In physics, the WKB (Wentzel-Kramers-Brillouin) approximation, also known as WKBJ (Wentzel-Kramers-Brillouin-Jeffreys) approximation, is the most familiar example of a semiclassical calculation in quantum mechanics in which the wavefunction is recast as an exponential function, semiclassically expanded, and then either the amplitude or the phase is taken to be slowly changing. A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... In physics, the adjective semiclassical has different precise meanings depending on the context. ... For a less technical and generally accessible introduction to the topic, see Introduction to quantum mechanics. ...

This method is named after physicists Wentzel, Kramers, and Brillouin, who all developed it in 1926. In 1923, mathematician Harold Jeffreys had developed a general method of approximating linear, second-order differential equations, which includes the Schrödinger equation. But since the Schrödinger equation was developed two years later, and Wentzel, Kramers, and Brillouin were apparently unaware of this earlier work, Jeffreys is often neglected credit. Early texts in quantum mechanics contain any number of combinations of their initials, including WBK, BWK, WKBJ and BWKJ. Gregor Wentzel (February 17, 1898, in Düsseldorf, Germany – August 12, 1978, in Ascona, Switzerland) was a German physicist known for development of quantum mechanics. ... Hendrik Anthony Kramers (Rotterdam, February 2, 1894 – Oegstgeest, April 24, 1952) was a Dutch physicist. ... Léon N. Brillouin ( August 7, 1889- 1969) was a French physicist. ... Year 1926 (MCMXXVI) was a common year starting on Friday (link will display the full calendar) of the Gregorian calendar. ... Year 1923 (MCMXXIII) was a common year starting on Monday (link will display the full calendar) of the Gregorian calendar. ... Sir Harold Jeffreys (22 April 1891 – 18 March 1989) was a mathematician, statistician, geophysicist, and astronomer. ... For a non-technical introduction to the topic, please see Introduction to quantum mechanics. ...

Derivation

The one dimensional, time-independent Schrödinger equation is For a non-technical introduction to the topic, please see Introduction to quantum mechanics. ...

,

which can be rewritten as

.

The wavefunction can be rewritten as the exponential of another function Φ (which is closely related to the action): In physics, the action is an integral quantity that is used to determine the evolution of a physical system between two defined states using the calculus of variations. ...

,

so that

,

where Φ' indicates the derivative of Φ with respect to x. The derivative Φ'(x) can be separated into real and imaginary parts by introducing the real functions A and B:

.

The amplitude of the wavefunction is then e^A(x) while the phase is B(x). The Schrödinger equation implies that these functions must satisfy:

and therefore, since the right hand side of the differential equation for Φ is real,

Next, the semiclassical approximation is invoked. This means that each function is expanded as a power series in . From the equations it can be seen that the power series must start with at least an order of to satisfy the real part of the equation. In order to achieve a good classical limit, it is necessary to start with as high a power of Planck's constant as possible. In physics, the adjective semiclassical has different precise meanings depending on the context. ...

To first order in this expansion, the conditions on A and B can be written.

If the amplitude varies sufficiently slowly as compared to the phase (A₀(x) = 0), it follows that

,

which is only valid when the total energy is greater than the potential energy, as is always the case in classical motion. After the same procedure on the next order of the expansion it follows that Classical mechanics (commonly confused with Newtonian mechanics, which is a subfield thereof) is used for describing the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies. ...

On the other hand, if it is the phase varies that varies slowly (as compared to the amplitude), (B₀(x) = 0) then

which is only valid when the potential energy is greater than the total energy (the regime in which quantum tunneling occurs). Grinding out the next order of the expansion yields Quantum tunneling is the quantum-mechanical effect of transitioning through a classically-forbidden energy state. ...

It is apparent from the denominator, that both of these approximate solutions 'blow up' near the classical turning point where E = V(x) and cannot be valid. These are the approximate solutions away from the potential hill and beneath the potential hill. Away from the potential hill, the particle acts similarly to a free wave - the phase is oscillating. Beneath the potential hill, the particle undergoes exponential changes in amplitude.

To be complete the derivation, the approximate solutions must be found everywhere and their coefficients matched to make a global approximate solution. The approximate solution near the classical turning points E = V(x) is yet to be found.

For a classical turning point x₁ and close to E = V(x₁), can be expanded in a power series.

To first order, one finds

.

This differential equation is known as the Airy equation, and the solution may be written in terms of Airy functions. In mathematics, the Airy function Ai(x) is a special function named after the British astronomer George Biddell Airy. ... In mathematics, the Airy function Ai(x) is a special function, i. ...

This solution should connect the far away and beneath solutions. Given the 2 coefficients on one side of the classical turning point, the 2 coefficients on the other side of the classical turning point can be determined by using this local solution to connect them. Thus, a relationship between C₀,θ and C ₊ ,C ₋ can be found.

Fortunately the Airy functions will asymptote into sine, cosine and exponential functions in the proper limits. The relationship can be found to be as follows (often referred to as "connection formulas"):

Now the global (approximate) solutions can be constructed.

See also

• Airy Function
• Langer correction
• Method of steepest descent / Laplace Method
• Perturbation methods
• Quantum tunneling

In mathematics, the Airy function Ai(x) is a special function, i. ... Langer correction is a correction when WKB approximation method is applied to three-dimensional problems with spherical symmetry. ... In mathematics, the steepest descent method or saddle-point approximation is a method used to approximate integrals of the form where f(x) is some twice-differentiable function, M is a large number, and the integral endpoints a and b could possibly be infinite. ... This article describes perturbation theory as a general mathematical method. ... Quantum tunneling is the quantum-mechanical effect of transitioning through a classically-forbidden energy state. ...

References

• Razavy, Moshen (2003). Quantum Theory of Tunneling. World Scientific. ISBN 981-238-019-1. 
• Griffiths, David J. (2004). Introduction to Quantum Mechanics (2nd ed.). Prentice Hall. ISBN 0-13-111892-7. 
• Liboff, Richard L. (2003). Introductory Quantum Mechanics (4th ed.). Addison-Wesley. ISBN 0-8053-8714-5. 
• Sakurai, J. J. (1993). Modern Quantum Mechanics. Addison-Wesley. ISBN 0-201-53929-2. 

Richard L. Liboff is a U.S. physicist who has authored five books and nearly 150 other publications in variety of fields, including plasma physics, planetary physics, cosmology, quantum chaos, and quantum billiards. ...

External links

• Richard Fitzpatrick, The W.K.B. Approximation (2002). (An application of the WKB approximation to the scattering of radio waves from the ionosphere.)
• Free WKB library for Microsoft Visual C v6 for some special functions