In
physicsPhysics is a natural science; it is the study of matter and its motion through spacetime and all that derives from these, such as energy and force...
, the
WKB approximation is the most familiar example of a
semiclassicalIn physics, the adjective semiclassical has different precise meanings depending on the context. All these meanings usually refer to some approximation, limit or situation that combines quantum and classical aspects in a given problem...
calculation (See
Old quantum theoryThe old quantum theory was a collection of results from the years 1900–1925 which predate modern quantum mechanics. The theory was never complete or self-consistent, but was a collection of heuristic prescriptions which are now understood to be the first quantum corrections to classical mechanics...
) in
quantum mechanicsQuantum mechanics is a set of principles describing the physical reality at the atomic level of matter and the subatomic . These descriptions include the simultaneous wave-like and particle-like behavior of both matter and radiation...
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.
The name of this method is an acronym for
Wentzel–Kramers–Brillouin approximation. Other often-used acronyms for the method include
JWKB approximation and
WKBJ approximation, where the "J" stands for Jeffreys.
This method is named after physicists
WentzelGregor Wentzel was a German physicist known for development of quantum mechanics...
,
KramersHendrik Anthony "Hans" Kramers was a Dutch physicist. He was the son of Hendrik Kramers, a physician, and Jeanne Susanne Breukelman. On October 25 1920 he was married to Anna Petersen...
, and
BrillouinLéon Nicolas Brillouin was a French physicist. He was born in Sèvres , France. His father, Marcel Brillouin, grandfather, Éleuthère Mascart, and great-grandfather, Charles Briot, were physicists as well...
, who all developed it in 1926.
In
physicsPhysics is a natural science; it is the study of matter and its motion through spacetime and all that derives from these, such as energy and force...
, the
WKB approximation is the most familiar example of a
semiclassicalIn physics, the adjective semiclassical has different precise meanings depending on the context. All these meanings usually refer to some approximation, limit or situation that combines quantum and classical aspects in a given problem...
calculation (See
Old quantum theoryThe old quantum theory was a collection of results from the years 1900–1925 which predate modern quantum mechanics. The theory was never complete or self-consistent, but was a collection of heuristic prescriptions which are now understood to be the first quantum corrections to classical mechanics...
) in
quantum mechanicsQuantum mechanics is a set of principles describing the physical reality at the atomic level of matter and the subatomic . These descriptions include the simultaneous wave-like and particle-like behavior of both matter and radiation...
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.
The name of this method is an acronym for
Wentzel–Kramers–Brillouin approximation. Other often-used acronyms for the method include
JWKB approximation and
WKBJ approximation, where the "J" stands for Jeffreys.
Brief history
This method is named after physicists
WentzelGregor Wentzel was a German physicist known for development of quantum mechanics...
,
KramersHendrik Anthony "Hans" Kramers was a Dutch physicist. He was the son of Hendrik Kramers, a physician, and Jeanne Susanne Breukelman. On October 25 1920 he was married to Anna Petersen...
, and
BrillouinLéon Nicolas Brillouin was a French physicist. He was born in Sèvres , France. His father, Marcel Brillouin, grandfather, Éleuthère Mascart, and great-grandfather, Charles Briot, were physicists as well...
, who all developed it in 1926. In 1923, mathematician
Harold JeffreysSir Harold Jeffreys, FRS was a mathematician, statistician, geophysicist, and astronomer.-Biography:...
had developed a general method of approximating solutions to linear, second-order differential equations, which includes the
Schrödinger equationIn physics, specifically quantum mechanics, the Schrödinger equation is an equation that describes how the quantum state of a physical system changes in time...
. 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, JWKB and BWKJ.
Earlier references to the method are:
CarliniFrancesco Carlini was an Italian astronomer. Born in Milan, he became director of the observatory there in 1832. He published Nuove tavole de moti apparenti del sole in 1832. In 1810, he had already published Esposizione di un nuovo metodo di construire le taole astronomiche applicato alle...
in 1817,
LiouvilleJoseph Liouville was a French mathematician.- Life and work :Liouville graduated from the École Polytechnique in 1827. After some years as an assistant at various institutions including the Ecole Centrale Paris, he was appointed as professor at the École Polytechnique in 1838...
in 1837,
GreenGeorge Green was a British mathematician and physicist, who wrote An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism...
in 1837, Rayleigh in 1912 and
GansRichard Martin Gans , German of Jewish origin, born in Hamburg, was the physicist whofounded the Physics Institute of the National University of La Plata, Argentina. He was its Director in two different periods....
in 1915. Liouville and Green may be called the founders of the method, in 1837, and it is also commonly referred to as the Liouville–Green or LG method.
The important contribution of Jeffreys, Wentzel, Kramers and Brillouin to the method was the inclusion of the treatment of
turning pointsIn mathematics, particularly in calculus, a stationary point is an input to a function where the derivative is zero : where the function "stops" increasing or decreasing ....
, connecting the
evanescentAn evanescent wave is a nearfield standing wave with an intensity that exhibits exponential decay with distance from the boundary at which the wave was formed. Evanescent waves are a general property of wave-equations, and can in principle occur in any context to which a wave-equation applies...
and
oscillatoryOscillation is the repetitive variation, typically in time, of some measure about a central value or between two or more different states. Familiar examples include a swinging pendulum and AC power...
solutions at either side of the turning point. For example, this may occur in the Schrödinger equation, due to a
potential energyPotential energy is energy stored within a physical system as a result of the position or configuration of the different parts of that system. It is called potential energy because it has the potential to be converted into other forms of energy, such as kinetic energy, and to do work in the process...
hill.
WKB method
Generally, WKB theory is a method for approximating the solution of a differential equation whose highest derivative is multiplied by a small parameter ε. The method of approximation is as follows:
For a differential equation
assume a solution of the form of an asymptotic series expansion
In the limit . Substitution of the above
ansatzAnsatz is a German noun with several meanings in the English language. The term "Ansatz" is used in describing solution methods for differential equations.-Definition:...
into the differential equation and canceling out the exponential terms allows one to solve for an arbitrary number of terms in the expansion. WKB theory is a special case of multiple scale analysis.
An example
Consider the second-order homogeneous linear differential equation
where . Plugging in
results in the equation
To leading order, (assuming, for the moment, the series will be asymptotically consistent) the above can be approximated as
In the limit , the dominant balance is given by
So δ is proportional to ε. Setting them equal and comparing powers renders
Which can be recognized as the
Eikonal equationThe eikonal equation is a non-linear partial differential equation encountered in problems of wave propagation, when the wave equation is approximated using the WKB theory...
, with solution
Looking at first-order powers of gives
Which is the unidimensional transport equation, which has the solution
And is an arbitrary constant. We now have a pair of approximations to the system (a pair because can take two signs); the first-order WKB-approximation will be a linear combination of the two:
Higher-order terms can be obtained by looking at equations for higher powers of ε. Explicitly
for . This example comes from Bender and Orszag's textbook (see references).
Application to Schrödinger equation
The one dimensional, time-independent
Schrödinger equationIn physics, specifically quantum mechanics, the Schrödinger equation is an equation that describes how the quantum state of a physical system changes in time...
is,
which can be rewritten as
.
The wavefunction can be rewritten as the exponential of another function Φ (which is closely related to the
actionIn physics, action is an attribute of the development of a physical system. It is a functional which takes the trajectory of the system as its argument and returns a real number as the result....
):
so that
where indicates the derivative of with respect to
x. The derivative can be separated into real and imaginary parts by introducing the real functions
A and
B:
The amplitude of the wavefunction is then , while the phase is . The real and imaginary parts of the Schrödinger equation then become
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.
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 , it follows that
which is only valid when the total energy is greater than the potential energy, as is always the case in
classical motionIn the fields of physics, classical mechanics is one of the two major sub-fields of study in the science of mechanics, which is concerned with the set of physical laws governing and mathematically describing the motions of bodies and aggregates of bodies geometrically distributed within a certain...
. After the same procedure on the next order of the expansion it follows that
On the other hand, if it is the phase varies that varies slowly (as compared to the amplitude), 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
It is apparent from the denominator, that both of these approximate solutions 'blow up' near the classical turning point where 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 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 is yet to be found.
For a classical turning point and close to , 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 functionIn the physical sciences, the Airy function Ai is a special function named after the British astronomer George Biddell Airy. The function Ai and the related function Bi, which is also called an Airy function, are solutions to the differential equationknown as the Airy equation or the Stokes equation...
s.
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 and 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.
Precision of the asymptotic series
The asymptotic series for is usually a divergent series whose general term starts to increase after a certain value . Therefore the smallest error achieved by the WKB method is at best of the order of the last included term. For the equation
with an analytic function, the value and the magnitude of the last term can be estimated as follows (see Winitzki 2005),
where is the point at which needs to be evaluated and is the (complex) turning point where , closest to . The number can be interpreted as the number of oscillations between and the closest turning point. If is a slowly-changing function,
the number will be large, and the minimum error of the asymptotic series will be exponentially small.
See also
- Airy Function
In the physical sciences, the Airy function Ai is a special function named after the British astronomer George Biddell Airy. The function Ai and the related function Bi, which is also called an Airy function, are solutions to the differential equationknown as the Airy equation or the Stokes equation...
- Field electron emission
- Langer correction
The Langer correction is a correction when WKB approximation method is applied to three-dimensional problems with spherical symmetry.When applying WKB approximation method to the radial Schrödinger equationwhere the effective potential is given by...
- Method of steepest descent
In mathematics, the steepest descent method or saddle-point approximation, is a method used to approximate integrals of the formwhere f is some twice-differentiable function, M is a large number, and the integral endpoints a and b could possibly be infinite...
/ Laplace Method
- Old quantum theory
The old quantum theory was a collection of results from the years 1900–1925 which predate modern quantum mechanics. The theory was never complete or self-consistent, but was a collection of heuristic prescriptions which are now understood to be the first quantum corrections to classical mechanics...
- Perturbation methods
- Quantum tunneling
- Slowly varying envelope approximation
In physics, the slowly varying envelope approximation is the assumption that the envelope of a forward-travelling wave pulse varies slowly in time and space compared to a period or wavelength...
External links