- #1
jcap
- 166
- 12
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##?
$$\gamma^\mu\big(i\partial_\mu-eA_\mu\big)\psi-m_e\psi=0.\tag{1}$$
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##?