Electron Spin in Nonrelativistic Quantum Mechanics

The spin angular momentum has homologous properties to the orbital angular momentum, namely, 

To match the two-valuedness of the electron spin and the commutating properties shown above, Pauli introduced spin matrices, a,-. Each component of spin can then be written in terms of the 2x2 spin matrices. 

Then the eigenvalue of the Oi is - -1. Because the spin matrices are 2x2 matrices, the internal state of electrons must be represented by two-column matrices, 

The complex conjugate of the Pauli spinor is given by a two-row matrix, 

Therefore, the spin operators must be 2 x 2 matrices. When the eigenvalue is either —1 or -pl assuming that the quantization is observed in the z direction, it is readily seen that 

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