11.3: Scattering in one dimension- Step function

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Firstly, we review the problem of scattering by a step function in one dimension. Consider a particle moving from a region \((x < 0)\) where the potential is \(V = 0\) to a region \((x > 0)\) where the potential is \(V = V_0\).

Assuming the particle energy \(E > V_0\), this is simply the free particle problem, the spatial solution to which is:

\[\Phi = A \text{ exp}(ikx) + B \text{ exp}(−ikx) \quad (x < 0); \quad \Phi = C \text{ exp}(ik'x) + D \text{ exp}(−ik'x) \quad (x > 0) \nonumber\]

where \(k = \sqrt{2mE/}\hbar\) and \(k' = \sqrt{2m(E − V_0)}/\hbar\)

11.1.PNG — Figure \(\PageIndex{1}\): Scattering at a step function.

From the boundary condition that all particles start from \(x = −\infty\), we can immediately set \(D=0\).

From the condition of continuity of \(\Phi\) and \(d\Phi /dx\) at \(x = 0\) we also require \(A + B = C\) and \(k(A − B) = k' C\)

This gives the reflected amplitude \(B/A = (k − k')/(k + k')\) and the transmitted amplitude \(C/A = 2k/(k + k' )\)

The reflected flux is thus

\[\frac{\hbar k}{m} A^2 \left( \frac{k − k'}{k + k'}\right)^2 \nonumber\]

and the transmitted flux is

\[\frac{\hbar k'}{m} A^2 \left(\frac{2k}{k + k'}\right)^2 \nonumber\]

Note that \(A^2 \neq B^2 + C^2\). The conserved quantity is the flux of particles, not the probability density. In this case the transmitted particles are moving more slowly than the incident ones.

Notice that if \(V_0\) is negative, the transmitted flux gets smaller as \(|V_0|\) gets larger: it is difficult to fall off a big cliff! This anomaly is due to the unphysical potential - the discontinuous first derivative at \(x = 0\).

We have not considered the case of \(E < V_0\). Now the square root is imaginary and \(\Phi (x > 0) = Ce^{-\kappa' x}\) where we define a real quantity \(\kappa' = ik' = \sqrt{2m(V_0 − E)}/\hbar\). The boundary conditions are then \(A+B = C\) and \(ik(A−B) = \kappa' C\), which gives the reflected amplitude \(B/A = (ik−\kappa' )/(ik+\kappa')\) and the transmitted amplitude \(C\kappa' /Ak = 2ik/(ik + \kappa' )\).

Now the reflected flux is equal to the incident flux, and although the wavefunction penetrates the region \(x > 0\), it decays exponentially and there is no propagating wave.