Potential Step -- from Eric Weisstein's World of Physics

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Potential Step

Consider the solution to the Schrödinger equation

where is h-bar, m is the mass of a particle, is the wavefunction, and E is the energy of a given state, for a potential step given by

Define

For , define

For x < 0,

which has a solution

For x > 0,

which has a solution

As , the wavefunction must decay to , so

Schrödinger Equation