The Dirac equation for an electron in the presence of an electromagnetic 4-potential $A_\mu$, where $\hbar=c=1$, is given by


I assume the Weyl basis so that

$$\psi=\begin{pmatrix}\psi_L\\\psi_R\end{pmatrix}\hbox{ and }\gamma_0=\begin{pmatrix}0&I\\I&0\end{pmatrix}.\tag{2}$$

I assume that the electron is stationary so that $${\bf\hat{p}}\psi=-i\nabla\psi=(0,0,0).\tag{3}$$

Finally I assume that an electric potential $\phi_{E}$ exists so that we have $$A_\mu=(-\phi_{E},0,0,0).\tag{4}$$

Substituting into the Dirac equation $(1)$ we find $$i\begin{pmatrix}0&I\\I&0\end{pmatrix}\frac{\partial}{\partial t}\begin{pmatrix}\psi_L\\\psi_R\end{pmatrix}+e\ \phi_E\begin{pmatrix}0&I\\I&0\end{pmatrix}\begin{pmatrix}\psi_L\\\psi_R\end{pmatrix}-m_e\begin{pmatrix}\psi_L\\\psi_R\end{pmatrix}=0.\tag{5}$$ Writing out the two equations for $\phi_L$ and $\phi_R$, contained in Eqn $(5)$, explicitly we obtain

$$\begin{eqnarray*} i\frac{\partial\psi_R}{\partial t} &=& m_e\ \psi_L - e\ \phi_E\ \psi_R\tag{6}\\ i\frac{\partial\psi_L}{\partial t} &=& -e\ \phi_E\ \psi_L + m_e\ \psi_R.\tag{7} \end{eqnarray*}$$

Adding and subtracting Eqns. $(6)$ and $(7)$ we obtain

$$\begin{eqnarray*} i\frac{\partial}{\partial t}\big(\psi_L+\psi_R\big) &=& \big(m_e\ - e\ \phi_E\big)\big(\psi_L+\psi_R\big)\tag{8}\\ i\frac{\partial}{\partial t}\big(\psi_L-\psi_R\big) &=& \big(-m_e\ - e\ \phi_E\big)\big(\psi_L-\psi_R\big).\tag{9} \end{eqnarray*}$$

It seems to me that Eqn. $(8)$ describes an electron with an effective rest mass/energy $M_e=m_e-e\phi_E$ and Eqn. $(9)$ describes a positron with an effective rest mass/energy $M_p=m_e+e\phi_E$.

If we can change the effective mass of electrons/positrons by changing the electric potential $\phi_E$ then can we change the dynamics of electrons in atoms by applying a large $\phi_E\sim m_e/e$?


I've just discovered the answer from an old Physics Forums post

Define: $$\psi_+=\psi_L+\psi_R$$ $$\psi_-=\psi_L-\psi_R$$ $$f=e\phi_E$$

Redefine: $$\chi_+=e^{-ift}\psi_+$$ $$\chi_-=e^{-ift}\psi_-$$

The electrostatic potential drops out to leave the standard equations for the electron/positron rest mass energy $m_e$:

$$i\frac{\partial\chi_+}{\partial t}=m_e\chi_+$$ $$i\frac{\partial\chi_-}{\partial t}=-m_e\chi_-$$


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