Ethan's Notes

    Delta Potential

    #Physics

    $\displaystyle U(x)=-\alpha \delta(x)$

    • $\displaystyle \alpha$ has units of energy times length
    • $\displaystyle \delta(x)$ has units of $\displaystyle \frac{1}{L}$

    Bounded: $\displaystyle E<0$

    $$

    \psi(x)=\begin{cases}
    Be^{kx}, & x\leq 0 \\Be^{-kx}, & x\geq 0
    \end{cases}
    $$

    • $\displaystyle B=\sqrt{ k }=\frac{\sqrt{ m\alpha }}{\hbar}$
      • Obtained by normalization
    • $\displaystyle k\equiv \frac{\sqrt{ -2mE}}{\hbar}=\frac{m\alpha}{\hbar ^{2}}$
      • Unrelated to wavenumber
      • Second equation is obtained by integrating the TISE over a small area around $\displaystyle x=0$

    $\displaystyle E=-\frac{m\alpha ^{2}}{2\hbar ^{2}}$

    Scattering: $\displaystyle E>0$

    $$

    \psi(x)=\begin{cases}
    Ae^{ikx}+Be^{-ikx}, & x<0 \\Fe^{ikx}+G^{-ikx}, & x>0
    \end{cases}
    $$

    • $\displaystyle \frac{k\equiv\sqrt{ 2mE }}{\hbar}$
    • For wave traveling rightward:
      • $\displaystyle G=0$ if the wave travels rightward since there would be no incident wave on the right of the delta potential traveling leftward
      • $\displaystyle F=A(1+2i\beta)-B(1-2i\beta)$
      • $\beta\equiv \frac{m\alpha}{\hbar ^{2}k}=\frac{m\alpha}{\hbar\sqrt{ 2mE }}$
      • $\displaystyle B=\frac{i\beta}{1-i\beta}A$
      • $\displaystyle F=\frac{1}{1-i\beta}A$
      • $\displaystyle R=\frac{\lvert B\rvert^{2}}{\lvert A\rvert^{2}}=\frac{\beta ^{2}}{1+\beta ^{2}}=\frac{1}{1+ \frac{2\hbar ^{2}E}{m\alpha ^{2}}}$
      • $\displaystyle T=\frac{\lvert F\rvert^{2}}{\lvert A\rvert^{2}}=\frac{1}{1+\beta ^{2}}=\frac{1}{1+ \frac{m\alpha ^{2}}{2\hbar ^{2}E}}$
      • Changing delta well to delta barrier doesn't change $\displaystyle R$ and $\displaystyle T$