In
mathematical physics
Mathematical physics is the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the de ... , the WKB approximation or WKB method is a technique for finding approximate solutions to
linear differential equations
In mathematics, a linear differential equation is a differential equation that is linear in the unknown function and its derivatives, so it can be written in the form
a_0(x)y + a_1(x)y' + a_2(x)y'' \cdots + a_n(x)y^ = b(x)
where and are arbi ... with spatially varying coefficients. It is typically used for a
semiclassical calculation in
quantum mechanics
Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ... in which the
wave function
In quantum physics, a wave function (or wavefunction) is a mathematical description of the quantum state of an isolated quantum system. The most common symbols for a wave function are the Greek letters and (lower-case and capital psi (letter) ... is recast as an exponential function, semiclassically expanded, and then either the amplitude or the phase is taken to be changing slowly.
The name is an initialism for Wentzel–Kramers–Brillouin. It is also known as the LG or Liouville–Green method. Other often-used letter combinations include JWKB and WKBJ, where the "J" stands for Jeffreys.
Brief history
This method is named after physicists
Gregor Wentzel
Gregor Wentzel (17 February 1898 – 12 August 1978) was a German physicist known for development of quantum mechanics. Wentzel, Hendrik Kramers, and Léon Brillouin developed the Wentzel–Kramers–Brillouin approximation in 1926. In his early y ... ,
Hendrik Anthony Kramers
Hendrik Anthony "Hans" Kramers (17 December 1894 – 24 April 1952) was a Dutch physicist who worked with Niels Bohr to understand how electromagnetic waves interact with matter and made important contributions to quantum mechanics and statistica ... , and
Léon Brillouin
Léon Nicolas Brillouin (; August 7, 1889 – October 4, 1969) was a French physicist. He made contributions to quantum mechanics, radio wave propagation in the atmosphere, solid-state physics, and information theory.
Early life
Brilloui ... , who all developed it in 1926.
[ In 1923,][ mathematician ]Harold Jeffreys
Sir Harold Jeffreys, FRS (22 April 1891 – 18 March 1989) was a British geophysicist who made significant contributions to mathematics and statistics. His book, ''Theory of Probability'', which was first published in 1939, played an importan ... Schrödinger equation
The Schrödinger equation is a partial differential equation that governs the wave function of a non-relativistic quantum-mechanical system. Its discovery was a significant landmark in the development of quantum mechanics. It is named after E ... Francesco Carlini
Francesco Carlini (January 7, 1783 – August 29, 1862) was an Italian astronomer. Born in Milan, he became director of the Brera Astronomical Observatory there in 1832. He published ''Nuove tavole de moti apparenti del sole'' in 1832. In 1810, ... [ ]Joseph Liouville
Joseph Liouville ( ; ; 24 March 1809 – 8 September 1882) was a French mathematician and engineer.
Life and work
He was born in Saint-Omer in France on 24 March 1809. His parents were Claude-Joseph Liouville (an army officer) and Thérès ... [ ]George Green in 1837,[ ]Lord Rayleigh
John William Strutt, 3rd Baron Rayleigh ( ; 12 November 1842 – 30 June 1919), was an English physicist who received the Nobel Prize in Physics in 1904 "for his investigations of the densities of the most important gases and for his discovery ... [ and ]Richard Gans
__NOTOC__
Richard Martin Gans (7 March 1880 – 27 June 1954), German of Jewish origin, born in Hamburg, was the physicist who founded the Physics Institute of the National University of La Plata, Argentina. He was its Director in two different ... [ 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
Evanescent may refer to:
* Evanescent (dermatology), a class of skin lesions
* "Evanescent" (song), a song by Vamps
* Evanescent wave
In electromagnetics, an evanescent field, or evanescent wave, is an oscillating electric and/or magnetic f ... oscillatory
Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendulum ... potential energy
In physics, potential energy is the energy of an object or system due to the body's position relative to other objects, or the configuration of its particles. The energy is equal to the work done against any restoring forces, such as gravity ... Formulation
Generally, WKB theory is a method for approximating the solution of a differential equation whose ''highest derivative is multiplied by a small parameter'' . The method of approximation is as follows.
For a differential equation
\varepsilon \frac + a(x)\frac + \cdots + k(x)\frac + m(x)y= 0,
assume a solution of the form of an asymptotic series
In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation t ... y(x) \sim \exp\left frac\sum_^ \delta^n S_n(x)\right /math>
in the limit . The asymptotic scaling of in terms of will be determined by the equation – see the example below.
Substituting the above ansatz
In physics and mathematics, an ansatz (; , meaning: "initial placement of a tool at a work piece", plural ansatzes or, from German, ansätze ; ) is an educated guess or an additional assumption made to help solve a problem, and which may later be ... multiple scale analysis . An example
This example comes from the text of Carl M. Bender and Steven Orszag
Steven Alan Orszag (February 27, 1943 – May 1, 2011) was an American mathematician.
Life and career
Orszag was born to a Jewish family in Manhattan, the son of Joseph Orszag, a lawyer. \epsilon^2 \frac = Q(x) y,
where Q(x) \neq 0 . Substituting
y(x) = \exp \left frac \sum_^\infty \delta^n S_(x)\right /math>
results in the equation
\epsilon^2\left frac \left(\sum_^\infty \delta^nS_^\right)^2 + \frac \sum_^\delta^n S_^\right = Q(x).
To leading order in ''ϵ'' (assuming, for the moment, the series will be asymptotically consistent), the above can be approximated as
\frac ^2 + \frac S_^ S_^ + \frac S_^ = Q(x).
In the limit , the dominant balance is given by
\frac ^2 \sim Q(x).
So is proportional to ''ϵ''. Setting them equal and comparing powers yields
\epsilon^0: \quad ^2 = Q(x),
which can be recognized as the eikonal equation
An eikonal equation (from Greek εἰκών, image) is a non-linear first-order partial differential equation that is encountered in problems of wave propagation.
The classical eikonal equation in geometric optics is a differential equation of t ... S_(x) = \pm \int_^x \sqrt\,dx'.
Considering first-order powers of fixes
\epsilon^1: \quad 2 S_^ S_^ + S_^ = 0.
This has the solution
S_(x) = -\frac \ln Q(x) + k_1,
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:
y(x) \approx c_1 Q^(x) \exp\left frac \int_^x \sqrt \, dt\right + c_2 Q^(x) \exp\left \frac \int_^x\sqrt \, dt\right
Higher-order terms can be obtained by looking at equations for higher powers of . Explicitly,
2S_^ S_^ + S^_ + \sum_^S^_ S^_ = 0
for .
Precision of the asymptotic series
The asymptotic series for is usually a divergent series
In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit.
If a series converges, the individual terms of the series mus ... \epsilon^2 \frac = Q(x) y,
with an analytic function, the value n_\max and the magnitude of the last term can be estimated as follows:
n_\max \approx 2\epsilon^ \left, \int_^ \sqrt\,dz \ ,
\delta^S_(x_0) \approx \sqrt \exp n_\max
where x_0 is the point at which y(x_0) needs to be evaluated and x_ is the (complex) turning point where Q(x_) = 0 , closest to x = x_0 .
The number can be interpreted as the number of oscillations between x_0 and the closest turning point.
If \epsilon^Q(x) is a slowly changing function,
\epsilon\left, \frac \ \ll Q^2 , ^
the number will be large, and the minimum error of the asymptotic series will be exponentially small.
Application in non relativistic quantum mechanics
Schrödinger equation
The Schrödinger equation is a partial differential equation that governs the wave function of a non-relativistic quantum-mechanical system. Its discovery was a significant landmark in the development of quantum mechanics. It is named after E ... -\frac \frac \Psi(x) + V(x) \Psi(x) = E \Psi(x),
which can be rewritten as
\frac \Psi(x) = \frac \left( V(x) - E \right) \Psi(x).
Approximation away from the turning points
The wavefunction can be rewritten as the exponential of another function (closely related to the action
Action may refer to:
* Action (philosophy), something which is done by a person
* Action principles the heart of fundamental physics
* Action (narrative), a literary mode
* Action fiction, a type of genre fiction
* Action game, a genre of video gam ... \Psi(\mathbf x) = e^,
so that its substitution in Schrödinger's equation gives:
i\hbar \nabla^2 S(\mathbf x) - (\nabla S(\mathbf x))^2 = 2m \left( V(\mathbf x) - E \right),
Next, the semiclassical approximation is used. This means that each function is expanded as a power series
In mathematics, a power series (in one variable) is an infinite series of the form
\sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots
where ''a_n'' represents the coefficient of the ''n''th term and ''c'' is a co ... S = S_0 + \hbar S_1 + \hbar^2 S_2 + \cdots
Substituting in the equation, and only retaining terms up to first order in , we get:
(\nabla S_0+\hbar \nabla S_1)^2-i\hbar(\nabla^2 S_0) = 2m(E-V(\mathbf x))
which gives the following two relations:
\begin
(\nabla S_0)^2= 2m (E-V(\mathbf x)) = (p(\mathbf x))^2\\
2\nabla S_0 \cdot \nabla S_1 - i \nabla^2 S_0 = 0
\end
which can be solved for 1D systems, first equation resulting in:S_0(x) = \pm \int \sqrt \,dx=\pm\int p(x) \,dx and the second equation computed for the possible values of the above, is generally expressed as:\Psi(x) \approx C_+ \frac + C_- \frac
Thus, the resulting wavefunction in first order WKB approximation is presented as,V(x) < E the integrand in the exponent is imaginary and the approximate wave function is oscillatory. In the classically forbidden region V(x) > E , the solutions are growing or decaying. It is evident in the denominator that both of these approximate solutions become singular near the classical turning points, where , and cannot be valid. (The turning points are the points where the classical particle changes direction.)
Hence, when E > V(x) , the wavefunction can be chosen to be expressed as:\Psi(x') \approx C \frac + D \frac and for V(x) > E ,\Psi(x') \approx \frac + \frac . The integration in this solution is computed between the classical turning point and the arbitrary position x'.
Validity of WKB solutions
From the condition:(S_0'(x))^2-(p(x))^2 + \hbar (2 S_0'(x)S_1'(x)-iS_0''(x)) = 0
It follows that: \hbar\mid 2 S_0'(x)S_1'(x)\mid+\hbar \mid i S_0''(x)\mid \ll \mid(S_0'(x))^2\mid +\mid (p(x))^2\mid
For which the following two inequalities are equivalent since the terms in either side are equivalent, as used in the WKB approximation:
\begin
\hbar \mid S_0''(x)\mid \ll \mid(S_0'(x))^2\mid\\
2\hbar \mid S_0'S_1' \mid \ll \mid(p'(x))^2\mid
\end
The first inequality can be used to show the following:
\begin
\hbar \mid S_0''(x)\mid \ll \mid(p(x))\mid^2\\
\frac\frac\left, \frac\ \ll , p(x), ^2\\
\lambda \left, \frac\ \ll \frac\\
\end
where , S_0'(x), = , p(x), is used and \lambda(x) is the local de Broglie wavelength
Matter waves are a central part of the theory of quantum mechanics, being half of wave–particle duality. At all scales where measurements have been practical, matter exhibits wave-like behavior. For example, a beam of electrons can be diffract ... E-V(x) or that of the momentum p(x) , over the wavelength \lambda , being much smaller than 1 .
Similarly it can be shown that \lambda(x) also has restrictions based on underlying assumptions for the WKB approximation that:\left, \frac\ \ll 1 which implies that the de Broglie wavelength
Matter waves are a central part of the theory of quantum mechanics, being half of wave–particle duality. At all scales where measurements have been practical, matter exhibits wave-like behavior. For example, a beam of electrons can be diffract ... Behavior near the turning points
We now consider the behavior of the wave function near the turning points. For this, we need a different method. Near the first turning points, , the term \frac\left(V(x)-E\right) can be expanded in a power series,
\frac\left(V(x)-E\right) = U_1 \cdot (x - x_1) + U_2 \cdot (x - x_1)^2 + \cdots\;.
To first order, one finds
\frac \Psi(x) = U_1 \cdot (x - x_1) \cdot \Psi(x).
This differential equation is known as the Airy equation
In the physical sciences, the Airy function (or Airy function of the first kind) is a special function named after the British astronomer George Biddell Airy (1801–1892). The function Ai(''x'') and the related function Bi(''x''), are linearly ... Airy function
In the physical sciences, the Airy function (or Airy function of the first kind) is a special function named after the British astronomer George Biddell Airy (1801–1892). The function Ai(''x'') and the related function Bi(''x''), are Linear in ... \Psi(x) = C_A \operatorname\left( \sqrt
Although for any fixed value of \hbar , the wave function is bounded near the turning points, the wave function will be peaked there, as can be seen in the images above. As \hbar gets smaller, the height of the wave function at the turning points grows. It also follows from this approximation that:
\frac\int p(x) dx = \sqrt \int \sqrt\, dx = \frac 2 3 (\sqrt
Connection conditions
It now remains to construct a global (approximate) solution to the Schrödinger equation. For the wave function to be square-integrable, we must take only the exponentially decaying solution in the two classically forbidden regions. These must then "connect" properly through the turning points to the classically allowed region. For most values of , this matching procedure will not work: The function obtained by connecting the solution near +\infty to the classically allowed region will not agree with the function obtained by connecting the solution near -\infty to the classically allowed region. The requirement that the two functions agree imposes a condition on the energy , which will give an approximation to the exact quantum energy levels.x=x_1
and the second turning point, where potential is increasing over x, occur at x=x_2
. Given that we expect wavefunctions to be of the following form, we can calculate their coefficients by connecting the different regions using Airy and Bairy functions.
\begin
\Psi_ (x) \approx A \frac + B \frac \\
\Psi_(x) \approx C \frac + D \frac\\
\end
First classical turning point
For U_1 < 0 ie. decreasing potential condition or x=x_1
in the given example shown by the figure, we require the exponential function to decay for negative values of x so that wavefunction for it to go to zero. Considering Bairy functions to be the required connection formula, we get:\begin
\operatorname(u) \rightarrow -\frac\frac \sin \quad \textrm \quad u \rightarrow -\infty\\
\operatorname(u) \rightarrow \frac\frac e^ \quad \textrm \quad u \rightarrow +\infty \\
\end
We cannot use Airy function since it gives growing exponential behaviour for negative x. When compared to WKB solutions and matching their behaviours at \pm \infty , we conclude:
A=-D=N ,
B=C=0 and \alpha = \frac \pi 4 .
Thus, letting some normalization constant be N , the wavefunction is given for increasing potential (with x) as:\Psi_(x) = \begin
-\frac\exp & \text x < x_1\\
\frac \sin & \text x_2 > x > x_1 \\
\end
Second classical turning point
For U_1 > 0 ie. increasing potential condition or x=x_2
in the given example shown by the figure, we require the exponential function to decay for positive values of x so that wavefunction for it to go to zero. Considering Airy functions to be the required connection formula, we get:\begin
\operatorname (u)\rightarrow \frac\frac e^ \quad \textrm \quad u \rightarrow + \infty \\
\operatorname(u) \rightarrow \frac\frac \cos \quad \textrm \quad u \rightarrow -\infty\\
\end
We cannot use Bairy function since it gives growing exponential behaviour for positive x. When compared to WKB solutions and matching their behaviours at \pm \infty , we conclude:
2B=C=N' ,
D=A=0 and \alpha = \frac \pi 4 .
Thus, letting some normalization constant be N' , the wavefunction is given for increasing potential (with x) as:\Psi_(x) = \begin
\frac \cos & \text x_1 < x < x_2 \\
\frac\exp & \text x > x_2\\
\end
Common oscillating wavefunction
Matching the two solutions for region x_1, it is required that the difference between the angles in these functions is \pi(n+1/2) where the \frac \pi 2 phase difference accounts for changing cosine to sine for the wavefunction and n \pi difference since negation of the function can occur by letting N= (-1)^n N' . Thus:
\int_^ \sqrt\,dx = (n+1/2)\pi \hbar ,
Where ''n'' is a non-negative integer. This condition can also be rewritten as saying that:
::The area enclosed by the classical energy curve is 2\pi\hbar(n+1/2) .
Either way, the condition on the energy is a version of the Bohr–Sommerfeld quantization condition, with a " Maslov correction " equal to 1/2.
It is possible to show that after piecing together the approximations in the various regions, one obtains a good approximation to the actual eigenfunction. In particular, the Maslov-corrected Bohr–Sommerfeld energies are good approximations to the actual eigenvalues of the Schrödinger operator. Specifically, the error in the energies is small compared to the typical spacing of the quantum energy levels. Thus, although the "old quantum theory" of Bohr and Sommerfeld was ultimately replaced by the Schrödinger equation, some vestige of that theory remains, as an approximation to the eigenvalues of the appropriate Schrödinger operator.
General connection conditions
Thus, from the two cases the connection formula is obtained at a classical turning point, x=a
: \frac \sin \Longrightarrow - \frac\exp
and:
\frac \cos \Longleftarrow \frac\exp
The WKB wavefunction at the classical turning point away from it is approximated by oscillatory sine or cosine function in the classically allowed region, represented in the left and growing or decaying exponentials in the forbidden region, represented in the right. The implication follows due to the dominance of growing exponential compared to decaying exponential. Thus, the solutions of oscillating or exponential part of wavefunctions can imply the form of wavefunction on the other region of potential as well as at the associated turning point.
Probability density
One can then compute the probability density associated to the approximate wave function. The probability that the quantum particle will be found in the classically forbidden region is small. In the classically allowed region, meanwhile, the probability the quantum particle will be found in a given interval is approximately the ''fraction of time the classical particle spends in that interval'' over one period of motion. Since the classical particle's velocity goes to zero at the turning points, it spends more time near the turning points than in other classically allowed regions. This observation accounts for the peak in the wave function (and its probability density) near the turning points.
Applications of the WKB method to Schrödinger equations with a large variety of potentials and comparison with perturbation methods and path integrals are treated in Müller-Kirsten.
Examples in quantum mechanics
Although WKB potential only applies to smoothly varying potentials, Bound states for 1 rigid wall
The potential of such systems can be given in the form:
V(x) = \begin
V(x) & \text x \geq x_1\\
\infty & \text x < x_1 \\
\end
where x_1 < x_2 .
Finding wavefunction in bound region, ie. within classical turning points x_1 and x_2 , by considering approximations far from x_1 and x_2 respectively we have two solutions:
\Psi_(x) =
\frac\sin
\Psi_(x) =
\frac\cos
Since wavefunction must vanish near x_1 , we conclude \alpha = 0 . For airy functions near x_2 , we require \beta = - \frac \pi 4 . We require that angles within these functions have a phase difference \pi(n+1/2) where the \frac \pi 2 phase difference accounts for changing sine to cosine and n \pi allowing B= (-1)^n A .
\frac 1 \hbar \int_^ , p(x), dx = \pi \left(n + \frac 3 4\right) Where ''n'' is a non-negative integer.\pi(n-1/4) if n was only allowed to non-zero natural numbers.
Thus we conclude that, for n = 1,2,3,\cdots \int_^ \sqrt\,dx = \left(n-\frac 1 4\right)\pi \hbar In 3 dimensions with spherically symmetry, the same condition holds where the position x is replaced by radial distance r, due to its similarity with this problem.
Bound states within 2 rigid wall
The potential of such systems can be given in the form:
V(x) = \begin
\infty & \text x > x_2 \\
V(x) & \text x_2 \geq x \geq x_1\\
\infty & \text x < x_1 \\
\end
where x_1 < x_2 .
For E \geq V(x) between x_1 and x_2 which are thus the classical turning points, by considering approximations far from x_1 and x_2 respectively we have two solutions:
\Psi_(x) =
\frac\sin
\Psi_(x) =
\frac\sin
Since wavefunctions must vanish at x_1 and x_2 . Here, the phase difference only needs to account for n \pi which allows B= (-1)^n A . Hence the condition becomes:
\int_^ \sqrt\,dx = n\pi \hbar where n = 1,2,3,\cdots but not equal to zero since it makes the wavefunction zero everywhere. Quantum bouncing ball
Consider the following potential a bouncing ball is subjected to:
V(x) = \begin
mgx & \text x \geq 0\\
\infty & \text x < 0 \\
\end
The wavefunction solutions of the above can be solved using the WKB method by considering only odd parity solutions of the alternative potential V(x) = mg, x, . The classical turning points are identified x_1 = - and x_2 = . Thus applying the quantization condition obtained in WKB:
\int_^ \sqrt\,dx = (n_+1/2)\pi \hbar
Letting n_=2n-1 where n = 1,2,3,\cdots , solving for E with given V(x) = mg, x, , we get the quantum mechanical energy of a bouncing ball:E = (mg^2\hbar^2)^.
This result is also consistent with the use of equation from bound state of one rigid wall without needing to consider an alternative potential.
Quantum Tunneling
The potential of such systems can be given in the form:
V(x) = \begin
0 & \text x < x_1 \\
V(x) & \text x_2 \geq x \geq x_1\\
0 & \text x > x_2 \\
\end
where x_1 < x_2 .
Its solutions for an incident wave is given as
\psi(x) = \begin
A \exp( ) + B \exp() & \text x < x_1 \\
\frac\exp & \text x_2 \geq x \geq x_1\\
D \exp( ) & \text x > x_2 \\
\end
where the wavefunction in the classically forbidden region is the WKB approximation but neglecting the growing exponential. This is a fair assumption for wide potential barriers through which the wavefunction is not expected to grow to high magnitudes.
By the requirement of continuity of wavefunction and its derivatives, the following relation can be shown:\frac = \frac \frac\exp\left(-\frac 2 \hbar \int_^ , p(x'), dx'\right)
where a_1 = , p(x_1), and a_2 = , p(x_2), .
Using \mathbf J(\mathbf x,t) = \frac(\psi^* \nabla\psi-\psi\nabla\psi^*) we express the values without signs as:
J_ = \frac(\frac, A, ^2)
J_ = \frac(\frac, B, ^2)
J_ = \frac(\frac, D, ^2)
Thus, the transmission coefficient
The transmission coefficient is used in physics and electrical engineering when wave propagation in a medium containing discontinuities is considered. A transmission coefficient describes the amplitude, intensity, or total power of a transmitt ... T = \frac = \frac \frac\exp\left(-\frac 2 \hbar \int_^ , p(x'), dx'\right)
where p(x) = \sqrt , a_1 = , p(x_1), and a_2 = , p(x_2), . The result can be stated as T \sim ~ e^ where \gamma = \int_^ , p(x'), dx' . See also
* Airy function
In the physical sciences, the Airy function (or Airy function of the first kind) is a special function named after the British astronomer George Biddell Airy (1801–1892). The function Ai(''x'') and the related function Bi(''x''), are Linear in ... Einstein–Brillouin–Keller method
The Einstein–Brillouin–Keller (EBK) method is a semiclassical technique (named after Albert Einstein, Léon Brillouin, and Joseph B. Keller) used to compute eigenvalues in quantum-mechanical systems. EBK quantization is an improvement from ... Field electron emission
Field electron emission, also known as field-induced electron emission, field emission (FE) and electron field emission, is the emission of electrons from a material placed in an electrostatic field. The most common context is field emission from ... Instanton
An instanton (or pseudoparticle) is a notion appearing in theoretical and mathematical physics. An instanton is a classical solution to equations of motion with a finite, non-zero action, either in quantum mechanics or in quantum field theory. M ... Langer correction
* Maslov index In mathematics, the Lagrangian Grassmannian is the smooth manifold of Lagrangian subspaces of a real symplectic vector space ''V''. Its dimension is ''n''(''n'' + 1) (where the dimension of ''V'' is ''2n''). It may be identified with the homogeneous ... Method of dominant balance
* Method of matched asymptotic expansions
In mathematics, the method of matched asymptotic expansions is a common approach to finding an accurate approximation to the solution to an equation, or simultaneous equations, system of equations. It is particularly used when solving singular pert ... Method of steepest descent
In mathematics, the method of steepest descent or saddle-point method is an extension of Laplace's method for approximating an integral, where one deforms a contour integral in the complex plane to pass near a stationary point (saddle point), in ... Old quantum theory
The old quantum theory is a collection of results from the years 1900–1925, which predate modern quantum mechanics. The theory was never complete or self-consistent, but was instead a set of heuristic corrections to classical mechanics. The th ... Perturbation methods
In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middle ... Quantum tunneling
In physics, a quantum (: quanta) is the minimum amount of any physical entity (physical property) involved in an interaction. The fundamental notion that a property can be "quantized" is referred to as "the hypothesis of quantization". This me ... Slowly varying envelope approximation
* Supersymmetric WKB approximation
References
Further reading
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External links
* {{cite web, first=Richard , last=Fitzpatrick , url=http://farside.ph.utexas.edu/teaching/jk1/lectures/node70.html , title= The W.K.B. Approximation, year=2002 (An application of the WKB approximation to the scattering of radio waves from the ionosphere.)
Approximations
Asymptotic analysis
Mathematical physics
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