Students’ difficulties with solving bound and scattering state problems in quantum mechanics

Tao Tu; Chuan-Feng Li; Jin-Shi Xu; Guang-Can Guo

doi:10.1103/physrevphyseducres.17.020142

Students’ difficulties with solving bound and scattering state problems in quantum mechanics

Physical Review Physics Education Research ◽

10.1103/physrevphyseducres.17.020142 ◽

2021 ◽

Vol 17 (2) ◽

Author(s):

Tao Tu ◽

Chuan-Feng Li ◽

Jin-Shi Xu ◽

Guang-Can Guo

Keyword(s):

Quantum Mechanics ◽

Scattering State

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Do attractive scattering potentials concentrate particles at the origin in one, two and three dimensions? I. Potentials finite at the origin

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1983.0089 ◽

1983 ◽

Vol 388 (1795) ◽

pp. 401-418 ◽

Cited By ~ 3

Keyword(s):

Quantum Mechanics ◽

Classical Mechanics ◽

Three Dimensions ◽

Simple Extension ◽

Final State ◽

Central Potential ◽

Scattering State ◽

Square Well ◽

Semiclassical Regime ◽

High Wavenumbers

Paradoxically, in beta decay, for instance, the final-state Coulomb forces pulling the electron inwards accelerate the emission. Quantum mechanics (q. m. ) makes the rate proportional to α ≡ ρ 0 / ρ ∞ ; ρ 0, ∞ (and v 0, ∞ ) are the particle densities (and speeds) at r = 0 and far upstream in the scattering state which describes the electron. Hence, as regards the effects of finalstate interactions, one must base one’s physical intuition on this ratio α . It is shown that according to (non-relativistic) classical mechanics, if the origin is accessible, then any central potential U(r) where v 0 < ∞ (i. e. where U (0) > -∞) gives in 1, 2 and 3 dimensions, α 1 = v ∞ / v 0 , α 2 = 1, α 3 = v 0 / v ∞ ; the remaining course of U(r) is irrelevant to α . The same results hold also in q. m. in the semiclassical regime, i. e. in the W. K. B. approximation which for such potentials becomes valid at high wavenumbers; in 2D it needs rather careful formulation, and in 3D one must avoid the Langer modification. (The W. K. B. results apply even if d U / d r diverges at r = 0, provided U (0) remains finite; these cases are covered by a simple extension of the argument. ) The square-well and exponential potentials are discussed as examples. Potentials which diverge at the origin are treated in the following paper.

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