In mathematical physics, the WKB approximation or WKB method is a technique 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 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. 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. 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. 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 The WKB method approximates the solution of a differential equation whose highest derivative is multiplied by a small parameter ε {\displaystyle \varepsilon } . For a differential equation ε d n y d x n + a ( x ) d n − 1 y d x n − 1 + ⋯ + k ( x ) d y d x + m ( x ) y = 0 , {\displaystyle \varepsilon {\frac {d^{n}y}{dx^{n}}}+a(x){\frac {d^{n-1}y}{dx^{n-1}}}+\cdots +k(x){\frac {dy}{dx}}+m(x)y=0,} assume a solution of the form of an asymptotic series expansion y ( x ) ∼ exp [ 1 δ ∑ n = 0 ∞ δ n S n ( x ) ] {\displaystyle y(x)\sim \exp \left[{\frac {1}{\delta }}\sum _{n=0}^{\infty }\delta ^{n}S_{n}(x)\right]} in the limit δ → 0 {\displaystyle \delta \rightarrow 0} . The asymptotic scaling of δ {\displaystyle \delta } in terms of ε {\displaystyle \varepsilon } 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 S n ( x ) {\displaystyle S_{n}(x)} in the expansion. WKB theory is a special case of multiple scale analysis.
An example This example comes from the text of Carl M. Bender and Steven Orszag. Consider the second-order homogeneous linear differential equation ε 2 d 2 y d x 2 = Q ( x ) y , {\displaystyle \varepsilon ^{2}{\frac {d^{2}y}{dx^{2}}}=Q(x)\,y,} where Q ( x ) ≠ 0 {\displaystyle Q(x)\neq 0} . Substituting y ( x ) = exp [ 1 δ ∑ n = 0 ∞ δ n S n ( x ) ] {\displaystyle y(x)=\exp \left[{\frac {1}{\delta }}\sum _{n=0}^{\infty }\delta ^{n}S_{n}(x)\right]} results in the equation ε 2 [ 1 δ 2 ( ∑ n = 0 ∞ δ n S n ′ ) 2 + 1 δ ∑ n = 0 ∞ δ n S n ′ ′ ] = Q ( x ) . {\displaystyle \varepsilon ^{2}\left[{\frac {1}{\delta ^{2}}}\left(\sum _{n=0}^{\infty }\delta ^{n}S_{n}^{\prime }\right)^{2}+{\frac {1}{\delta }}\sum _{n=0}^{\infty }\delta ^{n}S_{n}^{\prime \prime }\right]=Q(x).} To leading order in ε {\displaystyle \varepsilon } (assuming, for the moment, the series will be asymptotically consistent), the above can be approximated as ε 2 δ 2 S 0 ′ 2 + 2 ε 2 δ S 0 ′ S 1 ′ + ε 2 δ S 0 ′ ′ = Q ( x ) . {\displaystyle {\frac {\varepsilon ^{2}}{\delta ^{2}}}{S_{0}^{\prime }}^{2}+{\frac {2\varepsilon ^{2}}{\delta }}S_{0}^{\prime }S_{1}^{\prime }+{\frac {\varepsilon ^{2}}{\delta }}S_{0}^{\prime \prime }=Q(x).} In the limit δ → 0 {\displaystyle \delta \rightarrow 0} , the dominant balance is given by ε 2 δ 2 S 0 ′ 2 ∼ Q ( x ) . {\displaystyle {\frac {\varepsilon ^{2}}{\delta ^{2}}}{S_{0}^{\prime }}^{2}\sim Q(x).} So δ {\displaystyle \delta } is proportional to ε {\displaystyle \varepsilon } . Setting them equal and comparing powers yields ε 0 : S 0 ′ 2 = Q ( x ) , {\displaystyle \varepsilon ^{0}:\quad {S_{0}^{\prime }}^{2}=Q(x),} which can be recognized as the eikonal equation, with solution S 0 ( x ) = ± ∫ x 0 x Q ( x ′ ) d x ′ . {\displaystyle S_{0}(x)=\pm \int _{x_{0}}^{x}{\sqrt {Q(x')}}\,dx'.} Considering first-order powers of ε {\displaystyle \varepsilon } fixes ε 1 : 2 S 0 ′ S 1 ′ + S 0 ′ ′ = 0. {\displaystyle \varepsilon ^{1}:\quad 2S_{0}^{\prime }S_{1}^{\prime }+S_{0}^{\prime \prime }=0.} This has the solution S 1 ( x ) = − 1 4 ln Q ( x ) + k 1 , {\displaystyle S_{1}(x)=-{\frac {1}{4}}\ln Q(x)+k_{1},} where k 1 {\displaystyle k_{1}} is an arbitrary constant. We now have a pair of approximations to the system (a pair, because S 0 {\displaystyle S_{0}} can take two signs); the first-order WKB approximation will be a linear combination of the two: y ( x ) ≈ c 1 Q − 1 4 ( x ) exp ( 1 ε ∫ x 0 x Q ( t ) d t ) + c 2 Q − 1 4 ( x ) exp ( − 1 ε ∫ x 0 x Q ( t ) d t ) . {\displaystyle y(x)\approx c_{1}Q^{-{\frac {1}{4}}}(x)\exp \left({\frac {1}{\varepsilon }}\int _{x_{0}}^{x}{\sqrt {Q(t)}}\,dt\right)+c_{2}Q^{-{\frac {1}{4}}}(x)\exp \left(-{\frac {1}{\varepsilon }}\int _{x_{0}}^{x}{\sqrt {Q(t)}}\,dt\right).} Higher-order terms can be obtained by looking at equations for higher powers of δ {\displaystyle \delta } . Explicitly, 2 S 0 ′ S n ′ + S n − 1 ′ ′ + ∑ j = 1 n − 1 S j ′ S n − j ′ = 0 {\displaystyle 2S_{0}^{\prime }S_{n}^{\prime }+S_{n-1}^{\prime \prime }+\sum _{j=1}^{n-1}S_{j}^{\prime }S_{n-j}^{\prime }=0} for n ≥ 2 {\displaystyle n\geq 2} .
Precision of the asymptotic series The asymptotic series for y(x) is usually a divergent series, whose general term δ n S n ( x ) {\displaystyle \delta ^{n}S_{n}(x)} starts to increase after a certain value n = n max {\displaystyle n=n_{\textrm {max}}} . Therefore, the smallest error achieved by the WKB method is at best of the order of the last included term. For the equation ε 2 d 2 y d x 2 = Q ( x ) y , {\displaystyle \varepsilon ^{2}{\frac {d^{2}y}{dx^{2}}}=Q(x)y,}
with Q ( x ) < 0 {\displaystyle Q(x)<0} an analytic function, the value n max {\displaystyle n_{\max }} and the magnitude of the last term can be estimated as follows: n max ≈ 2 ε | ∫ x 0 x ∗ − Q ( z ) d z | , {\displaystyle n_{\max }\approx {\frac {2}{\varepsilon }}\left|\int _{x_{0}}^{x_{\ast }}{\sqrt {-Q(z)}}\,dz\right|,}
δ n max S n max ( x 0 ) ≈ 2 π n max e − n max , {\displaystyle \delta ^{n_{\max }}S_{n_{\max }}(x_{0})\approx {\sqrt {\frac {2\pi }{n_{\max }}}}e^{-n_{\max }},} where x 0 {\displaystyle x_{0}} is the point at which y ( x 0 ) {\displaystyle y(x_{0})} needs to be evaluated and x ∗ {\displaystyle x_{\ast }} is the (complex) turning point where Q ( x ∗ ) = 0 {\displaystyle Q(x_{\ast })=0} , closest to x = x 0 {\displaystyle x=x_{0}} . The number n max {\displaystyle n_{\max }} can be interpreted as the number of oscillations between x 0 {\displaystyle x_{0}} and the closest turning point. If ε − 1 Q ( x ) {\displaystyle \varepsilon ^{-1}Q(x)} is a slowly changing function,
ε | d Q d x | ≪ Q 2 , [might be Q 3 / 2 ?] {\displaystyle \varepsilon \left|{\frac {dQ}{dx}}\right|\ll Q^{2},^{{\text{[might be }}Q^{3/2}{\text{?]}}}}
the number n max {\displaystyle n_{\max }} will be large, and the minimum error of the asymptotic series will be exponentially small.
Application in non-relativistic quantum mechanics
The above example may be applied specifically to the one-dimensional, time-independent Schrödinger equation, − ℏ 2 2 m d 2 d x 2 Ψ ( x ) + V ( x ) Ψ ( x ) = E Ψ ( x ) , {\displaystyle -{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}}{dx^{2}}}\Psi (x)+V(x)\Psi (x)=E\Psi (x),} which can be rewritten as d 2 d x 2 Ψ ( x ) = 2 m ℏ 2 ( V ( x ) − E ) Ψ ( x ) . {\displaystyle {\frac {d^{2}}{dx^{2}}}\Psi (x)={\frac {2m}{\hbar ^{2}}}\left(V(x)-E\right)\Psi (x).} Approximation away from the turning points The wave function can be rewritten as the exponential of another function S (closely related to the action), which could be complex,
Ψ ( x ) = e i S ( x ) / ℏ , {\displaystyle \Psi (\mathbf {x} )=e^{iS(\mathbf {x} )/\hbar },} so that its substitution in Schrödinger's equation gives: i ℏ ∇ 2 S ( x ) − ( ∇ S ( x ) ) 2 = 2 m ( V ( x ) − E ) , {\displaystyle i\hbar \nabla ^{2}S(\mathbf {x} )-\left(\nabla S(\mathbf {x} )\right)^{2}=2m\left(V(\mathbf {x} )-E\right),} Next, the semi-classical approximation is used. This means that each function is expanded as a power series in ℏ {\displaystyle \hbar } : S = S 0 + ℏ S 1 + ℏ 2 S 2 + ⋯ {\displaystyle S=S_{0}+\hbar S_{1}+\hbar ^{2}S_{2}+\cdots } Substituting in the equation, and only retaining terms up to first order in ℏ {\displaystyle \hbar } , one gets ( ∇ S 0 + ℏ ∇ S 1 ) 2 − i ℏ ( ∇ 2 S 0 ) = 2 m ( E − V ( x ) ) , {\displaystyle \left(\nabla S_{0}+\hbar \nabla S_{1}\right)^{2}-i\hbar \left(\nabla ^{2}S_{0}\right)=2m\left(E-V(\mathbf {x} )\right),} which gives the following two relations: ( ∇ S 0 ) 2 = 2 m ( E − V ( x ) ) = ( p ( x ) ) 2 2 ∇ S 0 ⋅ ∇ S 1 − i ∇ 2 S 0 = 0. {\displaystyle {\begin{aligned}\left(\nabla S_{0}\right)^{2}=2m\left(E-V(\mathbf {x} )\right)&=\left(p(\mathbf {x} )\right)^{2}\\[1ex]2\nabla S_{0}\cdot \nabla S_{1}-i\nabla ^{2}S_{0}&=0.\end{aligned}}} These can be solved for one-dimension systems. The first equation can be solved by S 0 ( x ) = ± ∫ 2 m ( E − V ( x ) ) d x = ± ∫ p ( x ) d x , {\displaystyle S_{0}(x)=\pm \int {\sqrt {2m\left(E-V(x)\right)}}\,dx=\pm \int p(x)\,dx,} and the second equation computed for the possible values of the above, is generally expressed as: Ψ ( x ) ≈ C + e + i ℏ ∫ p ( x ) d x | p ( x ) | + C − e − i ℏ ∫ p ( x ) d x | p ( x ) | {\displaystyle \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)|}}}} Thus, the resulting wave function in first order WKB approximation is presented as, In the classically allowed region, namely the region where V ( x ) < E {\displaystyle 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 {\displaystyle 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 E = V ( x ) {\displaystyle E=V(x)} , and cannot be valid. (The turning points are the points where the classical particle changes direction.) Hence, when E > V ( x ) {\displaystyle E>V(x)} , the wave function can be chosen to be expressed as: Ψ ( x ′ ) ≈ 1 | p ( x ) | [ C cos ( 1 ℏ ∫ | p ( x ) | d x + α ) + D sin ( − 1 ℏ ∫ | p ( x ) | d x + α ) ] {\displaystyle \Psi (x')\approx {\frac {1}{\sqrt {|p(x)|}}}\left[C\cos \left({\frac {1}{\hbar }}\int \left|p(x)\right|dx+\alpha \right)+D\sin \left(-{\frac {1}{\hbar }}\int \left|p(x)\right|dx+\alpha \right)\right]} and for V ( x ) > E {\displaystyle V(x)>E} , Ψ ( x ′ ) ≈ C + e − 1 ℏ ∫ | p ( x ) | d x | p ( x ) | + C − e + 1 ℏ ∫ | p ( x ) | d x | p ( x ) | . {\displaystyle \Psi (x')\approx {\frac {C_{+}e^{-{\frac {1}{\hbar }}\int |p(x)|\,dx}}{\sqrt {|p(x)|}}}+{\frac {C_{-}e^{+{\frac {1}{\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 ′ {\displaystyle x'} .
Validity of WKB solutions From the condition: ( S 0 ′ ( x ) ) 2 − ( p ( x ) ) 2 + ℏ ( 2 S 0 ′ ( x ) S 1 ′ ( x ) − i S 0 ″ ( x ) ) = 0 {\displaystyle \left(S_{0}'(x)\right)^{2}-\left(p(x)\right)^{2}+\hbar \left(2S_{0}'(x)S_{1}'(x)-iS_{0}''(x)\right)=0}
It follows that: ℏ | 2 S 0 ′ ( x ) S 1 ′ ( x ) | + ℏ | i S 0 ″ ( x ) | ≪ | ( S 0