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, we solve for a stationary state with energy ${\displaystyle E>V_{0}}$ The solution to the time independent Schrödinger equation:

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

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

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

${\displaystyle k_{1}}$ 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 ${\displaystyle T}$, we set ${\displaystyle B_{3}=0}$ to correspond 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 ${\displaystyle x=0}$ and ${\displaystyle x=L}$, we find relations between the coefficients, which allows us to find ${\displaystyle T}$ as

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

We see that, the Transmission co-efficient ${\displaystyle T}$ reaches its maximum value of 1, when :

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

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 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 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 Transmission co-efficient are the same.

A Plot Of Transmission co-efficient against (E/V₀) for shape factor of 30

A Plot Of Transmission co-efficient against (E/V₀) for shape factor of 13

As a function of the length of square well (${\displaystyle L}$), the Transmission coefficient swings between its maximum of 1 and minimum of ${\displaystyle \left[1+{\frac {V_{0}^{2}}{4E(E-V_{0})}}\right]^{-1}}$, with a period of ${\displaystyle {\frac {\pi }{k_{2}}}}$. As a function of energy, the first term in the denominator dominates the oscillating term for ${\displaystyle E>>V_{0}}$ and therefore, ${\displaystyle T\rightarrow 1}$. Sharper resonances occur at lower energies,where the oscillating term in the denominator controls the behaviour of ${\displaystyle T}$. The resonances become flat at higher energies, because the minimas of ${\displaystyle T}$ get higher with ${\displaystyle E}$ as the effect of the oscillatory term in the denominator diminishes. This is demonstrated in plots of Transmission coefficient against incident particle energy for fixed values of the shape factor, defined as ${\displaystyle {\sqrt {\frac {2mV_{0}L^{2}}{\hbar ^{2}}}}}$