Spin in the Nonrelativistic Hamiltonian

One of the missing features in the usual form of the Schrodinger equation is spin, and it is often stated that this is because it is a relativistic phenomenon. This is at best a tenuous statement, and we will show that there is a method for introducing spin into the nonrelativistic equation. This demonstration also serves to introduce some quantities that will turn up later in a fully relativistic equation that accounts for spin. Among these are the 2 X 2 Pauli spin matrices  

This demonstration was first made by Levy-Leblond (1967, 1970). 

Similar relations for the other products of Pauli matrices may be derived by cyclic permutation of the indices. In fact, the Pauli matrices anticommute  

We will often be dealing with the scalar product of a with various vectors u that is, with (ff u). Using the commutation properties of the Pauli matrices, (4.13), it may be shown that 

These 2x2 matrices can be regarded as operators on a space of two-component vector functions, 

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