deletions | additions       diff --git a/However_experimentally_no_threshold_voltage__.tex b/However_experimentally_no_threshold_voltage__.tex  index a96939a..2c94132 100644  --- a/However_experimentally_no_threshold_voltage__.tex  +++ b/However_experimentally_no_threshold_voltage__.tex 
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\begin{equation}  \kappa_b(x) = \frac{1}{\hbar} \sqrt{2m^*[V(x)-\hbar \omega]}  \end{equation}  The current density in one channel will be $J(\omega) $J_{1}(\omega)  = N T(\omega) e v$ where $N$ is the number of electrons injected toward the potential and $v$ is the average velocity of the electrons through the potential. Secondly, consider the classical equation of motion for classical single particle in equation (2) and replace $V_{applied} = V = V_{ac} \space cos(\omega t)$. Equation (2) can be solved numerically and giving the frequency dependent velocity amplitude $|v| = |\frac{dx}{dt}| = v(\omega)$. When the frequency of applied voltage $\omega$ is approximately equal to the characteristic frequency $\omega_p$, resonance takes place with the largest velocity amplitude. The current density in this case is $J_{2}(\omega) = Nev(\omega)$.