11.4: Scattering in one dimension - Square Well

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The square well potential has \(V (x < 0) = V (x > a) = 0; V (0 < x < a) = V_0\). As with the step function, we can write the wavefunction as a plane wave in each of the three regions.

\[\Phi (x < 0) = A \text{ exp}(ikx) + B \text{ exp}(−ikx) \nonumber\]

\[\Phi (0 < x < a) = F \text{ exp}(ik0x) + G \text{ exp}(−ik0x) \nonumber\]

\[\Phi (x > a) = C \text{ exp}(ikx) + D \text{ exp}(−ikx) \nonumber\]

Once again there is no wave coming back from \(x = \infty (D = 0)\).

There are now four boundary conditions from continuity of the wave function and its derivative at x=0 and x=a. The solving of four equations in four unknowns is straightforward but tedious. Eventually one can obtain ratios for reflected and transmitted flux:

\[B/A = \frac{(k^2 − k^{'2} )(1 − e^{2ik'a} )}{(k + k' )^2 − (k − k' )^2 e^{2ik'a}} \nonumber\]

\[C/A = \frac{4kk' e^{i(k' − k)a}}{(k + k' )^2 − (k − k' )^2 e^{2ik'a}} \nonumber\]

where \(k^2 = 2mE/\hbar^2\) and \(k^{'2} = 2m(E − V_0)/\hbar^2\). Since the wavenumber is the same on both sides of the barrier, the reflection and transmission coefficients are just:

\[|B/A|^2 = \left[ 1 + \frac{4k^2 k^{'2}}{(k^2 − k^{'2} )^2 \sin^2 k' a} \right]^{−1} = \left[ 1 + \frac{4E(E − V_0)}{V^2_0 \sin^2 k' a} \right]^{−1} \nonumber\]

\[|C/A|^2 = \left[ 1 + \frac{(k^2 − k^{'2} )^2 \sin^2 k' a}{4k^2 k^{'2}} \right]^{−1} = \left[ 1 + \frac{V^2_0 \sin^2 k' a}{4E(E − V_0)} \right]^{-1} \nonumber\]

We get complete transmission when \(k' a = n\pi\), i.e. when an exact number of half waves fit in the well.

Assuming that \(E > V_0\). Looking at the limits of this, we see that as \(E \rightarrow V_0\) then \(\sin^2 (k' a) \rightarrow k' a\) and the transmission coefficient

\[|C/A|^2 \rightarrow \left[ 1 + \frac{mV_0a^2}{2\hbar^2} \right]^{-1} \nonumber\]

As the incoming particle energy is increased, the transmission oscillates between \(\left[1 + \frac{V^2_0}{4E(E−V_0)} \right]^{−1}\) and 1 at \(k' a = n\pi\). The lower limit itself increases to 1 as E increases.

For the tunnelling case where \(E < V_0\) we can use these solutions for B/A and C/A, except that \(k'\) is now imaginary. This gives

\[|C/A|^2 = \left[ 1 + \frac{4E(E − V_0)}{V^2_0 \sinh^2 |k' |a} \right]^{−1} \nonumber\]

which decreases monotonically with decreasing E. Thus a small change in \(V_0\) can give a large change in \(|C/A|^2\). This is the principle on which the transistor and the tunnelling electron microscope are based.

Note that the transmitted wave \(\Phi (x > a) = C \text{ exp}(ikx)\), differs from the incident wave only by a phase - it has the same wavevector. Thus the only effect of the potential on the transmitted particles is to change their phase, an idea we shall meet again.

11.2.PNG — Figure \(\PageIndex{1}\): Forward moving wavefunctions passing a square well potential