In a recent paper by Eu and Tsien on the uniform WKB theory of inelastic collision, the Magnus-approximation formula for the transition probability was applied to the ${\mathrm{He}}^{+}$ -Ne inelastic-scattering process. In this Comment, we report that for the system studied, i.e., the ${\mathrm{He}}^{+}$ -Ne system, the uniform WKB Magnus approximation underestimates the cross sections by about 50% and thus overestimates the coupling constant ${a}_{4}$ by about 30% above what it should be, while the distorted-wave theory underestimates the coupling constant by about 5%. The uniform-WKB Magnus-approximation formula for the transition probability is further assessed by making comparisons with the results of the exact theory, uniform-WKB numerical-solution method, and the Landau-Zener-St\"uckelberg theory, and is found to behave correctly in the strong-coupling regime as ${a}_{4}$ increases. The uniform-WKB Magnus approximation therefore appears to be complementary to the distorted-wave theory which is better in the weak-coupling regime, but breaks down in the strong-coupling regime. The uniform-WKB Magnus approximation is generally better behaved than the Landau-Zener-St\"uckelberg theory over all values of ${a}_{4}$ for the system studied.In a recent paper by Eu and Tsien on the uniform WKB theory of inelastic collision, the Magnus-approximation formula for the transition probability was applied to the ${\mathrm{He}}^{+}$-Ne inelastic-scattering process. In this Comment, we report that for the system studied, i.e., the ${\mathrm{He}}^{+}$-Ne system, the uniform WKB Magnus approximation underestimates the cross sections by about 50% and thus overestimates the coupling constant ${a}_{4}$ by about 30% above what it should be, while the distorted-wave theory underestimates the coupling constant by about 5%. The uniform-WKB Magnus-approximation formula for the transition probability is further assessed by making comparisons with the results of the exact theory, uniform-WKB numerical-solution method, and the Landau-Zener-St\"uckelberg theory, and is found to behave correctly in the strong-coupling regime as ${a}_{4}$ increases. The uniform-WKB Magnus approximation therefore appears to be complementary to the distorted-wave theory which is better in the weak-coupling regime, but breaks down in the strong-coupling regime. The uniform-WKB Magnus approximation is generally better behaved than the Landau-Zener-St\"uckelberg theory over all values of ${a}_{4}$ for the system studied.

Physical Review AFeb 1, 19747 citations

Multidimensional Theory of Tunneling Splitting

Hiroki Nakamura, Gennady Mil'nikov

In this section, generalizations of the instanton theory and the modiﬁed WKB theory described in Chapter 2 to multidimensional space are presented [43]. Those who are not interested in the generalization of the instanton approach and the proof of its equivalence to the modiﬁed WKB theory can skip Section 6.1.1 and Section 6.1.2. In Section 6.1.3 a general WKB formulation for a general Hamiltonian in curved space is provided and its ﬁnal expression of tunneling splitting can be directly applied to any real systems.In this section, generalizations of the instanton theory and the modiﬁed WKB theory described in Chapter 2 to multidimensional space are presented [43]. Those who are not interested in the generalization of the instanton approach and the proof of its equivalence to the modiﬁed WKB theory can skip Section 6.1.1 and Section 6.1.2. In Section 6.1.3 a general WKB formulation for a general Hamiltonian in curved space is provided and its ﬁnal expression of tunneling splitting can be directly applied to any real systems.

CRC Press eBooksJul 18, 20131 citation

Quasilinear and WKB Solutions in Quantum Mechanics

R. Krivec, V. B. Mandelzaweig, Frank Tabakin A

Asia-Pacific Conference on Few-Body Problems in Physics (APFB)Jan 1, 20072 citations

WKB properties of the time-dependent Schrödinger system

Lochlainn O'Raifeartaigh, Andreas Wipf

Foundations of PhysicsMar 1, 19888 citations

In a recent paper by Eu and Tsien on the uniform WKB theory of inelastic collision, the Magnus-approximation formula for the transition probability was applied to the

{\mathrm{He}}^{+}

-Ne inelastic-scattering process. In this Comment, we report that for the system studied, i.e., the

{\mathrm{He}}^{+}

-Ne system, the uniform WKB Magnus approximation underestimates the cross sections by about 50% and thus overestimates the coupling constant

{a}_{4}

by about 30% above what it should be, while the distorted-wave theory underestimates the coupling constant by about 5%. The uniform-WKB Magnus-approximation formula for the transition probability is further assessed by making comparisons with the results of the exact theory, uniform-WKB numerical-solution method, and the Landau-Zener-St\"uckelberg theory, and is found to behave correctly in the strong-coupling regime as

{a}_{4}

increases. The uniform-WKB Magnus approximation therefore appears to be complementary to the distorted-wave theory which is better in the weak-coupling regime, but breaks down in the strong-coupling regime. The uniform-WKB Magnus approximation is generally better behaved than the Landau-Zener-St\"uckelberg theory over all values of

{a}_{4}

for the system studied.