Whether something is a scalar, vector, or tensor depends on how it is viewed by two observers whose coordinate frames are related to each other by a rotation.
Alternatively, one may ask how, for a single observer, a physical quantity transforms if the state of the system is rotated.
The kinetic energy on the other hand must be represented by a scalar operator, whose expected value must be the same in the initial and the rotated states.
where J = (Jx, Jy, Jz) are the rotation generators (also the angular momentum matrices):
of the orbital angular momentum operator L and solutions of Laplace's equation on a 3d sphere are spherical harmonics:
Spherical harmonics are functions of the polar and azimuthal angles, ϕ and θ respectively, which can be conveniently collected into a unit vector n(θ, ϕ) pointing in the direction of those angles, in the Cartesian basis it is:
Any observable vector quantity of a quantum mechanical system should be invariant of the choice of frame of reference.
where the RHS is due to the rotation transformation acting on the vector formed by expectation values.
From the above relation for infinitesimal rotations and the Baker Hausdorff lemma, by equating coefficients of order
The above commutator rule can also be used as an alternative definition for vector operators which can be shown by using the Baker Hausdorff lemma.
[3] In the above example, we will show that the 9 independent tensor components can be divided into a set of 1, 3 and 5 combination of operators that each form irreducible invariant subspaces.
[4] These three terms are irreducible, which means they cannot be decomposed further and still be tensors satisfying the defining transformation laws under which they must be invariant.
Each of the irreducible representations T(0), T(1), T(2) ... transform like angular momentum eigenstates according to the number of independent components.
For example, the quadrupole moment tensor is already symmetric and traceless, and hence has only 5 independent components to begin with.
Hence, the above commutation relations and the transformation property are equivalent definitions of spherical tensor operators.
[5] One way is to specify how spherical tensors transform under a physical rotation - a group theoretical definition.
Using the infinitesimal rotation operator and its Hermitian conjugate, one can derive the commutation relation in the spherical basis:
.^ This may include constant due to normalization from spherical harmonics which is meaningless in context of operators.
which raise or lower the orbital magnetic quantum number mℓ by one unit.
which raise or lower the spin magnetic quantum number ms by one unit.
The transition amplitude is proportional to matrix elements of the dipole operator between the initial and final states.
The position operator r has three components, and the initial and final levels consist of 2ℓ + 1 and 2ℓ′ + 1 degenerate states, respectively.
where, we have multiplied each Y1m by the radius r. On the right hand side we see the spherical components rq of the position vector r. The results can be summarized by,
This equation reveals a relationship between vector operators and the angular momentum value ℓ = 1, something we will have more to say about presently.
We see that all the dependence on the three magnetic quantum numbers (m′,q,m) is contained in the angular part of the integral.
The radial integral is independent of the three magnetic quantum numbers (m′, q, m), and the trick we have just used does not help us to evaluate it.
The example we have just given of simplifying the calculation of matrix elements for a dipole transition is really an application of the Wigner–Eckart theorem, which we take up later in these notes.
The spherical tensor formalism provides a common platform for treating coherence and relaxation in nuclear magnetic resonance.
In NMR and EPR, spherical tensor operators are employed to express the quantum dynamics of particle spin, by means of an equation of motion for the density matrix entries, or to formulate dynamics in terms of an equation of motion in Liouville space.
The Liouville space equation of motion governs the observable averages of spin variables.