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Other One-Dimensional Problems

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Relativistic and Non-Relativistic Quantum Mechanics


In this Chapter, we continue to solve unrealistic one-dimensional problems, but we improve our models of submicroscopic phenomena by exploring the use of more realistic potentials. This chapter contains a novel and ambitious study of the consequences of special theory of relativity for quantum mechanics. This includes the study of the relativistic harmonic oscillator, and a pedagogical presentation of Klein’s paradox based on solving the Schrödinger-like PPGP equation for a step potential.

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  1. W. Greiner, Relativistic quantum mechanics: wave equations (Spring-Verlag, New York, 1990)

    Book  MATH  Google Scholar 

  2. D.J. Griffiths, Introduction to quantum mechanics (Prentice Hall, USA, 1995)

    MATH  Google Scholar 

  3. D. Bohm, Quantum theory, 11th edn. (Prentice -Hall, USA, 1964)

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  4. L.A. Poveda, L. Grave de Peralta, J. Pittman, B. Poirier, A non-relativistic approach to relativistic quantum mechanics: the case of the harmonic oscillator. Found. Phys. 52, 29 (2022)

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  5. L. Grave de Peralta, Quasi-relativistic description of a quantum particle moving through one-dimensional piecewise constant potentials. Results Phys. 18, 103318 (2020)

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  6. Klein’s paradox (Wikipedia). Accessed 10 Apr 2023

  7. Step potential, non-relativistic case, (Wikipedia). Accessed 10 Apr 2023

  8. Rectangular potential barrier, non-relativistic case, (Wikipedia). Accessed 10 Apr 2023

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Correspondence to Luis Grave de Peralta .

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Grave de Peralta, L., Fernández Lozada, M., Farooq, H., Eichman, G., Singh, A., Prime, G. (2023). Other One-Dimensional Problems. In: Relativistic and Non-Relativistic Quantum Mechanics. Undergraduate Lecture Notes in Physics. Springer, Cham.

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