WKB approximation
Encyclopedia
In mathematical physics
, the WKB approximation or WKB method is a method for finding approximate solutions to linear partial differential equations with spatially varying coefficients. It is typically used for 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.
The name is an acronym for Wentzel–Kramers–Brillouin. It is also known as the LG or Liouville–Green method. Other often-used acronyms for the method include JWKB and WKBJ, where the "J" stands for Jeffreys.
, Kramers
, and Brillouin
, who all developed it in 1926. In 1923, mathematician Harold Jeffreys
had developed a general method of approximating solutions to linear, second-order differential equations, which includes the Schrödinger equation
. Even though the Schrödinger equation was developed two years later, Wentzel, Kramers, and Brillouin were apparently unaware of this earlier work, so 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: Carlini
in 1817, Liouville
in 1837, Green
in 1837, Rayleigh in 1912 and Gans
in 1915. Liouville and Green may be said to have founded 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 points
, connecting the evanescent
and oscillatory
solutions at either side of the turning point. For example, this may occur in the Schrödinger equation, due to a potential energy
hill.
For a differential equation
assume a solution of the form of an asymptotic series expansion
In the limit
. Substitution of the above ansatz
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.
where
. Substituting
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 equation
, with solution
Looking at first-order powers of
gives
This is the unidimensional transport equation, having the solution
where
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).
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.
is
which can be rewritten as
The wavefunction can be rewritten as the exponential of another function Φ (which is closely related to the action
):
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 zeroth-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 motion
. After the same procedure on the next order of the expansion it follows that
On the other hand, if it is the phase 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). Finding the next order of the expansion yields
It is apparent from the denominator, that both of these approximate solutions become singular 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 wave-function 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 
, the term 
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 function
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.
Mathematical physics
Mathematical physics refers to development of mathematical methods for application to problems in physics. The Journal of Mathematical Physics defines this area as: "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and...
, the WKB approximation or WKB method is a method for finding approximate solutions to linear partial differential equations with spatially varying coefficients. It is typically used for a semiclassical calculation in quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
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 is an acronym for Wentzel–Kramers–Brillouin. It is also known as the LG or Liouville–Green method. Other often-used acronyms for the method include JWKB and WKBJ, where the "J" stands for Jeffreys.
Brief history
This method is named after physicists WentzelGregor Wentzel
Gregor Wentzel was a German physicist known for development of quantum mechanics. Wentzel, Hendrik Kramers, and Léon Brillouin developed the Wentzel–Kramers–Brillouin approximation in 1926...
, Kramers
Hendrik Anthony Kramers
Hendrik Anthony "Hans" Kramers was a Dutch physicist.-Background and education:...
, and Brillouin
Léon Brillouin
Léon Nicolas Brillouin was a French physicist. He made contributions to quantum mechanics, radio wave propagation in the atmosphere, solid state physics, and information theory.-Early life:...
, who all developed it in 1926. In 1923, mathematician Harold Jeffreys
Harold Jeffreys
Sir Harold Jeffreys, FRS was a mathematician, statistician, geophysicist, and astronomer. His seminal book Theory of Probability, which first appeared in 1939, played an important role in the revival of the Bayesian view of probability.-Biography:Jeffreys was born in Fatfield, Washington, County...
had developed a general method of approximating solutions to linear, second-order differential equations, which includes the Schrödinger equation
Schrödinger equation
The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....
. Even though the Schrödinger equation was developed two years later, Wentzel, Kramers, and Brillouin were apparently unaware of this earlier work, so 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: Carlini
Francesco Carlini
Francesco 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, Liouville
Joseph Liouville
- 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, Green
George Green
George Green was a British mathematical physicist who wrote An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism...
in 1837, Rayleigh in 1912 and Gans
Richard Gans
Richard Martin Gans , German of Jewish origin, born in Hamburg, was the physicist who founded the Physics Institute of the National University of La Plata, Argentina...
in 1915. Liouville and Green may be said to have founded 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 points
Stationary point
In 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 evanescent
Evanescent wave
An 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 oscillatory
Oscillation
Oscillation 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. The term vibration is sometimes used more narrowly to mean a mechanical oscillation but sometimes...
solutions at either side of the turning point. For example, this may occur in the Schrödinger equation, due to a potential energy
Potential energy
In physics, potential energy is the energy stored in a body or in a system due to its position in a force field or due to its configuration. The SI unit of measure for energy and work is the Joule...
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
Ansatz is a German noun with several meanings in the English language.It is widely encountered in physics and mathematics literature.Since ansatz is a noun, in German texts the initial a of this word is always capitalised.-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 equationwhere
. Substitutingresults 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 bySo δ is proportional to ε. Setting them equal and comparing powers renders
which can be recognized as the Eikonal equation
Eikonal equation
The 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
givesThis is the unidimensional transport equation, having the solution
where
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).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 equationwith
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.Application to Schrödinger equation
The one dimensional, time-independent Schrödinger equationSchrödinger equation
The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it 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 action
Action (physics)
In physics, action is an attribute of the dynamics of a physical system. It is a mathematical functional which takes the trajectory, also called path or history, of the system as its argument and has a real number as its result. Action has the dimension of energy × time, and its unit is...
):
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 becomeNext, 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 zeroth-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 thatwhich is only valid when the total energy is greater than the potential energy, as is always the case in classical motion
Classical mechanics
In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...
. After the same procedure on the next order of the expansion it follows that
On the other hand, if it is the phase that varies slowly (as compared to the amplitude), (
) thenwhich is only valid when the potential energy is greater than the total energy (the regime in which quantum tunneling occurs). Finding the next order of the expansion yields
It is apparent from the denominator, that both of these approximate solutions become singular 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 wave-function 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 , the term 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 function
Airy function
In the physical sciences, the Airy function Ai is a special function named after the British astronomer George Biddell Airy...
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.
See also
- InstantonInstantonAn instanton is a notion appearing in theoretical and mathematical physics. Mathematically, a Yang–Mills instanton is a self-dual or anti-self-dual connection in a principal bundle over a four-dimensional Riemannian manifold that plays the role of physical space-time in non-abelian gauge theory...
- Airy FunctionAiry functionIn the physical sciences, the Airy function Ai is a special function named after the British astronomer George Biddell Airy...
- Field electron emission
- Langer correction
- Method of steepest descent / Laplace Method
- Old quantum theoryOld 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...
- Perturbation methods
- Quantum tunneling
- Slowly varying envelope approximationSlowly varying envelope approximationIn 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...