Abstract
AbstractA general solution to the one‐dimensional time‐independent Schrödinger equation is derived using the properties of the Laplace transform. The derivation assumes that the potential function is real and that it can be expressed as a Fourier series with a finite number of terms, which includes, but is not limited to, equations of the type of Hill's or Mathieu's equations. In the case where the Laplace transform of the potential function contains only simple first‐order poles, then a complete closed‐form solution is derived. In other cases, an approximate solution is derived and the symbolic algebra program MACSYMA is used to carry out the inverse Laplace transform. The MACSYMA implementation of this general solution is discussed and is applied to a sample problem, namely, the solution of eigenvalues of Mathieu's equation. Comparison is made between the eigenvalues obtained by this method and those obtained by another analytical solution of this equation, by Hill's method.
Published Version
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