11.4: Scattering in one dimension - Square Well
( \newcommand{\kernel}{\mathrm{null}\,}\)
The square well potential has V(x<0)=V(x>a)=0;V(0<x<a)=V0. As with the step function, we can write the wavefunction as a plane wave in each of the three regions.
Φ(x<0)=A exp(ikx)+B exp(−ikx)
Φ(0<x<a)=F exp(ik0x)+G exp(−ik0x)
Φ(x>a)=C exp(ikx)+D exp(−ikx)
Once again there is no wave coming back from x=∞(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=(k2−k′2)(1−e2ik′a)(k+k′)2−(k−k′)2e2ik′a
C/A=4kk′ei(k′−k)a(k+k′)2−(k−k′)2e2ik′a
where k2=2mE/ℏ2 and k′2=2m(E−V0)/ℏ2. Since the wavenumber is the same on both sides of the barrier, the reflection and transmission coefficients are just:
|B/A|2=[1+4k2k′2(k2−k′2)2sin2k′a]−1=[1+4E(E−V0)V20sin2k′a]−1
|C/A|2=[1+(k2−k′2)2sin2k′a4k2k′2]−1=[1+V20sin2k′a4E(E−V0)]−1
We get complete transmission when k′a=nπ, i.e. when an exact number of half waves fit in the well.
Assuming that E>V0. Looking at the limits of this, we see that as E→V0 then sin2(k′a)→k′a and the transmission coefficient
|C/A|2→[1+mV0a22ℏ2]−1
As the incoming particle energy is increased, the transmission oscillates between [1+V204E(E−V0)]−1 and 1 at k′a=nπ. The lower limit itself increases to 1 as E increases.
For the tunnelling case where E<V0 we can use these solutions for B/A and C/A, except that k′ is now imaginary. This gives
|C/A|2=[1+4E(E−V0)V20sinh2|k′|a]−1
which decreases monotonically with decreasing E. Thus a small change in V0 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 Φ(x>a)=C 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.
Figure 11.4.1: Forward moving wavefunctions passing a square well potential