Vector operator

In quantum mechanics, a vector operator is an operator that transforms like a vector under rotations . It is a special case of a tensor operator.

In the angular momentum algebra of quantum mechanics, expectation values of vector operators (and generally of tensor operators) can be reduced to a few reduced matrix elements with the help of the Wigner-Eckart theorem .

The abstract mathematical definition is explained in more detail below. A vector operator creates morphisms between state vectors and has a special transformation behavior under rotations. The state vector space is the Hilbert space and the rotating group is the . ${\displaystyle {\mathcal {H}}:={\mathcal {L}}^{2}(\mathbb {R} ^{3};\mathbb {C} )}$ ${\displaystyle \operatorname {SO} (3)}$

Formal definition

The rotation group operates canonically (covariant) on , on and on its tensor product . A vector operator is then a morphism of representations ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle {\mathcal {H}}}$ ${\displaystyle A}$

${\displaystyle A\colon {\mathcal {H}}\to \mathbb {R} ^{3}\otimes {\mathcal {H}}}$,

d. H. a vector space homomorphism that commutes with rotations.

properties

If the canonical base of is , one can write: ${\displaystyle \{e_{i}\}}$ ${\displaystyle \mathbb {R} ^{3}}$

${\displaystyle A\colon \psi \mapsto \sum _{i}e_{i}\otimes A_{i}\psi }$.

If you suppress all structure, it becomes:

${\displaystyle A\psi =(A_{1}\psi ,A_{2}\psi ,A_{3}\psi )\in {\mathcal {H}}^{3}}$.

If one conjugates with a rotation (that is the natural operation of rotations on such morphisms), then this yields the identity in this notation: ${\displaystyle A}$

${\displaystyle R.A_{i}=\sum _{k}A_{k}R_{ki}=\sum _{k}R_{ik}^{-1}A_{k}}$, which is used as a definition in some places.

Because it is . ${\displaystyle R.A=D(R)\circ A\circ D(R^{-1})\colon \psi \mapsto D(R)\left(\sum _{i}e_{i}\otimes A_{i}\psi \right)\circ D(R^{-1})}$ ${\displaystyle =\sum _{i}Re_{i}\otimes A_{i}\psi =\sum _{ik}R_{ki}e_{i}\otimes A_{k}(\psi )=\sum _{i}e_{i}\otimes \left(\sum _{k}A_{k}R_{ki}\right)(\psi )}$

Examples

Angular momentum operator ${\displaystyle {\hat {\mathbf {J} }}=({\hat {J}}_{x},{\hat {J}}_{y},{\hat {J}}_{z})}$

Spin operator ${\displaystyle {\hat {\mathbf {S} }}=({\hat {S}}_{x},{\hat {S}}_{y},{\hat {S}}_{z})}$

Transition dipole moment ${\displaystyle {\hat {\mathbf {M} }}=({\hat {M}}_{x},{\hat {M}}_{y},{\hat {M}}_{z})}$

Generalizations

A level tensor operator ${\displaystyle k}$ is a morphism of representations

${\displaystyle A\colon {\mathcal {H}}\to \mathbb {R} ^{3k}\otimes {\mathcal {H}}}$,

where the rotating group operates on as on . ${\displaystyle \mathbb {R} ^{3k}}$ ${\displaystyle \mathbb {R} ^{3^{\oplus k}}}$

This gives the equation in implicit notation

${\displaystyle R.T_{I}=D(R)T_{i_{1},\dots ,i_{k}}D(R^{-1})=\sum T_{j_{1},\dots ,j_{k}}R_{j_{1},i_{1}}\dots R_{j_{k},i_{k}}}$