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2.3 Simple example

We can define all these concepts (velocity, momentum, potential) in one dimensionas well as in three dimensions. Let us look at the example for a barrier

V(x) = $\displaystyle \left\{\vphantom{ \begin{array}{ll} 0& \vert x\vert>a\\  V_{0} & \vert x\vert<a \end{array}}\right.$$\displaystyle \begin{array}{ll} 0& \vert x\vert>a\\  V_{0} & \vert x\vert<a \end{array}$ $\displaystyle \left.\vphantom{ \begin{array}{ll} 0& \vert x\vert>a\\  V_{0} & \vert x\vert<a \end{array}}\right\}$ (2.5)

We can't find a solution for E less than 0 (no solution for v). For energy less then V0 the particles can move left or right from the barrier, with constant velocity, but will make a hard bounce at the barrier (sign of v is not determined from energy). For energies higher than V0 particles can move from one side to the other, but will move slower if they are above the barrier.

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