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{{short description|Solution method for linear differential equations}}{{Redirect2|WKB|WKBJ|other uses|WKB (disambiguation)|the television station in Live Oak, Florida|WKBJ-LD}}In mathematical physics, the WKB approximation or WKB method is a method for finding approximate solutions to linear 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 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, Hendrik Anthony Kramers, and Léon Brillouin, who all developed it in 1926.{{harvnb|Hall|2013}} Section 15.1 In 1923, mathematician Harold Jeffreys had developed a general method of approximating solutions to linear, second-order differential equations, a class that includes the Schrödinger equation. The Schrödinger equation itself was not developed until two years later, and 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. An authoritative discussion and critical survey has been given by Robert B. Dingle.BOOK, Robert Balson, Dingle, Asymptotic Expansions: Their Derivation and Interpretation, Academic Press, 1973, 0-12-216550-0, Earlier appearances of essentially equivalent methods are: Francesco Carlini in 1817, Joseph Liouville in 1837, George Green in 1837, Lord Rayleigh in 1912 and Richard 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.BOOK BOOK 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.

Formulation

Generally, WKB theory is a method for approximating the solution of a differential equation whose highest derivative is multiplied by a small parameter {{mvar|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 {{math|δ → 0}}. The asymptotic scaling of {{mvar|δ}} in terms of {{mvar|ε}} will be determined by the equation – see the example below.Substituting the above ansatz into the differential equation and cancelling out the exponential terms allows one to solve for an arbitrary number of terms {{math|Sn(x)}} in the expansion.WKB theory is a special case of multiple scale analysis.BOOK BOOK

An example

This example comes from the text of Carl M. Bender and Steven Orszag. Consider the second-order homogeneous linear differential equation where Q(x) neq 0. Substitutingy(x) = exp left[frac{1}{delta} sum_{n=0}^infty delta^n S_n(x)right]results in the equationepsilon^2left[frac{1}{delta^2} left(sum_{n=0}^infty delta^nS_n'right)^2 + frac{1}{delta} sum_{n=0}^{infty}delta^n S_n''right] = Q(x).To leading order in ϵ (assuming, for the moment, the series will be asymptotically consistent), the above can be approximated asfrac{epsilon^2}{delta^2} S_0'^2 + frac{2epsilon^2}{delta} S_0' S_1' + frac{epsilon^2}{delta} S_0'' = Q(x).In the limit {{math|δ → 0}}, the dominant balance is given byfrac{epsilon^2}{delta^2} S_0'^2 sim Q(x).So {{mvar|δ}} is proportional to ϵ. Setting them equal and comparing powers yieldsepsilon^0: quad S_0'^2 = Q(x),which can be recognized as the eikonal equation, with solutionS_0(x) = pm int_{x_0}^x sqrt{Q(x')},dx'.Considering first-order powers of {{mvar|ϵ}} fixesepsilon^1: quad 2 S_0' S_1' + S_0'' = 0.This has the solutionS_1(x) = -frac{1}{4} ln Q(x) + k_1,where {{math|k1}} is an arbitrary constant.We now have a pair of approximations to the system (a pair, because {{math|S0}} can take two signs); the first-order WKB-approximation will be a linear combination of the two:y(x) approx c_1 Q^{-frac{1}{4}}(x) expleft[frac{1}{epsilon} int_{x_0}^x sqrt{Q(t)} , dtright] + c_2 Q^{-frac{1}{4}}(x) expleft[-frac{1}{epsilon} int_{x_0}^xsqrt{Q(t)} , dtright].Higher-order terms can be obtained by looking at equations for higher powers of {{mvar|δ}}. Explicitly, for {{math|n ≥ 2}}.

Precision of the asymptotic series

The asymptotic series for {{math|y(x)}} is usually a divergent series, whose general term {{math|δ'n S'n(x)}} starts to increase after a certain value {{math|1=n = nmax}}. Therefore, the smallest error achieved by the WKB method is at best of the order of the last included term.For the equation with {{math|Q(x) E,Psi(x') approx frac{ C_{+} e^{+ frac{i}{hbar} int |p(x)|,dx}}{sqrt{|p(x)|}} + frac{ C_{-} e^{- frac{i}{hbar} int |p(x)|,dx} }{ sqrt{|p(x)|} } . 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: hbarmid 2 S_0'(x)S_1'(x)mid+hbar mid i S_0''(x)mid ll mid(S_0'(x))^2mid +mid (p(x))^2mid For which the following two inequalities are equivalent since the terms in either side are equivalent, as used in the WKB approximation:begin{align}hbar mid S_0''(x)mid ll mid(S_0'(x))^2mid2hbar mid S_0'S_1' mid ll mid(p'(x))^2midend{align} The first inequality can be used to show the following:begin{align}hbar mid S_0''(x)mid ll mid(p(x))mid^2frac{1}{2}frac{hbar}{|p(x)|}left|frac{dp^2}{dx}right| ll |p(x)|^2lambda left|frac{dV}{dx}right| ll frac{|p|^2}{m}end{align} where |S_0'(x)|= |p(x)| is used and lambda(x) is the local de Broglie wavelength of the wavefunction. The inequality implies that the variation of potential is assumed to be slowly varying.WEB, Zwiebach, Barton, Semiclassical approximation,weblink This condition can also be restated as the fractional change of E-V(x) or that of the momentum p(x) , over the wavelength lambda , being much smaller than 1 .BOOK, Bransden, B. H.,weblink Physics of Atoms and Molecules, Joachain, Charles Jean, 2003, Prentice Hall, 978-0-582-35692-4, 140-141, en, Similarly it can be shown that lambda(x) also has restrictions based on underlying assumptions for the WKB approximation that:left|frac{dlambda}{dx}right| ll 1 which implies that the de Broglie wavelength of the particle is slowly varying.

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, {{math|x1}}, the term frac{2m}{hbar^2}left(V(x)-Eright) can be expanded in a power series,frac{2m}{hbar^2}left(V(x)-Eright) = U_1 cdot (x - x_1) + U_2 cdot (x - x_1)^2 + cdots;.To first order, one findsfrac{d^2}{dx^2} Psi(x) = U_1 cdot (x - x_1) cdot Psi(x).This differential equation is known as the Airy equation, and the solution may be written in terms of Airy functions,{{harvnb|Hall|2013}} Section 15.5Psi(x) = C_A operatorname{Ai}left( sqrt[3]{U_1} cdot (x - x_1) right) + C_B operatorname{Bi}left( sqrt[3]{U_1} cdot (x - x_1) right)= C_A operatorname{Ai}left( u right) + C_B operatorname{Bi}left( u right).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{1}{hbar}int p(x) dx = sqrt{U_1} int sqrt{x-a}, dx = frac 2 3 (sqrt[3]{U_1} (x-a))^{frac 3 2} = frac 2 3 u^{frac 3 2}

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 {{math|E}}, 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 {{math|E}}, which will give an approximation to the exact quantum energy levels.File:WKB approximation example.svg|thumb|WKB approximation to the indicated potential. Vertical lines show the energy level and its intersection with potential shows the turning points with dotted lines. The problem has two classical turning points with U_1 < 0 at x=x_1 .The wavefunction's coefficients can be calculated for a simple problem shown in the figure. Let the first turning point, where the potential is decreasing over x, occur at x=x_1 . 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{align} Psi_{V>E} (x) approx A frac{ e^{frac 2 3 u^frac{3}{2}}}{sqrt[4]{u}} + B frac{ e^{-frac 2 3 u^frac{3}{2}} }{sqrt[4]{u}} Psi_{E>V}(x) approx C frac{cos{(frac 2 3 u^frac{3}{2} - alpha ) } }{sqrt[4]{u} } + D frac{ sin{(frac 2 3 u^frac{3}{2} - alpha)}}{sqrt[4]{u} } end{align}

First classical turning point

For U_1 < 0 ie. decreasing potential condition or x=x_1 begin{align}operatorname{Bi}(u) rightarrow -frac{1}{sqrt pi}frac{1}{sqrt[4]{u}} sin{left(frac 2 3 |u|^{frac 3 2} - frac pi 4right)} quad textrm{where,} quad u rightarrow -inftyoperatorname{Bi}(u) rightarrow frac{1}{sqrt pi}frac{1}{sqrt[4]{u}} e^{frac 2 3 u^{frac 3 2}} quad textrm{where,} quad u rightarrow +infty end{align} 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:B=-D=N ,A=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_{text{WKB}}(x) = begin{cases}-frac{N}{sqrt{|p(x)|}}exp{(-frac 1 hbar int_{x}^{x_1} |p(x)| dx )} & text{if } x < x_1

Second classical turning point

For U_1 > 0 ie. increasing potential condition or x=x_2 begin{align}operatorname{Ai} (u)rightarrow frac{1}{2sqrt pi}frac{1}{sqrt[4]{u}} e^{-frac 2 3 u^{frac 3 2}} quad textrm{where,} quad u rightarrow + infty operatorname{Ai}(u) rightarrow frac{1}{sqrt pi}frac{1}{sqrt[4]{u}} cos{left(frac 2 3 |u|^{frac 3 2} - frac pi 4right)} quad textrm{where,} quad u rightarrow -inftyend{align} 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:2A=C=N' ,D=B=0 and alpha = frac pi 4 . Thus, letting some normalization constant be N' , the wavefunction is given for increasing potential (with x) as:Psi_{text{WKB}}(x) = begin{cases}

Common oscillating wavefunction

Matching the two solutions for region x_1 x_2 V(x) & text{if } x_2 geq x geq x_1 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_{text{WKB}}(x) = frac{A}{sqrt{|p(x)|}}sin{left(frac 1 hbar int_{x}^{x_1} |p(x)| dx right)} Psi_{text{WKB}}(x) = frac{B}{sqrt{|p(x)|}}sin{left(frac 1 hbar int_{x}^{x_2} |p(x)| dx right)} 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_{x_1}^{x_2} sqrt{2m left( E-V(x)right)},dx = npi 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{cases}mgx & text{if } x geq 0 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 = - {E over mg} and x_2 = {E over mg} . Thus applying the quantization condition obtained in WKB:int_{x_1}^{x_2} sqrt{2m left( E-V(x)right)},dx = (n_{text{odd}}+1/2)pi hbarLetting n_{text{odd}}=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:BOOK, Sakurai, Jun John, Modern quantum mechanics, Napolitano, Jim, 2021, Cambridge University Press, 978-1-108-47322-4, 3rd, Cambridge, E = {left(3left(n-frac 1 4right)piright)^{frac 2 3} over 2}(mg^2hbar^2)^{frac 1 3}. 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{cases}V(x) & text{if } x_2 geq x geq x_1 where x_1 < x_2 . It's solutions for an incident wave is given:V(x) = begin{cases}A exp({ i p_0 x over hbar} ) + B exp({- i p_0 x over hbar}) & text{if } x < x_1 D exp({ i p_0 x over hbar} ) & text{if } x > x_2 Where the wavefunction in the classically forbidden region is the WKB approximation but neglecting the growing exponential, which 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 {|E|^2} {|A|^2} = frac{4}{(1+{a_1^2}/{p_0^2} )} frac{a_1}{a_2}expleft(-frac 2 hbar int_{x_1}^{x_2} |p(x')| dx'right) where a_1 = |p(x_1)| and a_2 = |p(x_2)| .Using mathbf J(mathbf x,t) = frac{ihbar}{2m}(psi^* nablapsi-psinablapsi^*) we express the values without signs as:J_{text{inc.}} = frac{hbar}{2m}(frac{2p_0}{hbar}|A|^2) J_{text{ref.}} = frac{hbar}{2m}(frac{2p_0}{hbar}|B|^2) J_{text{trans.}} = frac{hbar}{2m}(frac{2p_0}{hbar}|E|^2) Thus, the transmission coefficient is found to be:T = frac {|E|^2} {|A|^2} = frac{4}{(1+{a_1^2}/{p_0^2} )} frac{a_1}{a_2}expleft(-frac 2 hbar int_{x_1}^{x_2} |p(x')| dx'right) where p(x) = sqrt {2m( E - V(x))} , a_1 = |p(x_1)| and a_2 = |p(x_2)| . The result can be stated as T sim ~ e^{-2gamma} where gamma = int_{x_1}^{x_2} |p(x')| dx' .

See also

{{Div col|colwidth=20em}}

• Airy function
• Einstein–Brillouin–Keller method
• Field electron emission
• Instanton
• Langer correction
• Maslov index
• Method of dominant balance
• Method of matched asymptotic expansions
• Method of steepest descent
• Old quantum theory
• Perturbation methods
• Quantum tunneling
• Slowly varying envelope approximation
• Supersymmetric WKB approximation

{{div col end}}

References

{{Reflist}}

Modern references

• BOOK, Carl M. Bender, Bender, Carl, Steven A. Orszag, Orszag, Steven, Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill, 1978, 0-07-004452-X,
• BOOK, Child, M. S., Semiclassical mechanics with molecular applications, 1991, Clarendon Press, Oxford, 0-19-855654-3,
• BOOK, Fröman, N., Fröman, P.-O., JWKB Approximation: Contributions to the Theory, North-Holland, Amsterdam, 1965,
• BOOK, Griffiths, David J., Introduction to Quantum Mechanics, 2nd, Prentice Hall, 2004, 0-13-111892-7,
• {{citation|first=Brian C.|last=Hall | title=Quantum Theory for Mathematicians | series=Graduate Texts in Mathematics| volume=267 | publisher=Springer|year=2013| isbn=978-1461471158}}
• BOOK, Liboff, Richard L., Introductory Quantum Mechanics, 4th, Addison-Wesley, 2003, 0-8053-8714-5, Liboff, Richard L,
• BOOK, Olver, Frank William John, Frank William John Olver, Asymptotics and Special Functions,weblink registration, Academic Press, 1974, 0-12-525850-X,
• BOOK, Razavy, Mohsen, Quantum Theory of Tunneling,weblink registration, World Scientific, 2003, 981-238-019-1,
• BOOK, Sakurai, J. J., Modern Quantum Mechanics, Addison-Wesley, 1993, 0-201-53929-2,

Historical references

• BOOK, Carlini, Francesco, 1817, Ricerche sulla convergenza della serie che serva alla soluzione del problema di Keplero, Milano, Francesco Carlini,
• JOURNAL, Liouville, Joseph, 1837, Sur le développement des fonctions et séries.., Journal de Mathématiques Pures et Appliquées, 1, 16–35, Joseph Liouville,
• JOURNAL, Green, George, 1837, On the motion of waves in a variable canal of small depth and width, Transactions of the Cambridge Philosophical Society, 6, 457–462, George Green (mathematician),
• JOURNAL, Rayleigh, Lord (John William Strutt), 1912, On the propagation of waves through a stratified medium, with special reference to the question of reflection, Proceedings of the Royal Society A, 86, 207–226, 10.1098/rspa.1912.0014, 1912RSPSA..86..207R, 586, Lord Rayleigh, free,
• JOURNAL, Gans, Richard, 1915, Fortplantzung des Lichts durch ein inhomogenes Medium, Annalen der Physik, 47, 14, 709–736, 10.1002/andp.19153521402, 1915AnP...352..709G,weblink Richard Gans,
• JOURNAL, Jeffreys, Harold, 1924, On certain approximate solutions of linear differential equations of the second order, Proceedings of the London Mathematical Society, 23, 428–436, 10.1112/plms/s2-23.1.428, Harold Jeffreys,
• JOURNAL, Brillouin, Léon, 1926, La mécanique ondulatoire de Schrödinger: une méthode générale de resolution par approximations successives, Comptes Rendus de l'Académie des Sciences, 183, 24–26, Léon Brillouin,
• JOURNAL, Kramers, Hendrik A., 1926, Wellenmechanik und halbzahlige Quantisierung, Zeitschrift für Physik, 39, 828–840, 10.1007/BF01451751, 1926ZPhy...39..828K, 10–11, 122955156, Hendrik Anthony Kramers,
• JOURNAL, Wentzel, Gregor, 1926, Eine Verallgemeinerung der Quantenbedingungen für die Zwecke der Wellenmechanik, Zeitschrift für Physik, 38, 518–529, 10.1007/BF01397171, 1926ZPhy...38..518W, 6–7, 120096571, Gregor Wentzel,

External links

• WEB, Richard, Fitzpatrick,weblink The W.K.B. Approximation, 2002, (An application of the WKB approximation to the scattering of radio waves from the ionosphere.)