Solvable models of the Schrödinger equation are important models of quantum systems because they are idealistic approximations of real quantum systems and much insight into real quantum systems can be gained from the exact solutions of the solvable models. In this paper we show that a universal Laplace transform scheme can be used to solve the Schrödinger equations in closed form for all known solvable models. The work demonstrates how to apply the Laplace transform to differential equations with non-constant coefficients, which is useful in many branches of physics in addition to quantum mechanics. The advantages of the Laplace transform over the power expansion method and its connection with the methods of supersymmetry shape-invariant potentials and quantum canonical transformation, which also give closed-form solutions for solvable models, are elucidated.
As an intense laser pulse propagates through an underdense plasma, the strong ponderomotive force pushes away the electrons and produces a trailing plasma bubble. In the meantime the pulse itself undergoes extreme nonlinear evolution that results in strong spectral broadening toward the long-wavelength side. By experiment we demonstrate that this process can be utilized to generate ultrashort midinfrared pulses with an energy three orders of magnitude larger than that produced by crystal-based nonlinear optics. The infrared pulse is encapsulated in the bubble before exiting the plasma, hence is not absorbed by the plasma. The process is analyzed experimentally with laser-plasma tomographic measurements and numerically with three-dimensional particle-in-cell simulation. Good agreement is found between theoretical estimation, numerical simulation, and experimental results.
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