11.4: Scattering in one dimension - Square Well
- Page ID
- 28679
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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.
Figure \(\PageIndex{1}\): Forward moving wavefunctions passing a square well potential