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In the 3rd edition of Sakurai Quantum Mechanics, section 8.2 on Dirac's equation, Sakurai writes that the Dirac equation satisfies the continuity equation $$\frac{\partial\rho}{\partial t}+\nabla\cdot\boldsymbol{j}=0,$$ where $$\boldsymbol{j}=\Psi^\dagger\alpha\Psi,$$ and says this is easy to prove by the Schrodinger's equation and its adjoint.

My question is: aren't we doing the Dirac's equation? Why does Sakurai says to use the Schrodinger's equation if the two equation acts on different type of objects? In the previous page Sakurai also confuses me when he says, "which form of the Dirac equation we use etc", he mentions equation 8.1 which is $i\hbar\frac{\partial}{\partial t}|\psi(t)\rangle=H|\psi(t)\rangle$ the Schrodinger's equation.

Is there a "Schrodinger's equation" for the Dirac's equation somehow? Thank you for your help, I'm just starting on relativistic QM.

$\textbf{Never mind, I see what he means now. }$

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  • $\begingroup$ Sorry. I saw your edit after I posted my answer. $\endgroup$
    – joseph h
    Oct 24, 2021 at 4:31

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He may be referring to the quantity $${\tag1}\boldsymbol{j}^\mu=\bf\Psi^\dagger\gamma^\mu\Psi$$ from the Dirac treatment being analogous to the quantity $$j=\psi^*\psi$$ in the Schrodinger treatment. In each case, they both satisfy a continuity equation $$\frac{\partial\rho}{\partial t}+\nabla\cdot\boldsymbol{j}=0$$ where each $j$ is a current density.

He may be referring to which equation corresponds to which situation. The $\bf\Psi$ appearing in the Dirac equation $$(i\gamma^\mu\partial_{\mu} -m)\Psi=0$$

is not a wavefunction and it is also not a vector in a Hilbert space (but when it acts on the vacuum it produces a one-particle state that is a vector in a Hilbert space). Its square in equation (1) is not a probability density as in the Schrodinger case (it could be a charge density, mass density, etc). The $\bf\Psi$ field is an operator and the Dirac equation reduces to the Schrodinger equation in the non-relativistic limit.

As such, there is no "Schrodinger's equation for the Dirac's equation" but two analogous equations that apply to the relativistic (with spin)$^1$ and non-relativistic regimes.

$^1$ Of course the Klein-Gordon equation is also relativistically covariant but for spinless (scalar) fields.

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