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The difference between the Pauli equation and the Schrödinger equation for a charged particle in an electric field is that one has the Pauli matrix and the other doesn't. How do you combine the Pauli matrix and the Hamiltonian to get the Schrodinger-Pauli equation? $$H=\frac{1}{2m}\left(p-\frac{q}{c}A\right)^2+q\phi$$ $$\left[\frac{1}{2m}\left(\sigma\cdot\left(p-\frac{q}{c}A\right)^2\right)+q\phi\right]|\psi\rangle=i\frac{h}{2\pi}\frac{\partial}{\partial t}|\psi\rangle$$

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The Schrödinger equation is always the time evolution equation in the Schrödinger picture and is given by $$ \mathrm{i}\hbar\partial_t\lvert\psi(t)\rangle = H\lvert \psi(t)\rangle.$$ This means that the Pauli equation is obviously not the evolution equation for the Hamiltonian $H$ you write down - because that Hamiltonian is the Hamiltonian of a spinless charged particle, while the Pauli equation describes a spin-1/2 charged particle. The correct Hamiltonian for such a particle is simply the l.h.s. of the Pauli equation, i.e. $$ H_P = \frac{1}{2m}(\vec\sigma\cdot(\vec p - q\vec A))^2 + q\phi.$$ The correct form of this Hamiltonian, like all Hamiltonians, is eventually guessed by matching it to experiment, in this case to the Stern-Gerlach experiment, as described on the Wikipedia page for the Pauli equation.

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