Resonances in scattering from potentials

A one-dimensional finite square potential is given by

${\displaystyle V(x)={\begin{cases}V_{0},&0<x<L,\\0,&{\text{otherwise,}}\end{cases}},}$

The sign of ${\displaystyle V_{0}}$ determines whether the square potential is a well or a barrier. To study the phenomena of resonance, the time-independent Schrödinger equation for a stationary state of a massive particle with energy ${\displaystyle E>V_{0}}$ is solved:

${\displaystyle -{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}\psi }{dx^{2}}}+V(x)\psi =E\psi }$

The wave function solutions for the three regions ${\displaystyle x<0,0<x<L,x>L}$ are

${\displaystyle \psi _{1}(x)={\begin{cases}A_{1}e^{ik_{1}x}+B_{1}e^{-ik_{1}x},&x<0,\\A_{2}e^{ik_{2}x}+B_{2}e^{-ik_{2}x},&0<x<L,\\A_{3}e^{ik_{1}x}+B_{3}e^{-ik_{1}x},&x>L,\end{cases}}}$

Here, ${\displaystyle k_{1}}$ and ${\displaystyle k_{2}}$ are the wave numbers in the potential-free region and within the potential respectively:

${\displaystyle k_{1}={\frac {\sqrt {2mE}}{\hbar }},}$

${\displaystyle k_{2}={\frac {\sqrt {2m(E-V_{0})}}{\hbar }},}$

To calculate ${\displaystyle T}$, a coefficient in the wave function is set as ${\displaystyle B_{3}=0}$, which corresponds to the fact that there is no wave incident on the potential from the right. Imposing the condition that the wave function ${\displaystyle \psi (x)}$ and its derivative ${\displaystyle {\frac {d\psi }{dx}}}$ should be continuous at the well/barrier boundaries ${\displaystyle x=0}$ and ${\displaystyle x=L}$, the relations between the coefficients are found, which allows ${\displaystyle T}$ to be found as:

${\displaystyle T={\frac {|A_{3}|^{2}}{|A_{1}|^{2}}}={\frac {4E(E-V_{0})}{4E(E-V_{0})+V_{0}^{2}\sin ^{2}\left[{\sqrt {2m(E-V_{0})}}{\frac {L}{\hbar }}\right]}}}$

It follows that the transmission coefficient ${\displaystyle T}$ reaches its maximum value of 1 when:

${\displaystyle \sin ^{2}\left[{\sqrt {2m(E-V_{0})}}{\frac {L}{\hbar }}\right]=0{\text{, or }}{\sqrt {2m(E-V_{0})}}={\frac {n\pi \hbar }{L}}}$

for any integer value ${\displaystyle n}$. This is the resonance condition, which leads to the peaking of ${\displaystyle T}$ to its maxima, called resonance.

From the above expression, resonance occurs when the distance covered by the particle in traversing the well and back (${\displaystyle 2L}$) is an integral multiple of the De Broglie wavelength of a particle inside the potential (${\displaystyle \lambda ={\frac {2\pi }{k}}}$). For ${\displaystyle E>V_{0}}$, reflections at potential discontinuities are not accompanied by any phase change.[1] Therefore, resonances correspond to the formation of standing waves within the potential barrier/well. At resonance, the waves incident on the potential at ${\displaystyle x=0}$ and the waves reflecting between the walls of the potential are in phase, and reinforce each other. Far from resonances, standing waves can't be formed. Then, waves reflecting between both walls of the potential (at ${\displaystyle x=0}$ and ${\displaystyle x=L}$) and the wave transmitted through ${\displaystyle x=0}$ are out of phase, and destroy each other by interference. The physics is similar to that of transmission in Fabry–Pérot interferometer in optics, where the resonance condition and functional form of the transmission coefficient are the same.

Plot of the transmission co-efficient against (E/V₀) for a shape factor of 30

Plot of the transmission co-efficient against (E/V₀) for a shape factor of 13

The transmission coefficient swings between its maximum of 1 and minimum of ${\displaystyle \left[1+{\frac {V_{0}^{2}}{4E(E-V_{0})}}\right]^{-1}}$as a function of the length of square well (${\displaystyle L}$) with a period of ${\displaystyle {\frac {\pi }{k_{2}}}}$. The minima of the transmission tend to ${\displaystyle 1}$ in the limit of large energy ${\displaystyle E>>V_{0}}$, resulting in more shallow resonances, and inversely tend to ${\displaystyle 0}$ in the limit of low energy ${\displaystyle E<<V_{0}}$, resulting in sharper resonances. This is demonstrated in plots of transmission coefficient against incident particle energy for fixed values of the shape factor, defined as ${\displaystyle {\sqrt {\frac {2mV_{0}L^{2}}{\hbar ^{2}}}}}$