One of the most elemental quantum mechanical transport problems is the solution of the time independent Schrödinger equation in a one-dimensional system where one of the two half spaces has a higher potential energy than the other. The analytical solution is readily obtained using a scattering matrix approach where wavefunction amplitude and slope are matched at the interface between the two half-spaces. Of particular interest are the wave/particle injection from the lower potential energy half-space. In a classical system a particle will suffer from complete reflection at the half-space border if its kinetic energy is not larger than the potential energy difference at the barrier. The classical particle will be completely transmitted if its kinetic energy exceeds the potential barrier difference. A quantum mechanical particle or wave however exhibits a few interesting features: 1) it can penetrate into the potential barrier when ints kinetic energy is lower than the potential step energy, and 2) transmission over the barrier is not complete and energy dependent. Incomplete transmission implies a reflection probability for the wave even though its kinteci energy exceeds the potential barrier difference. This simple example shos the extended nature of wavefunctions and the non-local effects of local potential variations in its simplest form.
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