique to gener-ate tensor of any higher rank, for example, we can consider the formation of the components

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of a rank 2 tensor:

Y ±22 =

√1532π

e±2iφsin 2θ

=

√1532π

x2 − y2 ± 2ixy

r2

=⇒ T ±22 = c (Vx ± iVy)2 (10c)

The following theorem helps generate an arbitrary tensor T kq of rank k from tensors X k1

q1

and Z k2q2

with ranks k1 and k2 respectively:

T kq =

∑q1

∑q2

〈k1, k2; q1, q2|k1, k2; k, q〉X k1q1

Z k2q2

(11)

the indices q1 and q2 in (11) run from to −k1 to k1 and from −k2 to k2 respectively. More-over the inner product 〈k1, k2; q1, q2|k1, k2; k, q〉 can be recognized to be the Clebsch-Gordoncoefficient C(j1, j2,m1,m2, J,M) with j1 = k1, j2 = k2, m1 = q1, m2 = q2, J = k andM = q. This co-efficient will be non-zero only if q = q1 + q2. The similarity of (11) withthe relationship between the direct product and total angular momentum bases must beappreciated:

|J,M〉 =∑m1

∑m2

〈j1, j2;m1,m2|j1, j2; J,M〉 |j1, j2; m1,m2〉 (12)

It follows from (11) and (12), that higher rank tensors can be obtained by combining two lowrank tensors in almost the same way in which two individual angular momenta are added.Both methods are essentially the same, employing the same C-G co-efficients.

9. Spherical Tensor Operator Basis for NMR Different operator bases for an NMR systemcan be constructed. Each basis provides its own set of merits and demerits. For an n-dimensional system we need n2(= 4N ) operators in the operator space, where N is thenumber of spins, if we are only considering 1/2 spins. (See the document ”Superoperatorsin NMR” for a discussion of operators and superoperators). One useful basis is the setof spherical tensor operators. For a system of spin with a single spin I, we shall obtainbasis tensor operators of ranks 0, 1, . . . , 2I. For a single spin 1/2 the tensor operators in theZeeman basis are simply:

T 00 =

12

(1 00 1

)=

121 = E

T 10 =

12

(1 00 −1

)= −Iz

T 11 =

12

(0 −10 0

)= −I+

T 1−1 =

12

(0 01 0

)= I− (13)

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While constructing spherical tensor operators, we make note of the fact that the operatorsmust form an orthonormal set, defined as:

Tr(U†i Uj) = δij (14)

Moreover the following phase convention is respected for tensor operators, in accordancewith the Condon-Shortley phase convention for spherical harmonics:

(T kq )† = (−1)qT k

−q (15)

Equipped with the conditions set forth in (14) and (15), and the commutation relations givenin (8), we can derive all spherical tensor basis operators. For example, a single spin 1 willhave a set of 9 basis operators, explicitly written in the Zeeman eigenbasis as:

T 00 =

13

1 0 00 1 00 0 1

=

131

T 11 = − 1√

2

0 1 00 0 10 0 0

= −1

2I+

T 10 =

1√2

1 0 00 0 00 0 −1

=

1√2

Iz

T 1−1 =

1√2

0 0 01 0 00 1 0

=

12

I− (16)

Likewise we also have the following rank 2 operators:

T 22 =

0 0 10 0 00 0 0

T 21 =

1√2

0 1 00 0 −10 0 0

T 20 =

1√6

0 1 00 −2 00 0 1

T 2−1 = − 1√

2

0 0 01 0 00 −1 0

= (−1)−1 (T 2

1 )†

T 2−2 =

0 0 00 0 01 0 0

= (−1)2 (T 2

2 )† (16 contd.)

As an additional example, let us construct the spherical operators for the two-spin case,using (11). We need 16 tensor operators. Inspection shows that they will be of ranksk = 0, 1, 2 and 3.

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References

1. Sakurai J.J., Modern Quantum Mechanics, Addison-Wesley Publishing, 1994 (Ch3, pp152).

2. Tannoudji C. et. al, Quantum Mechanics Vols. 1 and 2., (Ch6, 10).

3. Messiah A.A., Potter J., Quantum Mechanics, North Holland Publishing Compnay, 1961,Vol. 2, Ch. 8.

4. Schiff L.I., Quantum Mechanics, McGraw Hill Book Company, 1968, Ch.7, Sec. 28.

5. Mayne C.L., Liouville Equation of Motion, Encycl. Nucl. Mag. Res., 2717-2730.

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