# Changing effective mass of electron using electric potential?

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

$$\gamma^\mu\big(i\partial_\mu-eA_\mu\big)\psi-m_e\psi=0.\tag{1}$$

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 $$\phi_E\sim m_e/e$$?

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