Ethan's Notes

    Transmission Coefficient

    #Physics

    Electromagnetism

    $\displaystyle T=\frac{I_{T}}{I_{I}}=\left( \frac{E_{0_{T}}}{E_{0_{I}}} \right)^{2}$

    Quantum Mechanics

    $\displaystyle T=\left( 1+\frac{\sinh ^{2}(k_{\text{below}}a)}{4\eta(1-\eta)} \right)^{-1}, ,,,E<V_{0}$

    • Probability that a particle will reflect off of a potential barrier for a finite square well
    • $\displaystyle E$ is the kinetic energy of the particle
    • $\displaystyle V_{0}$ is the potential energy of the barrier
    • $\displaystyle \sinh$ is one of the hyperbolic trig functions
    • $\displaystyle \eta=\frac{E}{V_{0}}$
    • $\displaystyle k_{\text{below}}=\frac{\sqrt{ 2mV_{0}(1-\eta) }}{\hbar}$, where $\displaystyle \hbar$ is the reduced Planck constant in $\displaystyle 4.135\cdot 10^{-15},\mathrm{eV\cdot s}$, $\displaystyle m$ is given in $\displaystyle \mathrm{\frac{eV}{c^{2}}}$. $\displaystyle m_{e}=0.51,\mathrm{\frac{MeV}{c^{2}}}=0.0017,eV\cdot\left( \frac{s}{m} \right)^{2}$
    • Based on Professor Dave Explains Video Pt. 2
    • Desmos Demo

    $\displaystyle T=\left( 1+\frac{\sin ^{2}(k_{\text{below}}a)}{4\eta(\eta-1)} \right)^{-1}, ,,,E>V_{0}$

    • $\displaystyle k_{\text{below}}=\frac{\sqrt{ 2mV_{0}(\eta-1) }}{\hbar}$
    • Desmos Demo