Relativity theory, quantum mechanics and spin

The time-dependent Schrodinger equation (2.43) presents a serious problem from the point of view of relativity theory it does not treat space and time in a symmetric way, because second-order derivatives of the wavefunction with respect to spatial coordinates are accompanied by a first-order derivative with respect to time. One way out, as actually proposed by Schrodinger and later known as the Klein-Gordon equation, would be to have also second-order derivatives with respect to time. However, that would lead to a total probability for the particle under consideration which would be a function of time, and to a variation of the number of particles of the universe (which, at the time, was completely unacceptable). In 1928, Dirac sought the solution for this problem, by accepting first-order derivation in the case of time and forcing the spatial derivatives to also be first order. This requires the wavefunction to have four components (functions of the spatial coordinates alone), often called a four-component spinor . 

In addition, Dirac s theory provides a direct explanation for the fact that the electron magnetic dipole moment is about twice the value expected classically on the basis of a spherical charged particle rotating around one 

It has been noted that spin occurs naturally in Dirac s theory not because it is a relativistic phenomenon, in the usual sense of this expression, but because the wavefunction has the correct properties. In some interpretations, it is not even a property of the electron (and other particles) but a manifestation of the interaction between the electron and the vacuum, whose field acquires angular momentum, besides the angular momentum the particle may have (see ref. 24). 

For the case of an electron, or any other particle of spin 1/2 (a fermion) considered free in a one-dimensional (the z direction, say) potential well of infinite walls, the time-independent relativistic wave equation 

Problem 2.14 Identify the Pauli matrix above in the following form of the spin angular momentum along the z direction 

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