The path integral formulation can reproduce the right energy spectrum of the harmonic oscillator potential, but it cannot resolve the Coulomb potential problem. This is because the path integral cannot properly take into account the boundary condition, which is due to the presence of the scattering states in the Coulomb potential system. On the other hand, the Sommerfeld quantization can reproduce the right energy spectrum of both harmonic oscillator and Coulomb potential cases since the boundary condition is effectively taken into account in this semiclassical treatment. The basic difference between the two schemes should be that no constraint is imposed on the wave function in the path integral while the Sommerfeld quantization rule is derived by requiring that the state vector should be a single-valued function. The limitation of the semiclassical method is also clarified in terms of the square well and $δ(x)$ function potential models.The path integral formulation can reproduce the right energy spectrum of the harmonic oscillator potential, but it cannot resolve the Coulomb potential problem. This is because the path integral cannot properly take into account the boundary condition, which is due to the presence of the scattering states in the Coulomb potential system. On the other hand, the Sommerfeld quantization can reproduce the right energy spectrum of both harmonic oscillator and Coulomb potential cases since the boundary condition is effectively taken into account in this semiclassical treatment. The basic difference between the two schemes should be that no constraint is imposed on the wave function in the path integral while the Sommerfeld quantization rule is derived by requiring that the state vector should be a single-valued function. The limitation of the semiclassical method is also clarified in terms of the square well and $δ(x)$ function potential models.

arXiv:1809.04416Aug 18, 20180 citations

On the semiclassical quantization condition

Luis Gottdiener

American Journal of PhysicsAug 1, 19770 citations

Radial power-like potentials: from the Bohr-Sommerfeld $\ell \neq 0$ state energies to the exact onesRadial power-like potentials: from the Bohr-Sommerfeld $\ell \neq 0$ state energies to the exact ones

J. C. del Valle, A. V. Turbiner

Two new Bohr-Sommerfeld (B-S) type quantization conditions for $d$ -dimensional radial potentials are proposed for non-singular and singular potentials, respectively. The power-like potentials \hbox{ $V=\operatorname{sign}(m)\,r^m$ } for $m=-3/2,-1,-1/2, 1,3,4,6$ and $V=\log r$ (that corresponds to $m=0$ ) are studied in detail for radial quantum numbers $n_r \in [0,100]$ and for angular momenta $\ell=1,2,3,4,5$ in dimensions $d=2,3$ . It is shown that for large $n_r$ the B-S energies $E_{BS}$ always tend to the exact ones. For $m=2$ (the radial harmonic oscillator) and $m=-1$ (the $d$ -dimensional analogue of the $3D$ Coulomb potential) the B-S spectrum coincides with the exact one for the first and second quantization conditions, respectively. In general, up to a small parametric shift of the radial quantum number, the analytic formulas for the energy spectrum can be written for arbitrary power-like potential. A comparison of $E_{BS}$ with the energies $E^{(L)}_{BS}$ , obtained for the radial potentials modified by the Langer correction for $d=3$ , are in good agreement in general: in detail, it was checked for the quartic oscillator $V=r^4$ . It is demonstrated that introducing a dimensionless {\it WKB correction} $γ$ into the r.h.s. of the proposed B-S type quantization condition allows us to reproduce the exact energies with $γ$ being very small for positive degrees $m \geq 0$ and sufficiently small for non-positive ones $m < 0$ for all $n_r, \ell$ studied.Two new Bohr-Sommerfeld (B-S) type quantization conditions for $d$-dimensional radial potentials are proposed for non-singular and singular potentials, respectively. The power-like potentials \hbox{$V=\operatorname{sign}(m)\,r^m$} for $m=-3/2,-1,-1/2, 1,3,4,6$ and $V=\log r$ (that corresponds to $m=0$) are studied in detail for radial quantum numbers $n_r \in [0,100]$ and for angular momenta $\ell=1,2,3,4,5$ in dimensions $d=2,3$. It is shown that for large $n_r$ the B-S energies $E_{BS}$ always tend to the exact ones. For $m=2$ (the radial harmonic oscillator) and $m=-1$ (the $d$-dimensional analogue of the $3D$ Coulomb potential) the B-S spectrum coincides with the exact one for the first and second quantization conditions, respectively. In general, up to a small parametric shift of the radial quantum number, the analytic formulas for the energy spectrum can be written for arbitrary power-like potential. A comparison of $E_{BS}$ with the energies $E^{(L)}_{BS}$, obtained for the radial potentials modified by the Langer correction for $d=3$, are in good agreement in general: in detail, it was checked for the quartic oscillator $V=r^4$. It is demonstrated that introducing a dimensionless {\it WKB correction} $γ$ into the r.h.s. of the proposed B-S type quantization condition allows us to reproduce the exact energies with $γ$ being very small for positive degrees $m \geq 0$ and sufficiently small for non-positive ones $m < 0$ for all $n_r, \ell$ studied.

quant-pharXiv:2610.36421Oct 28, 20260 citations

Semiclassical quantization with a quantum adiabatic phase

Hiroshi Kuratsuji, Shinji Iida

Physics Letters ASep 1, 198523 citations

Uncertainty relations for modified isotropic harmonic oscillator and Coulomb potentials

S. H. Patil, K. D. Sen

Physics Letters ANov 29, 200650 citations

Classical position probability densities for spherically symmetric potentials

Lorenzo J. Curtis, David G. Ellis

A simple position probability density formulation is presented for the motion of a particle in a spherically symmetric potential. The approach provides an alternative to Newtonian methods for presentation in an elementary course, and requires only elementary algebra and one tabulated integral. The method is applied to compute the distributions for the Kepler-Coulomb and isotropic harmonic oscillator potentials. Formulas are also deduced for the average values for powers of the radial coordinate, and applied to describe perturbations to these systems. The classical results are also compared with quantum mechanical calculations using the Einstein-Brillouin-Keller semiclassical quantization.A simple position probability density formulation is presented for the motion of a particle in a spherically symmetric potential. The approach provides an alternative to Newtonian methods for presentation in an elementary course, and requires only elementary algebra and one tabulated integral. The method is applied to compute the distributions for the Kepler-Coulomb and isotropic harmonic oscillator potentials. Formulas are also deduced for the average values for powers of the radial coordinate, and applied to describe perturbations to these systems. The classical results are also compared with quantum mechanical calculations using the Einstein-Brillouin-Keller semiclassical quantization.

arXiv:physics/0501111Jan 20, 20050 citations

δ(x)

function potential models.

Radial power-like potentials: from the Bohr-Sommerfeld

\ell \neq 0

state energies to the exact ones

Two new Bohr-Sommerfeld (B-S) type quantization conditions for

d

-dimensional radial potentials are proposed for non-singular and singular potentials, respectively. The power-like potentials \hbox{

V=\operatorname{sign}(m)\,r^m

} for

m=-3/2,-1,-1/2, 1,3,4,6

and

V=\log r

(that corresponds to

m=0

) are studied in detail for radial quantum numbers

n_r \in [0,100]

and for angular momenta

\ell=1,2,3,4,5

in dimensions

d=2,3

. It is shown that for large

n_r

the B-S energies

E_{BS}

always tend to the exact ones. For

m=2

(the radial harmonic oscillator) and

m=-1

(the

d

-dimensional analogue of the

3D

Coulomb potential) the B-S spectrum coincides with the exact one for the first and second quantization conditions, respectively. In general, up to a small parametric shift of the radial quantum number, the analytic formulas for the energy spectrum can be written for arbitrary power-like potential. A comparison of

E_{BS}

with the energies

E^{(L)}_{BS}

, obtained for the radial potentials modified by the Langer correction for

d=3

, are in good agreement in general: in detail, it was checked for the quartic oscillator

V=r^4

. It is demonstrated that introducing a dimensionless {\it WKB correction}

γ

into the r.h.s. of the proposed B-S type quantization condition allows us to reproduce the exact energies with

γ

being very small for positive degrees

m \geq 0

and sufficiently small for non-positive ones

m < 0

for all

n_r, \ell

studied.