Abstract
Based on the general considerations in the previous Chapters, we consider the self-adjoint extension and spectral problems for the momentum and Hamiltonian of a free one-dimensional nonrelativistic particle moving on an interval of the real axis. The solution of these problems crucially depends on the type of the interval. It is shown how the correct treatment removes all the paradoxes presented in the Introduction.
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Notes
- 1.
Although in the general formulation of Theorem 3.4, the situation seems to be the opposite.
- 2.
This relation is considered evident in most physics textbooks and sometimes incorrectly extended to other intervals; we discuss this point below.
- 3.
We note in passing that the correctness of the calculation, which is very simple in this case, is confirmed by the fact that both necessary conditions rankE 1 ∕ 2, θ = 1 and \({E}_{1/2,\theta }^{+}\left (0\right )\mathcal{E}{E}_{1/2,\theta }\left (0\right ) = 0\) hold; see the remark following Theorem 4.17.
References
Gitman, D.M., Tyutin, I.V., Voronov, B.L.: Oscillator representations for self-adjoint Calogero Hamiltonians. Journ. Phys. A Math. Theor. 44 425204 (2011)
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Gitman, D.M., Tyutin, I.V., Voronov, B.L. (2012). Free One-Dimensional Particle on an Interval. In: Self-adjoint Extensions in Quantum Mechanics. Progress in Mathematical Physics, vol 62. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4662-2_6
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DOI: https://doi.org/10.1007/978-0-8176-4662-2_6
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