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Time-Evolution of a Wavepacket in a Square Well
Time-Evolution of a Wavepacket in a Square Well
This demonstrates time-evolution of an initially smeared out density in a square well potential. Because of the nonlinear dispersion relation that follows from the Schrödinger equation, a wave packet will spread out with time. And because the eigenvalues of the energy are commensurable, the time evolution is periodic. When the initial wave packet spreads out over the whole well, no classical solution exists (the homogeneous Dirichlet boundary conditions are incompatible with the nonhomogeneous initial conditions at the boundaries). As a result, the weak solution exhibits fractal behavior in space and time.