Free Particle

  • Alessandro Teta
Part of the UNITEXT for Physics book series (UNITEXTPH)


We consider a free particle in \(\mathbb {R}^d\) and discuss the properties of the quantum dynamics. Using Fourier transform, we construct the Hamiltonian, describe the spectrum, derive the explicit formula for the unitary group, and characterize its asymptotic behavior for large times. We analyze the time evolution of a wave packet and give a quantitative description of the interference effect between two wave packets. Finally, we briefly discuss the dynamics of a particle subject to a constant force field.


  1. 1.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics, II: Fourier Analysis, Self-Adjointness. Academic Press, New York (1975)Google Scholar
  2. 2.
    Teschl, G.: Mathematical Methods in Quantum Mechanics. American Mathematical Society, Providence (2009)Google Scholar
  3. 3.
    Thaller, B.: Visual Quantum Mechanics. Springer, New York (2000)Google Scholar
  4. 4.
    de Oliveira, C.R.: Intermediate Spectral Theory and Quantum Dynamics. Birkhäuser, Basel (2009)Google Scholar
  5. 5.
    Feynman, R., Leighton, R.B., Sands, M.L.: The Feynman Lectures on Physics, vol. III. Addison Wesley Publ. Com (2005)Google Scholar
  6. 6.
    Landau, L.D., Lifshitz, E.M.: Quantum Mechanics, 3th edn. Pergamon Press, Oxford (1977)Google Scholar
  7. 7.
    Lebedev, N.N.: Special Functions and Their Applications. Dover Publ, New York (1972)Google Scholar

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Dipartimento di Matematica Guido CastelnuovoUniversità degli Studi di Roma “La Sapienza”RomeItaly

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