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4 - The Schrödinger Equation

Published online by Cambridge University Press:  11 May 2023

Uri Peskin
Affiliation:
Technion - Israel Institute of Technology, Haifa
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Summary

The postulates of quantum mechanics associate the time-evolution of a system with its time-dependent Schrödinger equation. We start by examining different solutions to this equation for a particle, represented in terms of a Gaussian wave packet, in different scenarios: free, scattered from a potential energy barrier, or trapped in a potential energy well. In some cases, we encounter stationary solutions, in which the probability density does not change in time (a standing wave). These solutions are identified as eigenfunctions of a system Hamiltonian. The properties of the Hamiltonian as a Hermitian operator are introduced, and particularly, the fact that its proper eigenfunctions can compose an orthonormal set, and that the corresponding eigenvalues are real-valued. Learning that all operators that relate to measurables are Hermitian, and that their eigenvalues relate to the measured values, we conclude that the eigenvalues of the energy operator (Hamiltonian) are the energy levels of the quantum system.

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Chapter
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Publisher: Cambridge University Press
Print publication year: 2023

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References

Peskin, U. and Moiseyev, N., “The solution of the time-dependent Schrödinger equation by the (, ’) method: theory, computational algorithm and applications,” The Journal of Chemical Physics 99, 4590 (1993).CrossRefGoogle Scholar
Heller, E. J., “Time-dependent approach to semiclassical dynamics,” The Journal of Chemical Physics 62, 1544 (1975).Google Scholar
Cohen-Tannoudji, C., Diu, B. and Laloë, F., “Quantum Mechanics,” vols. 1–2: (John Wiley & Sons, 2020).Google Scholar
Arnoldi, W. E., “The principle of minimized iterations in the solution of the matrix eigenvalue problem,” Quarterly of Applied Mathematics 9, 17 (1951).CrossRefGoogle Scholar

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  • The Schrödinger Equation
  • Uri Peskin, Technion - Israel Institute of Technology, Haifa
  • Book: Quantum Mechanics in Nanoscience and Engineering
  • Online publication: 11 May 2023
  • Chapter DOI: https://doi.org/10.1017/9781108877787.005
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  • The Schrödinger Equation
  • Uri Peskin, Technion - Israel Institute of Technology, Haifa
  • Book: Quantum Mechanics in Nanoscience and Engineering
  • Online publication: 11 May 2023
  • Chapter DOI: https://doi.org/10.1017/9781108877787.005
Available formats
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Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • The Schrödinger Equation
  • Uri Peskin, Technion - Israel Institute of Technology, Haifa
  • Book: Quantum Mechanics in Nanoscience and Engineering
  • Online publication: 11 May 2023
  • Chapter DOI: https://doi.org/10.1017/9781108877787.005
Available formats
×