Angular Momentum

Abstract

Physical properties or combinations of them are symbolized by operators X, Y, ... obeying algebraic relations, X + Y = Z, XY = Z; states are symbolized by vectors <|, | >, with algebraic relations, $X\left| 1 \right\rangle = \langle 2|,\quad \left| 1 \right\rangle + \left| 2 \right\rangle = \left| 3 \right\rangle$ all this subject to the adjoint relations, such as A ⁺ = A for a Hermitian operator and $\langle 1|{X^ + } = \langle 2|$. There are numbers formed by the vectors and operators: $\left\langle {1|2} \right\rangle ,\left\langle {1|X|2} \right\rangle ,$ or equivalent traces, e. g., tr $\left\{ {X\left| 2 \right\rangle \langle 1|} \right\}$. Suppose one systematically redefines all vectors and operators:

$$\overline {\langle |} = \langle |U,\quad \overline {|\rangle } = {U^{ - 1}}|\rangle ,\quad \overline X = {U^{ - 1}}XU,$$

(3.1.1)

where U ⁺ = U ^-1 is a unitary operator. Then all algebraic adjoint and numerical relations are maintained

$$\left. {\begin{array}{*{20}{c}} {X + Y + Z} \\ {XY = Z} \\ {|1\rangle + |2\rangle = \;|3\rangle } \\ {X|1\rangle = |2\rangle } \\ {{{A}^{ + }} = A} \\ \end{array} } \right\} \to \left\{ {\begin{array}{*{20}{c}} {\overline X + \overline Y + \overline Z } \\ {\overline X \overline Y = \overline Z } \\ {\overline {|1\rangle } + \overline {|2\rangle } = \overline {\;|3\rangle } } \\ {\overline X \overline {|1\rangle } = \overline {|2\rangle } } \\ {{{{\overline A }}^{ + }} = \overline A } \\ \end{array} } \right.$$

(3.1.2)

and

$${\overline {\langle |} ^ + } = \overline {|\rangle } ,\quad \langle 1|2\rangle = \overline {\langle 1|2\rangle } = \overline {\langle 1|} \times \overline {|2\rangle } ,\quad \langle 1|X|2\rangle = \overline {\langle 1|} \overline X \overline {|2\rangle } $$

(3.1.3)