WKB approximation: Difference between revisions

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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 methodtechnique for finding approximate solutions to [[Linear differential equation|linear differential equations]] with spatially varying coefficients. It is typically used for a [[Semiclassical physics|semiclassical]] calculation in [[quantum mechanics]] in which the wavefunction[[wave function]] 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.<ref name=Wentzel-1926/><ref name=Kramers-1926/><ref name=Brillouin-1926/><ref>{{harvnb|Hall|2013}} Section 15.1 </ref> In 1923,<ref name=Jefferys-1924/> 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.<ref>{{cite book |first=Robert Balson |last=Dingle |title=Asymptotic Expansions: Their Derivation and Interpretation |publisher=Academic Press |year=1973 |isbn=0-12-216550-0 }}</ref>

Earlier appearances of essentially equivalent methods are: [[Francesco Carlini]] in 1817,<ref name=Carlini-1817/> [[Joseph Liouville]] in 1837,<ref name=Liouville/> [[George Green (mathematician)|George Green]] in 1837,<ref name=Green-1837/> [[Lord Rayleigh]] in 1912<ref name=Rayleigh-1912/> and [[Richard Gans]] in 1915.<ref name=Gans-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.<ref>{{cite book

| title = Atmosphere-ocean dynamics

| author = Adrian E. Gill

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}}</ref><ref>

{{Cite book

| author1=KevorkianHolmes, JM.

| title=Introduction to Perturbation Methods, 2nd Ed

| author2=Cole, J. D.

| year=19962013

| title=Multiple scale and singular perturbation methods

| year=1996

| publisher=Springer

| isbn=0978-3871-942024614-55476-2

}}</ref><ref name=":0">{{cite book

| first1=Carl M.

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<math display="block"> \epsilon^2 \frac{d^2 y}{dx^2} = Q(x) y, </math>

where <math>Q(x) \neq 0</math>. Substituting

<math display="block">y(x) = \exp \left[\frac{1}{\delta} \sum_{n=0}^\infty \delta^n S_nS_{n}(x)\right]</math>

results in the equation

<math display="block">\epsilon^2\left[\frac{1}{\delta^2} \left(\sum_{n=0}^\infty \delta^nS_n'nS_{n}^{\prime}\right)^2 + \frac{1}{\delta} \sum_{n=0}^{\infty}\delta^n S_n''S_{n}^{\prime\prime}\right] = Q(x).</math>

To [[leading-order|leading order]] in ''ϵ'' (assuming, for the moment, the series will be asymptotically consistent), the above can be approximated as

<math display="block">\frac{\epsilon^2}{\delta^2} S_0'{S_{0}^{\prime}}^2 + \frac{2\epsilon^2}{\delta} S_0'S_{0}^{\prime} S_1'S_{1}^{\prime} + \frac{\epsilon^2}{\delta} S_0''S_{0}^{\prime\prime} = Q(x).</math>

In the limit {{math|''δ'' → 0}}, the [[Method of dominant balance|dominant balance]] is given by

<math display="block">\frac{\epsilon^2}{\delta^2} S_0'{S_{0}^{\prime}}^2 \sim Q(x).</math>

So {{mvar|δ}} is proportional to ''ϵ''. Setting them equal and comparing powers yields

<math display="block">\epsilon^0: \quad S_0'{S_{0}^{\prime}}^2 = Q(x),</math>

which can be recognized as the [[eikonal equation]], with solution

<math display="block">S_0S_{0}(x) = \pm \int_{x_0}^x \sqrt{Q(x')}\,dx'.</math>

Considering first-order powers of {{mvar|ϵ}} fixes

<math display="block">\epsilon^1: \quad 2 S_0'S_{0}^{\prime} S_1'S_{1}^{\prime} + S_0''S_{0}^{\prime\prime} = 0.</math>

This has the solution

<math display="block">S_1S_{1}(x) = -\frac{1}{4} \ln Q(x) + k_1,</math>

where {{math|''k''₁}} is an arbitrary constant.

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Higher-order terms can be obtained by looking at equations for higher powers of {{mvar|δ}}. Explicitly,

<math display="block"> 2S_0'2S_{0}^{\prime} S_n'S_{n}^{\prime} + S''^{\prime\prime}_{n-1} + \sum_{j=1}^{n-1}S'_j^{\prime}_{j} S'^{\prime}_{n-j} = 0</math>

for {{math|''n'' ≥ 2}}.

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For the equation

<math display="block"> \epsilon^2 \frac{d^2 y}{dx^2} = Q(x) y, </math>

with {{math|''Q''(''x'') <0}} an [[analytic function]], the value <math>n_\max</math> and the magnitude of the last term can be estimated as follows:<ref>{{cite journal| last=Winitzki |first=S. |year=2005 |arxiv=gr-qc/0510001 |title=Cosmological particle production and the precision of the WKB approximation |journal=Phys. Rev. D |volume=72 |issue=10 |pages=104011, 14&nbsp;pp |doi=10.1103/PhysRevD.72.104011 |bibcode = 2005PhRvD..72j4011W |s2cid=119152049 }}</ref>

<math display="block">n_\max \approx 2\epsilon^{-1} \left| \int_{x_0}^{x_{\ast}} \sqrt{-Q(z)}\,dz \right| , </math>

<math display="block">\delta^{n_\max}S_{n_\max}(x_0) \approx \sqrt{\frac{2\pi}{n_\max}} \exp[-n_\max], </math>

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the number {{math|''n''_max}} will be large, and the minimum error of the asymptotic series will be exponentially small.

== Application in non -relativistic quantum mechanics ==

[[File:WKB approximation example.svg|thumb|WKB approximation to the indicated potential. Vertical lines show the turning points]]

[[File:WKB approximation to probability density.svg|thumb|Probability density for the approximate wave function. Vertical lines show the turning points]]

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2\nabla S_0 \cdot \nabla S_1 - i \nabla^2 S_0 = 0

\end{align}</math>

which can be solved for 1D systems, first equation resulting in:<math display="block">S_0(x) = \pm \int \sqrt{ \frac{2m}{\hbar^2} \left( E - V(x)\right) } \,dx=\pm\int p(x) \,dx </math>and the second equation computed for the possible values of the above, is generally expressed as:<math display="block">\Psi(x) \approx C_+ \frac{ e^{+ \frac i \hbar \int p(x)\,dx} }{\sqrt{|p(x)| }} + C_- \frac{ e^{- \frac i \hbar \int p(x)\,dx} }{\sqrt{|p(x)| }} </math>

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Hence, when <math>E > V(x)</math>, the wavefunction can be chosen to be expressed as:<math display="block">\Psi(x') \approx C \frac{\cos{(\frac 1 \hbar \int |p(x)|\,dx} + \alpha) }{\sqrt{|p(x)| }} + D \frac{ \sin{(- \frac 1 \hbar \int |p(x)|\,dx} +\alpha)}{\sqrt{|p(x)| }} </math>and for <math>V(x) > E</math>,<math display="block">\Psi(x') \approx \frac{ C_{+} e^{+- \frac{i1}{\hbar} \int |p(x)|\,dx}}{\sqrt{|p(x)|}} + \frac{ C_{-} e^{-+ \frac{i1}{\hbar} \int |p(x)|\,dx} }{ \sqrt{|p(x)|} } . </math>The integration in this solution is computed between the classical turning point and the arbitrary position x'.

=== Validity of WKB solutions ===

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==== First classical turning point ====

For <math>U_1 < 0</math> ie. decreasing potential condition or <math>x=x_1

</math> 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:<ref name=":3">{{Cite journal |last1=Ramkarthik |first1=M. S. |last2=Pereira |first2=Elizabeth Louis |date=2021-06-01 |title=Airy Functions Demystified — II |url=https://doi.org/10.1007/s12045-021-1179-z |journal=Resonance |language=en |volume=26 |issue=6 |pages=757–789 |doi=10.1007/s12045-021-1179-z |issn=0973-712X|url-access=subscription }}</ref>

<math display="block">\begin{align}

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Either way, the condition on the energy is a version of the [[Bohr–Sommerfeld quantization]] condition, with a "[[Lagrangian Grassmannian#Maslov index|Maslov correction]]" equal to 1/2.<ref>{{harvnb|Hall|2013}} Section 15.2</ref>

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.<ref>{{harvnb|Hall|2013}} Theorem 15.8</ref> 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 ====

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<math display="block">E = {\left(3\left(n-\frac 1 4\right)\pi\right)^{\frac 2 3} \over 2}(mg^2\hbar^2)^{\frac 1 3}. </math>

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 ===

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By the requirement of continuity of wavefunction and its derivatives, the following relation can be shown:<math display="block">\frac {|ED|^2} {|A|^2} = \frac{4}{(1+{a_1^2}/{p_0^2} )} \frac{a_1}{a_2}\exp\left(-\frac 2 \hbar \int_{x_1}^{x_2} |p(x')| dx'\right) </math>

where <math>a_1 = |p(x_1)|</math> and <math>a_2 = |p(x_2)| </math>.

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<math display="inline">J_{\text{ref.}} = \frac{\hbar}{2m}(\frac{2p_0}{\hbar}|B|^2) </math>

<math display="inline">J_{\text{trans.}} = \frac{\hbar}{2m}(\frac{2p_0}{\hbar}|ED|^2) </math>

Thus, the [[transmission coefficient]] is found to be:

<math display="block">T = \frac {|ED|^2} {|A|^2} = \frac{4}{(1+{a_1^2}/{p_0^2} )} \frac{a_1}{a_2}\exp\left(-\frac 2 \hbar \int_{x_1}^{x_2} |p(x')| dx'\right) </math>

where <math display="inline">p(x) = \sqrt {2m( E - V(x))} </math>, <math>a_1 = |p(x_1)|</math> and <math>a_2 = |p(x_2)| </math>. The result can be stated as <math display="inline">T \sim ~ e^{-2\gamma} </math> where <math display="inline">\gamma = \int_{x_1}^{x_2} |p(x')| dx' </math>.<ref name=":1" />

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==References==

{{Reflist}}|refs=

*<ref name=Carlini-1817>{{cite book | author=Carlini, Francesco | year=1817 | title=Ricerche sulla convergenza della serie che serva alla soluzione del problema di Keplero | publisher=Milano | author-link=Francesco Carlini }}</ref>

*<ref name=Liouville>{{cite journal | author=Liouville, Joseph | year=1837 | title=Sur le développement des fonctions et séries..| journal=Journal de Mathématiques Pures et Appliquées | volume=1 | pages=16–35 | author-link=Joseph Liouville }}</ref>

*<ref name=Green-1837>{{cite journal | author=Green, George | year=1837 | title=On the motion of waves in a variable canal of small depth and width | journal=Transactions of the Cambridge Philosophical Society | volume=6 | pages=457–462 | author-link=George Green (mathematician) }}</ref>

*<ref name=Rayleigh-1912>{{cite journal | author=Rayleigh, Lord (John William Strutt) | year=1912 | title=On the propagation of waves through a stratified medium, with special reference to the question of reflection | journal=[[Proceedings of the Royal Society A]] | volume=86 | pages=207–226 | doi=10.1098/rspa.1912.0014 |bibcode = 1912RSPSA..86..207R | issue=586 | author-link=Lord Rayleigh | doi-access=free }}</ref>

*<ref name=Gans-1915>{{cite journal | author=Gans, Richard | year=1915 | title=Fortplantzung des Lichts durch ein inhomogenes Medium | journal=Annalen der Physik | volume=47 | issue=14 | pages=709–736 | doi = 10.1002/andp.19153521402 |bibcode = 1915AnP...352..709G | url=https://zenodo.org/record/1447303 | author-link=Richard Gans }}</ref>

*<ref name=Jefferys-1924>{{cite journal | author=Jeffreys, Harold | year=1924 | title=On certain approximate solutions of linear differential equations of the second order | journal=Proceedings of the London Mathematical Society | volume=23 | pages=428–436 | doi=10.1112/plms/s2-23.1.428 | author-link=Harold Jeffreys }}</ref>

*<ref name=Brillouin-1926>{{cite journal | author=Brillouin, Léon | year=1926 | title=La mécanique ondulatoire de Schrödinger: une méthode générale de resolution par approximations successives | journal=Comptes Rendus de l'Académie des Sciences | volume=183 | pages=24–26 | author-link=Léon Brillouin }}</ref>

*<ref name=Kramers-1926>{{cite journal | author=Kramers, Hendrik A. | year=1926 | title=Wellenmechanik und halbzahlige Quantisierung | journal=Zeitschrift für Physik | volume=39 |pages=828–840 | doi=10.1007/BF01451751 |bibcode = 1926ZPhy...39..828K | issue=10–11 | s2cid=122955156 | author-link=Hendrik Anthony Kramers }}</ref>

*<ref name=Wentzel-1926>{{cite journal | author=Wentzel, Gregor | year=1926 | title=Eine Verallgemeinerung der Quantenbedingungen für die Zwecke der Wellenmechanik | journal=Zeitschrift für Physik | volume=38 | pages=518–529 | doi=10.1007/BF01397171 |bibcode = 1926ZPhy...38..518W | issue=6–7 | s2cid=120096571 | author-link=Gregor Wentzel }}</ref>

}}

===ModernFurther referencesreading===

*{{cite book | author1-link = Carl M. Bender | author1 = Bender, Carl | author2-link = Steven A. Orszag | author2 = Orszag, Steven | title=Advanced Mathematical Methods for Scientists and Engineers | publisher=McGraw-Hill | year=1978 | isbn=0-07-004452-X}}

*{{cite book | author=Child, M. S. | title=Semiclassical mechanics with molecular applications | year=1991 | publisher = Clarendon Press | ___location=Oxford | isbn=0-19-855654-3}}

* {{cite book| last1=Fröman|first1=N. |last2= Fröman |first2= P.-O.| title=JWKB Approximation: Contributions to the Theory | publisher=North-Holland |___location = Amsterdam | year=1965}}

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*{{cite book | author=Olver, Frank William John |author-link=Frank William John Olver | title=Asymptotics and Special Functions | url=https://archive.org/details/asymptoticsspeci0000olve | url-access=registration | publisher=Academic Press | year=1974 | isbn=0-12-525850-X}}

*{{cite book | author=Razavy, Mohsen | title=Quantum Theory of Tunneling | url=https://archive.org/details/quantumtheoryoft0000raza | url-access=registration | publisher=World Scientific | year=2003 | isbn=981-238-019-1}}

*{{cite book | author=Sakurai, J. J. | title=[[Modern Quantum Mechanics]] | publisher=Addison-Wesley |year=1993 |isbn=0-201-53929-2}}

===Historical references===

*{{cite book | author=Carlini, Francesco | year=1817 | title=Ricerche sulla convergenza della serie che serva alla soluzione del problema di Keplero | publisher=Milano | author-link=Francesco Carlini }}

*{{cite journal | author=Liouville, Joseph | year=1837 | title=Sur le développement des fonctions et séries..| journal=Journal de Mathématiques Pures et Appliquées | volume=1 | pages=16–35 | author-link=Joseph Liouville }}

*{{cite journal | author=Green, George | year=1837 | title=On the motion of waves in a variable canal of small depth and width | journal=Transactions of the Cambridge Philosophical Society | volume=6 | pages=457–462 | author-link=George Green (mathematician) }}

*{{cite journal | author=Rayleigh, Lord (John William Strutt) | year=1912 | title=On the propagation of waves through a stratified medium, with special reference to the question of reflection | journal=[[Proceedings of the Royal Society A]] | volume=86 | pages=207–226 | doi=10.1098/rspa.1912.0014 |bibcode = 1912RSPSA..86..207R | issue=586 | author-link=Lord Rayleigh | doi-access=free }}

*{{cite journal | author=Gans, Richard | year=1915 | title=Fortplantzung des Lichts durch ein inhomogenes Medium | journal=Annalen der Physik | volume=47 | issue=14 | pages=709–736 | doi = 10.1002/andp.19153521402 |bibcode = 1915AnP...352..709G | url=https://zenodo.org/record/1447303 | author-link=Richard Gans }}

*{{cite journal | author=Jeffreys, Harold | year=1924 | title=On certain approximate solutions of linear differential equations of the second order | journal=Proceedings of the London Mathematical Society | volume=23 | pages=428–436 | doi=10.1112/plms/s2-23.1.428 | author-link=Harold Jeffreys }}

*{{cite journal | author=Brillouin, Léon | year=1926 | title=La mécanique ondulatoire de Schrödinger: une méthode générale de resolution par approximations successives | journal=Comptes Rendus de l'Académie des Sciences | volume=183 | pages=24–26 | author-link=Léon Brillouin }}

*{{cite journal | author=Kramers, Hendrik A. | year=1926 | title=Wellenmechanik und halbzahlige Quantisierung | journal=Zeitschrift für Physik | volume=39 |pages=828–840 | doi=10.1007/BF01451751 |bibcode = 1926ZPhy...39..828K | issue=10–11 | s2cid=122955156 | author-link=Hendrik Anthony Kramers }}

*{{cite journal | author=Wentzel, Gregor | year=1926 | title=Eine Verallgemeinerung der Quantenbedingungen für die Zwecke der Wellenmechanik | journal=Zeitschrift für Physik | volume=38 | pages=518–529 | doi=10.1007/BF01397171 |bibcode = 1926ZPhy...38..518W | issue=6–7 | s2cid=120096571 | author-link=Gregor Wentzel }}

==External links==