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Physics LibreTexts

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=(k2k2)(1e2ika)(k+k)2(kk)2e2ika

C/A=4kkei(kk)a(k+k)2(kk)2e2ika

where k2=2mE/2 and k2=2m(EV0)/2. Since the wavenumber is the same on both sides of the barrier, the reflection and transmission coefficients are just:

|B/A|2=[1+4k2k2(k2k2)2sin2ka]1=[1+4E(EV0)V20sin2ka]1

|C/A|2=[1+(k2k2)2sin2ka4k2k2]1=[1+V20sin2ka4E(EV0)]1

We get complete transmission when ka=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 EV0 then sin2(ka)ka and the transmission coefficient

|C/A|2[1+mV0a222]1

As the incoming particle energy is increased, the transmission oscillates between [1+V204E(EV0)]1 and 1 at ka=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(EV0)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.

11.2.PNG

Figure 11.4.1: Forward moving wavefunctions passing a square well potential


This page titled 11.4: Scattering in one dimension - Square Well is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Graeme Ackland via source content that was edited to the style and standards of the LibreTexts platform.

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