Adding Angular Momenta 4

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Adding Angular Momenta

Please study Adding Angular Momenta before reading the page below.

Here L = J + K holds.

The J and K operators listed below obey the same rules that
the L operators obey in Angular Momentum Operators.

The Q_l operators listed below are defined in Singlet and Triplet Operators.
For a particular j, k pair, all the different Q_l should total to the identity matrix.

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Explicit Matrices for Operators in | m_J m_K > bases

| m_J m_K > is a short-hand for the | j m_J > | k m_K > state.

In the following matrices,

the left-most column and top-most row represent the state with m_J=+j and m_K=+k,

the right-most column and bottom-most row represent the state with m_J=-j and m_K=-k,

the value of m_K changes between each row and column, and

the value of m_J usually stays the same between each row and column.

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Case with j=1 and k=1

Operator	Matrix in \| m_J m_K > basis
J_x	0 0 0 1/sqrt(2) 0 0 0 0 0 0 0 0 0 1/sqrt(2) 0 0 0 0 0 0 0 0 0 1/sqrt(2) 0 0 0 1/sqrt(2) 0 0 0 0 0 1/sqrt(2) 0 0 0 1/sqrt(2) 0 0 0 0 0 1/sqrt(2) 0 0 0 1/sqrt(2) 0 0 0 0 0 1/sqrt(2) 0 0 0 1/sqrt(2) 0 0 0 0 0 0 0 0 0 1/sqrt(2) 0 0 0 0 0 0 0 0 0 1/sqrt(2) 0 0 0
K_x	0 1/sqrt(2) 0 0 0 0 0 0 0 1/sqrt(2) 0 1/sqrt(2) 0 0 0 0 0 0 0 1/sqrt(2) 0 0 0 0 0 0 0 0 0 0 0 1/sqrt(2) 0 0 0 0 0 0 0 1/sqrt(2) 0 1/sqrt(2) 0 0 0 0 0 0 0 1/sqrt(2) 0 0 0 0 0 0 0 0 0 0 0 1/sqrt(2) 0 0 0 0 0 0 0 1/sqrt(2) 0 1/sqrt(2) 0 0 0 0 0 0 0 1/sqrt(2) 0
L_x	0 1/sqrt(2) 0 1/sqrt(2) 0 0 0 0 0 1/sqrt(2) 0 1/sqrt(2) 0 1/sqrt(2) 0 0 0 0 0 1/sqrt(2) 0 0 0 1/sqrt(2) 0 0 0 1/sqrt(2) 0 0 0 1/sqrt(2) 0 1/sqrt(2) 0 0 0 1/sqrt(2) 0 1/sqrt(2) 0 1/sqrt(2) 0 1/sqrt(2) 0 0 0 1/sqrt(2) 0 1/sqrt(2) 0 0 0 1/sqrt(2) 0 0 0 1/sqrt(2) 0 0 0 1/sqrt(2) 0 0 0 0 0 1/sqrt(2) 0 1/sqrt(2) 0 1/sqrt(2) 0 0 0 0 0 1/sqrt(2) 0 1/sqrt(2) 0
L_x²	1 0 1/2 0 1 0 1/2 0 0 0 3/2 0 1 0 1 0 1/2 0 1/2 0 1 0 1 0 0 0 1/2 0 1 0 3/2 0 1/2 0 1 0 1 0 1 0 2 0 1 0 1 0 1 0 1/2 0 3/2 0 1 0 1/2 0 0 0 1 0 1 0 1/2 0 1/2 0 1 0 1 0 3/2 0 0 0 1/2 0 1 0 1/2 0 1
J_x K_x	0 0 0 0 1/2 0 0 0 0 0 0 0 1/2 0 1/2 0 0 0 0 0 0 0 1/2 0 0 0 0 0 1/2 0 0 0 0 0 1/2 0 1/2 0 1/2 0 0 0 1/2 0 1/2 0 1/2 0 0 0 0 0 1/2 0 0 0 0 0 1/2 0 0 0 0 0 0 0 1/2 0 1/2 0 0 0 0 0 0 0 1/2 0 0 0 0
	memorize

Operator	Matrix in \| m_J m_K > basis
J_y	0 0 0 -i/sqrt(2) 0 0 0 0 0 0 0 0 0 -i/sqrt(2) 0 0 0 0 0 0 0 0 0 -i/sqrt(2) 0 0 0 i/sqrt(2) 0 0 0 0 0 -i/sqrt(2) 0 0 0 i/sqrt(2) 0 0 0 0 0 -i/sqrt(2) 0 0 0 i/sqrt(2) 0 0 0 0 0 -i/sqrt(2) 0 0 0 i/sqrt(2) 0 0 0 0 0 0 0 0 0 i/sqrt(2) 0 0 0 0 0 0 0 0 0 i/sqrt(2) 0 0 0
K_y	0 -i/sqrt(2) 0 0 0 0 0 0 0 i/sqrt(2) 0 -i/sqrt(2) 0 0 0 0 0 0 0 i/sqrt(2) 0 0 0 0 0 0 0 0 0 0 0 -i/sqrt(2) 0 0 0 0 0 0 0 i/sqrt(2) 0 -i/sqrt(2) 0 0 0 0 0 0 0 i/sqrt(2) 0 0 0 0 0 0 0 0 0 0 0 -i/sqrt(2) 0 0 0 0 0 0 0 i/sqrt(2) 0 -i/sqrt(2) 0 0 0 0 0 0 0 i/sqrt(2) 0
L_y	0 -i/sqrt(2) 0 -i/sqrt(2) 0 0 0 0 0 i/sqrt(2) 0 -i/sqrt(2) 0 -i/sqrt(2) 0 0 0 0 0 i/sqrt(2) 0 0 0 -i/sqrt(2) 0 0 0 i/sqrt(2) 0 0 0 -i/sqrt(2) 0 -i/sqrt(2) 0 0 0 i/sqrt(2) 0 i/sqrt(2) 0 -i/sqrt(2) 0 -i/sqrt(2) 0 0 0 i/sqrt(2) 0 i/sqrt(2) 0 0 0 -i/sqrt(2) 0 0 0 i/sqrt(2) 0 0 0 -i/sqrt(2) 0 0 0 0 0 i/sqrt(2) 0 i/sqrt(2) 0 -i/sqrt(2) 0 0 0 0 0 i/sqrt(2) 0 i/sqrt(2) 0
L_y²	1 0 -1/2 0 -1 0 -1/2 0 0 0 3/2 0 1 0 -1 0 -1/2 0 -1/2 0 1 0 1 0 0 0 -1/2 0 1 0 3/2 0 -1/2 0 -1 0 -1 0 1 0 2 0 1 0 -1 0 -1 0 -1/2 0 3/2 0 1 0 -1/2 0 0 0 1 0 1 0 -1/2 0 -1/2 0 -1 0 1 0 3/2 0 0 0 -1/2 0 -1 0 -1/2 0 1
J_y K_y	0 0 0 0 -1/2 0 0 0 0 0 0 0 1/2 0 -1/2 0 0 0 0 0 0 0 1/2 0 0 0 0 0 1/2 0 0 0 0 0 -1/2 0 -1/2 0 1/2 0 0 0 1/2 0 -1/2 0 -1/2 0 0 0 0 0 1/2 0 0 0 0 0 1/2 0 0 0 0 0 0 0 -1/2 0 1/2 0 0 0 0 0 0 0 -1/2 0 0 0 0
	memorize

Operator	Matrix in \| m_J m_K > basis
J_z	1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 -1
K_z	1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1
L_z	2 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 -2
L_z²	4 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 4
J_z K_z	1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
	memorize

Operator	Matrix in \| m_J m_K > basis
L²	6 0 0 0 0 0 0 0 0 0 4 0 2 0 0 0 0 0 0 0 2 0 2 0 0 0 0 0 2 0 4 0 0 0 0 0 0 0 2 0 4 0 2 0 0 0 0 0 0 0 4 0 2 0 0 0 0 0 2 0 2 0 0 0 0 0 0 0 2 0 4 0 0 0 0 0 0 0 0 0 6
J • K	1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 -1 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 -1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1
Q₀	0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1/3 0 -1/3 0 1/3 0 0 0 0 0 0 0 0 0 0 0 0 0 -1/3 0 1/3 0 -1/3 0 0 0 0 0 0 0 0 0 0 0 0 0 1/3 0 -1/3 0 1/3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Q₁	0 0 0 0 0 0 0 0 0 0 1/2 0 -1/2 0 0 0 0 0 0 0 1/2 0 0 0 -1/2 0 0 0 -1/2 0 1/2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1/2 0 -1/2 0 0 0 -1/2 0 0 0 1/2 0 0 0 0 0 0 0 -1/2 0 1/2 0 0 0 0 0 0 0 0 0 0
Q₂	1 0 0 0 0 0 0 0 0 0 1/2 0 1/2 0 0 0 0 0 0 0 1/6 0 1/3 0 1/6 0 0 0 1/2 0 1/2 0 0 0 0 0 0 0 1/3 0 2/3 0 1/3 0 0 0 0 0 0 0 1/2 0 1/2 0 0 0 1/6 0 1/3 0 1/6 0 0 0 0 0 0 0 1/2 0 1/2 0 0 0 0 0 0 0 0 0 1
	memorize

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