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6.2 Potential step

Consider a potential step

V(x) = $\displaystyle \left\{\vphantom{\begin{array}{ll}V_0 & x <0 \\  V_1& x>0 \end{array} }\right.$$\displaystyle \begin{array}{ll}V_0 & x <0 \\  V_1& x>0 \end{array}$ $\displaystyle \left.\vphantom{\begin{array}{ll}V_0 & x <0 \\  V_1& x>0 \end{array} }\right.$ (6.3)

  
Figure 6.1: The step potential discussed in the text
\includegraphics[width=6.0cm]{Figures/step1.eps}

Let me define

k0 = $\displaystyle \sqrt{\frac{2m}{\hbar^2}(E-V_0)}$, (6.4)
k1 = $\displaystyle \sqrt{\frac{2m}{\hbar^2}(E-V_1)}$. (6.5)

I assume a beam of particles comes in from the left,

$\displaystyle \phi$(x) = A0eik0x,    x < 0. (6.6)

At the potential step the particles either get reflected back to region I, or are transmitted to region II. There can thus only be a wave moving to the right in region II, but in region I we have both the incoming and a reflected wave,
$\displaystyle \phi_{I}^{}$(x) = A0eik0x + B0e-ik0x, (6.7)
$\displaystyle \phi_{II}^{}$(x) = A1eik1x. (6.8)

We define a transmission and reflection coefficient as the ratio of currents between reflected or transmitted wave and the incoming wave, where we have cancelled a common factor

R = $\displaystyle {\frac{\vert B_0\vert^2}{\vert A_0\vert^2}}$      T = $\displaystyle {\frac{k_1\vert A_1\vert^2}{k_0\vert A_0\vert^2}}$. (6.9)

Even though we have given up normalisability, we still have the two continuity conditions. At x = 0 these imply, using continuity of $ \phi$ and $ {\frac{d}{dx}}$$ \phi$,
A0 + B0 = A1, (6.10)
ik0(A0 - B0) = ik1A1. (6.11)

We thus find
A1 = $\displaystyle {\frac{2k_0}{k_0+k_1}}$A0, (6.12)
B0 = $\displaystyle {\frac{k_0-k_1}{k_0+k_1}}$A0, (6.13)

and the reflection and transmission coefficients can thus be expressed as
R = $\displaystyle \left(\vphantom{\frac{k_0-k_1}{k_0+k_1}}\right.$$\displaystyle {\frac{k_0-k_1}{k_0+k_1}}$ $\displaystyle \left.\vphantom{\frac{k_0-k_1}{k_0+k_1}}\right)^{2}$, (6.14)
T = $\displaystyle {\frac{4 k_1 k_0}{(k_0+k_1)^2}}$. (6.15)

Notice that R + T = 1!

  
Figure 6.2: The transmission and reflection coefficients for a square barrier.
\includegraphics[width=5.0cm]{Figures/stepTR.ps}

In Fig. 6.2 we have plotted the behaviour of the transmission and reflection of a beam of Hydrogen atoms impinging on a barrier of height 2 meV.


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Next: 6.3 Square barrier Up: 6. Scattering from potential Previous: 6.1 Non-normalisable wave functions

© 1998 Niels Walet, UMIST
Email Niels.Walet@umist.ac.uk