Spherical tensor operator

In pure and applied mathematics, quantum mechanics and computer graphics, a tensor operator generalizes the notion of operators which are scalars and vectors. A special class of these are spherical tensor operators which apply the notion of the spherical basis and spherical harmonics. The spherical basis closely relates to the description of angular momentum in quantum mechanics and spherical harmonic functions. The coordinate-free generalization of a tensor operator is known as a representation operator.

In quantum mechanics, physical observables that are scalars, vectors, and tensors, must be represented by scalar, vector, and tensor operators, respectively. 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. Consider, for example, a system consisting of a molecule of mass ${\displaystyle M}$, traveling with a definite center of mass momentum, ${\displaystyle p{\mathbf {\hat {z}} }}$, in the ${\displaystyle z}$ direction. If we rotate the system by ${\displaystyle 90^{\circ }}$ about the ${\displaystyle y}$ axis, the momentum will change to ${\displaystyle p{\mathbf {\hat {x}} }}$, which is in the ${\displaystyle x}$ direction. The center-of-mass kinetic energy of the molecule will, however, be unchanged at ${\displaystyle p^{2}/2M}$. The kinetic energy is a scalar and the momentum is a vector, and these two quantities must be represented by a scalar and a vector operator, respectively. By the latter in particular, we mean an operator whose expected values in the initial and the rotated states are ${\displaystyle p{\mathbf {\hat {z}} }}$ and ${\displaystyle p{\mathbf {\hat {x}} }}$. 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.

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