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.

In the same way, tensor quantities must be represented by tensor operators. An example of a tensor quantity (of rank two) is the electrical quadrupole moment of the above molecule. Likewise, the octupole and hexadecapole moments would be tensors of rank three and four, respectively.

Other examples of scalar operators are the total energy operator (more commonly called the Hamiltonian), the potential energy, and the dipole-dipole interaction energy of two atoms. Examples of vector operators are the momentum, the position, the orbital angular momentum, ${\displaystyle {\mathbf {L} }}$, and the spin angular momentum, ${\displaystyle {\mathbf {S} }}$. (Fine print: Angular momentum is a vector as far as rotations are concerned, but unlike position or momentum it does not change sign under space inversion, and when one wishes to provide this information, it is said to be a pseudovector.)

Scalar, vector and tensor operators can also be formed by products of operators. For example, the scalar product ${\displaystyle {\mathbf {L} }\cdot {\mathbf {S} }}$ of the two vector operators, ${\displaystyle {\mathbf {L} }}$ and ${\displaystyle {\mathbf {S} }}$, is a scalar operator, which figures prominently in discussions of the spin–orbit interaction. Similarly, the quadrupole moment tensor of our example molecule has the nine components

$${\displaystyle Q_{ij}=\sum _{\alpha }q_{\alpha }\left(3r_{\alpha ,i}r_{\alpha ,j}-r_{\alpha }^{2}\delta _{ij}\right).}$$Here, the indices ${\displaystyle i}$ and ${\displaystyle j}$ can independently take on the values 1, 2, and 3 (or ${\displaystyle x}$, ${\displaystyle y}$, and ${\displaystyle z}$) corresponding to the three Cartesian axes, the index ${\displaystyle \alpha }$ runs over all particles (electrons and nuclei) in the molecule, ${\displaystyle q_{\alpha }}$ is the charge on particle ${\displaystyle \alpha }$, and ${\displaystyle r_{\alpha ,i}}$ is the ${\displaystyle i}$-th component of the position of this particle. Each term in the sum is a tensor operator. In particular, the nine products ${\displaystyle r_{\alpha ,i}r_{\alpha ,j}}$ together form a second rank tensor, formed by taking the outer product of the vector operator ${\displaystyle {\mathbf {r} }_{\alpha }}$ with itself.

The rotation operator about the unit vector n (defining the axis of rotation) through angle θ is

$${\displaystyle U[R(\theta ,{\hat {\mathbf {n} }})]=\exp \left(-{\frac {i\theta }{\hbar }}{\hat {\mathbf {n} }}\cdot \mathbf {J} \right)}$$