Semiclassical quantization is exact only for the so called \emph{solvable} potentials, such as the harmonic oscillator. In the \emph{nonsolvable} case the semiclassical phase, given by a series in $\hbar$ , yields more or less approximate results and eventually diverges due to the asymptotic nature of the expansion. A quantum phase is derived to bypass these shortcomings. It achieves exact quantization of nonsolvable potentials and allows to obtain the quantum wavefunction while locally approaching the best pre-divergent semiclassical expansion. An iterative procedure allowing to implement practical calculations with a modest computational cost is also given. The theory is illustrated on two examples for which the limitations of the semiclassical approach were recently highlighted: cold atomic collisions and anharmonic oscillators in the nonperturbative regime.Semiclassical quantization is exact only for the so called \emph{solvable} potentials, such as the harmonic oscillator. In the \emph{nonsolvable} case the semiclassical phase, given by a series in $\hbar$, yields more or less approximate results and eventually diverges due to the asymptotic nature of the expansion. A quantum phase is derived to bypass these shortcomings. It achieves exact quantization of nonsolvable potentials and allows to obtain the quantum wavefunction while locally approaching the best pre-divergent semiclassical expansion. An iterative procedure allowing to implement practical calculations with a modest computational cost is also given. The theory is illustrated on two examples for which the limitations of the semiclassical approach were recently highlighted: cold atomic collisions and anharmonic oscillators in the nonperturbative regime.

arXiv:quant-ph/0411084Nov 11, 20040 citations

Semiclassical wavefunctions and Schrodinger equation

Maurice A. de Gosson

CRC Press eBooksMar 6, 20030 citations

On Non-Linear Quantum Mechanics and the Measurement Problem II. The Random Part of the Wavefunction

W. David Wick

In the first paper of this series, I introduced a non-linear, Hamiltonian, generalization of Schroedinger's theory that blocks formation of macroscopic dispersion ("cats"). But that theory was entirely deterministic, and so the origin of random outcomes in experiments such as Stern-Gerlach or EPRB was left open. Here I propose that Schroedinger's wavefunction has a random component and demonstrate that such an improvised stochastic theory can violate Bell's inequality. Repeated measurements and the back-reaction on the microsystem are discussed in a toy example. Experiments that might falsify the theory are described.In the first paper of this series, I introduced a non-linear, Hamiltonian, generalization of Schroedinger's theory that blocks formation of macroscopic dispersion ("cats"). But that theory was entirely deterministic, and so the origin of random outcomes in experiments such as Stern-Gerlach or EPRB was left open. Here I propose that Schroedinger's wavefunction has a random component and demonstrate that such an improvised stochastic theory can violate Bell's inequality. Repeated measurements and the back-reaction on the microsystem are discussed in a toy example. Experiments that might falsify the theory are described.

arXiv:1710.03800Oct 10, 20171 citation

Schrödinger representation in quantum field theory

Martin Lüscher

Nuclear Physics BJan 1, 198579 citations

Method of the semiclassical representation of quantum theory

Eugeni P. Gordov, Stanislav D. Tvorogov

Jan 1, 19845 citations

Semiclassical quantization is exact only for the so called \emph{solvable} potentials, such as the harmonic oscillator. In the \emph{nonsolvable} case the semiclassical phase, given by a series in

\hbar

, yields more or less approximate results and eventually diverges due to the asymptotic nature of the expansion. A quantum phase is derived to bypass these shortcomings. It achieves exact quantization of nonsolvable potentials and allows to obtain the quantum wavefunction while locally approaching the best pre-divergent semiclassical expansion. An iterative procedure allowing to implement practical calculations with a modest computational cost is also given. The theory is illustrated on two examples for which the limitations of the semiclassical approach were recently highlighted: cold atomic collisions and anharmonic oscillators in the nonperturbative regime.