© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

Physical Review

On the Wentzel-Brillouin-Kramers Approximate Solution of the Wave Equation

Lloyd A. Young, G. E. Uhlenbeck

Published Oct 1, 1930• 42 citations

Abstract

A discussion of the Wentzel-Brillouin-Kramers method of obtaining an approximate solution of the wave equation is given from the point of view that it forms a link between the quantum theory of Bohr and the new quantum mechanics. This becomes clear when one compares the probability distributions (see Figs. 2 and 3) and calculates the mean values $\stackrel{-}{{r}^{k}}$ (see Table I). It is shown that for high quantum numbers these become the classical values. The method leads also, as shown by Zwaan, to quantization by the classical phase integrals with the use of half-integer quantum numbers. In the last section the method is applied to the charged shell atom model. It is shown that the condition of smooth joining of the wave function is practically equivalent to half-integer quantization of the sum of the inner and outer phase integrals. Of course there is no longer a sharp distinction between penetrating and non-penetrating orbits. The Land\'e formula for the doublet separations is derived. The value of ${\ensuremath{\psi}}^{2}(0)$ (0), which occurs in the calculation of the hyperfine separations in $^{2}S$ states, is also given.

DOI: 10.1103/physrev.36.1154

Related papers

The Wentzel-Brillouin-Kramers Method of Solving the Wave Equation

J. L. Dunham

A more general treatment than has been available of the Wentzel-Brillouin-Kramers method of solving Schr\"odinger's equation for one degree of freedom is given. Wentzel's original energy-level condition is shown to be an asymptotic expression, good for large masses and large values of the quantum number. The connection between this method and that of Young is discussed. Finally the formulas are written in a form convenient for application to the calculation of energy levels from actual potential functions.

Physical Review•Sep 15, 1932•608 citations

The Wentzel–Kramers–Brillouin approximation method applied to the Wigner function

Jaromir Tosiek, R. Cordero, F. J. Turrubiates

An adaptation of the Wentzel–Kramers–Brilluoin method in the deformation quantization formalism is presented with the aim to obtain an approximate technique of solving the eigenvalue problem for energy in the phase space quantum approach. A relationship between the phase σ(r→) of a wave function expiħσ(r→) and its respective Wigner function is derived. Formulas to calculate the Wigner function of a product and of a superposition of wave functions are proposed. Properties of a Wigner function of interfering states are also investigated. Examples of this quasi–classical approximation in deformation quantization are analysed. A strict form of the Wigner function for states represented by tempered generalised functions has been derived. Wigner functions of unbound states in the Poeschl–Teller potential have been found.

arXiv:1502.05497•Journal of Mathematical Physics•Jun 1, 2016•5 citations

Alternating Wentzel-Kramers-Brillouin Approximation to the Schrödinger Equation: Rediscover the Bremmers series and beyond

Tsai, Yu-An, Sheng Der Chao

We propose an extension of the Wentzel-Kramers-Brillouin (WKB) approximation for solving the Schrödinger equation. Based on an ansatz for the wave function, subject to an auxiliary condition on its first derivative, we obtain a set of coupled differential equations that decouple, via an alternating perturbation method, into the well-known Bremmer series. The perturbation improves the wave function amplitudes alternately, and its phase is refined through recursive diagonalization. From this construction, we derive a general quantization formula that encodes the geometric-optics-like physics of the system. When the ratio of the differential reflection coefficient to the classical momentum remains constant, the formula reduces to a closed-form quantization condition consistent with that obtained by re-summing the WKB series to all orders.

arXiv:2207.00935•Jul 3, 2022•0 citations

Approximation Methods III: WKB and Asymptotic Methods

S. Rajasekar, R. Velusamy

The WKB approximation is a physically intuitive method and makes many connections with intuition about waves and classical mechanics as well as with results from the early days of quantum theory as it leads to the Bohr quantization condition. The WKB method is due to Gregor Wentzel, Hendrik Anthony Kramers and Leon Nicolas Brillouin. The method is applicable for slowly varying potentials, that is, the potentials that change not more than a few wavelengths. The WKB approximation is a physically intuitive method and makes many connections with intuition about waves and classical mechanics as well as with results from the early days of quantum theory as it leads to the Bohr quantization condition. The Schrödinger equation is solved to find the penetration of a square potential barrier. The WKB method is applied to obtain a transmission coefficient for penetration through a barrier.

CRC Press eBooks•Sep 29, 2022•0 citations

Application of WKB Method in Approximating Wave Functions in Square Potential Wells

Haohui Guo

Abstract This paper introduces a widely used approximation method, the Wentzel-Kramers-Brillouin (WKB) method, for solving the Schrodinger equation, sets up a theoretical framework to derive the general solutions of the method, and carries out the application of the approximation method in specific cases to verify the validity and practicality of the method. Throughout the study, numerical methods, such as the Simpsons method, and programming tools, such as Python, are used to process sophisticated calculations. The final approximated solutions for the Schrodinger equation are able to demonstrate the state of a particle in an isolated quantum system. All the presented studies are based upon the knowledge and skillsets of a high school student.

Journal of Physics Conference Series•Jun 1, 2022•3 citations

The double-well potential in quantum mechanics: a simple, numerically exact formulation

Vedran Jelic, F. Marsiglio

The double-well potential is arguably one of the most important potentials in quantum mechanics, because the solution contains the notion of a state as a linear superposition of 'classical' states, a concept which has become very important in quantum information theory. It is therefore desirable to have solutions to simple double-well potentials that are accessible to the undergraduate student. We describe a method for obtaining the numerically exact eigenenergies and eigenstates for such a model, along with the energies obtained through the Wentzel–Kramers–Brillouin (WKB) approximation. The exact solution is accessible with elementary mathematics, though numerical solutions are required. We also find that the WKB approximation is remarkably accurate, not just for the ground state, but also for the excited states.

arXiv:1209.2521•European Journal of Physics•Sep 14, 2012•93 citations

A discussion of the Wentzel-Brillouin-Kramers method of obtaining an approximate solution of the wave equation is given from the point of view that it forms a link between the quantum theory of Bohr and the new quantum mechanics. This becomes clear when one compares the probability distributions (see Figs. 2 and 3) and calculates the mean values

\stackrel{-}{{r}^{k}}

(see Table I). It is shown that for high quantum numbers these become the classical values. The method leads also, as shown by Zwaan, to quantization by the classical phase integrals with the use of half-integer quantum numbers. In the last section the method is applied to the charged shell atom model. It is shown that the condition of smooth joining of the wave function is practically equivalent to half-integer quantization of the sum of the inner and outer phase integrals. Of course there is no longer a sharp distinction between penetrating and non-penetrating orbits. The Land'e formula for the doublet separations is derived. The value of

{\ensuremath{\psi}}^{2}(0)

(0), which occurs in the calculation of the hyperfine separations in

^{2}S

states, is also given.

A more general treatment than has been available of the Wentzel-Brillouin-Kramers method of solving Schr\"odinger's equation for one degree of freedom is given. Wentzel's original energy-level condition is shown to be an asymptotic expression, good for large masses and large values of the quantum number. The connection between this method and that of Young is discussed. Finally the formulas are written in a form convenient for application to the calculation of energy levels from actual potential functions.

An adaptation of the Wentzel–Kramers–Brilluoin method in the deformation quantization formalism is presented with the aim to obtain an approximate technique of solving the eigenvalue problem for energy in the phase space quantum approach. A relationship between the phase σ(r→) of a wave function expiħσ(r→) and its respective Wigner function is derived. Formulas to calculate the Wigner function of a product and of a superposition of wave functions are proposed. Properties of a Wigner function of interfering states are also investigated. Examples of this quasi–classical approximation in deformation quantization are analysed. A strict form of the Wigner function for states represented by tempered generalised functions has been derived. Wigner functions of unbound states in the Poeschl–Teller potential have been found.

We propose an extension of the Wentzel-Kramers-Brillouin (WKB) approximation for solving the Schrödinger equation. Based on an ansatz for the wave function, subject to an auxiliary condition on its first derivative, we obtain a set of coupled differential equations that decouple, via an alternating perturbation method, into the well-known Bremmer series. The perturbation improves the wave function amplitudes alternately, and its phase is refined through recursive diagonalization. From this construction, we derive a general quantization formula that encodes the geometric-optics-like physics of the system. When the ratio of the differential reflection coefficient to the classical momentum remains constant, the formula reduces to a closed-form quantization condition consistent with that obtained by re-summing the WKB series to all orders.

The WKB approximation is a physically intuitive method and makes many connections with intuition about waves and classical mechanics as well as with results from the early days of quantum theory as it leads to the Bohr quantization condition. The WKB method is due to Gregor Wentzel, Hendrik Anthony Kramers and Leon Nicolas Brillouin. The method is applicable for slowly varying potentials, that is, the potentials that change not more than a few wavelengths. The WKB approximation is a physically intuitive method and makes many connections with intuition about waves and classical mechanics as well as with results from the early days of quantum theory as it leads to the Bohr quantization condition. The Schrödinger equation is solved to find the penetration of a square potential barrier. The WKB method is applied to obtain a transmission coefficient for penetration through a barrier.

Abstract This paper introduces a widely used approximation method, the Wentzel-Kramers-Brillouin (WKB) method, for solving the Schrodinger equation, sets up a theoretical framework to derive the general solutions of the method, and carries out the application of the approximation method in specific cases to verify the validity and practicality of the method. Throughout the study, numerical methods, such as the Simpsons method, and programming tools, such as Python, are used to process sophisticated calculations. The final approximated solutions for the Schrodinger equation are able to demonstrate the state of a particle in an isolated quantum system. All the presented studies are based upon the knowledge and skillsets of a high school student.

The double-well potential is arguably one of the most important potentials in quantum mechanics, because the solution contains the notion of a state as a linear superposition of 'classical' states, a concept which has become very important in quantum information theory. It is therefore desirable to have solutions to simple double-well potentials that are accessible to the undergraduate student. We describe a method for obtaining the numerically exact eigenenergies and eigenstates for such a model, along with the energies obtained through the Wentzel–Kramers–Brillouin (WKB) approximation. The exact solution is accessible with elementary mathematics, though numerical solutions are required. We also find that the WKB approximation is remarkably accurate, not just for the ground state, but also for the excited states.