Physics – Quantum Physics
Scientific paper
2004-03-24
Physics
Quantum Physics
12 pages, including 4 figures
Scientific paper
Analytical solutions to the time-dependent Shr\"{o}dinger equation in one dimension are developed for time-independent potentials, one consisting of an infinite wall and a repulsive delta function. An exact solution is obtained by means of a convolution of time-independent solutions spanning the given Hilbert space with appropriately chosen spectral functions. Square-integrability and the boundary conditions are satisfied. The probability for the particle to be found inside the potential well is calculated and shown to exhibit non-exponential decay decreasing at large times as $t^{-3}$. The result is generalized for all square-integrable solutions to this problem.
Luban Marshall
Petridis Athanasios N.
Staunton Lawrence P.
Vermedahl Jon
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