Angular Momentum As The Generator of Rotations
See also: Total angular momentum quantum numberThe most general and fundamental definition of angular momentum is as the generator of rotations. More specifically, let be a rotation operator, which rotates any quantum state about axis by angle . As, the operator approaches the identity operator, because a rotation of 0° maps all states to themselves. Then the angular momentum operator about axis is defined as:
where 1 is the identity operator. As a consequence,
where exp is matrix exponential.
In simpler terms, the total angular momentum operator characterizes how a quantum system is changed when it is rotated. The relationship between angular momentum operators and rotation operators is the same as the relationship between Lie algebras and Lie groups in mathematics, as discussed further below.
Just as J is the generator for rotation operators, L and S are generators for modified partial rotation operators. The operator
rotates the position (in space) of all particles and fields, without rotating the internal (spin) state of any particle. Likewise, the operator
rotates the internal (spin) state of all particles, without moving any particles or fields in space. The relation J=L+S comes from:
i.e. if the positions are rotated, and then the internal states are rotated, then altogether the complete system has been rotated.
Read more about this topic: Angular Momentum Operator
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