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50-456: 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. 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

100-408: A d x = 2 3 ( U 1 3 ( x − a ) ) 3 2 = 2 3 u 3 2 {\displaystyle {\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}}} It now remains to construct

150-849: 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

200-461: 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 E , this matching procedure will not work: The function obtained by connecting

250-1327: 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}{\epsilon }}\int _{x_{0}}^{x}{\sqrt {Q(t)}}\,dt\right]+c_{2}Q^{-{\frac {1}{4}}}(x)\exp \left[-{\frac {1}{\epsilon }}\int _{x_{0}}^{x}{\sqrt {Q(t)}}\,dt\right].} Higher-order terms can be obtained by looking at equations for higher powers of δ . Explicitly, 2 S 0 ′ S n ′ + S n − 1 ″ + ∑ j = 1 n − 1 S j ′ S n − j ′ = 0 {\displaystyle 2S_{0}'S_{n}'+S''_{n-1}+\sum _{j=1}^{n-1}S'_{j}S'_{n-j}=0} for n ≥ 2 . The asymptotic series for y ( x )

300-430: A mathematical or physical problem or solution. It typically provides an initial estimate or framework to the solution of a mathematical problem, and can also take into consideration the boundary conditions (in fact, an ansatz is sometimes thought of as a "trial answer" and an important technique in solving differential equations ). After an ansatz, which constitutes nothing more than an assumption, has been established,

350-830: A power series, 2 m ℏ 2 ( V ( x ) − E ) = U 1 ⋅ ( x − x 1 ) + U 2 ⋅ ( x − x 1 ) 2 + ⋯ . {\displaystyle {\frac {2m}{\hbar ^{2}}}\left(V(x)-E\right)=U_{1}\cdot (x-x_{1})+U_{2}\cdot (x-x_{1})^{2}+\cdots \;.} To first order, one finds d 2 d x 2 Ψ ( x ) = U 1 ⋅ ( x − x 1 ) ⋅ Ψ ( x ) . {\displaystyle {\frac {d^{2}}{dx^{2}}}\Psi (x)=U_{1}\cdot (x-x_{1})\cdot \Psi (x).} This differential equation

400-485: 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 . The asymptotic scaling of δ in terms of ε will be determined by

450-412: A tool at a work piece", plural ansatzes or, from German, ansätze / ˈ æ n s ɛ t s ə / ; German: [ˈʔanzɛtsə] ) is an educated guess or an additional assumption made to help solve a problem, and which may later be verified to be part of the solution by its results. An ansatz is the establishment of the starting equation(s), the theorem(s), or the value(s) describing

500-684: 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 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

550-444: Is available. Given a set of experimental data that looks to be clustered about a line, a linear ansatz could be made to find the parameters of the line by a least squares curve fit. Variational approximation methods use ansätze and then fit the parameters. Another example could be the mass, energy, and entropy balance equations that, considered simultaneous for purposes of the elementary operations of linear algebra , are

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600-1112: Is computed between the classical turning point and the arbitrary position x'. From the condition: ( S 0 ′ ( x ) ) 2 − ( p ( x ) ) 2 + ℏ ( 2 S 0 ′ ( x ) S 1 ′ ( x ) − i S 0 ″ ( x ) ) = 0 {\displaystyle (S_{0}'(x))^{2}-(p(x))^{2}+\hbar (2S_{0}'(x)S_{1}'(x)-iS_{0}''(x))=0} It follows that: ℏ ∣ 2 S 0 ′ ( x ) S 1 ′ ( x ) ∣ + ℏ ∣ i S 0 ″ ( x ) ∣ ≪ ∣ ( S 0 ′ ( x ) ) 2 ∣ + ∣ ( p ( x ) ) 2 ∣ {\textstyle \hbar \mid 2S_{0}'(x)S_{1}'(x)\mid +\hbar \mid iS_{0}''(x)\mid \ll \mid (S_{0}'(x))^{2}\mid +\mid (p(x))^{2}\mid } For which

650-971: Is known as the Airy equation , and the solution may be written in terms of Airy functions , Ψ ( x ) = C A Ai ⁡ ( U 1 3 ⋅ ( x − x 1 ) ) + C B Bi ⁡ ( U 1 3 ⋅ ( x − x 1 ) ) = C A Ai ⁡ ( u ) + C B Bi ⁡ ( u ) . {\displaystyle \Psi (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 ℏ {\displaystyle \hbar } ,

700-481: 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 can be interpreted as

750-469: Is used and λ ( x ) {\textstyle \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. This condition can also be restated as the fractional change of E − V ( x ) {\textstyle E-V(x)} or that of the momentum p ( x ) {\textstyle p(x)} , over

800-560: Is usually a divergent series , whose general term δ S n ( x ) starts to increase after a certain value n = n 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 \epsilon ^{2}{\frac {d^{2}y}{dx^{2}}}=Q(x)y,} with Q ( x ) <0 an analytic function,

850-738: The Wentzel–Kramers–Brillouin approximation , also known as the WKB approximation , classical approach , and phase integral method . Wentzel is also known for his contributions to photoemission and scattering theory. Late career work includes contributions to the discussion of gauge invariant theories of superconductivity. In 1975, Wentzel was awarded the Max Planck Medal . Ansatz In physics and mathematics , an ansatz ( / ˈ æ n s æ t s / ; German: [ˈʔanzats] , meaning: "initial placement of

900-435: The de Broglie wavelength of the particle is slowly varying. 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, x 1 , the term 2 m ℏ 2 ( V ( x ) − E ) {\displaystyle {\frac {2m}{\hbar ^{2}}}\left(V(x)-E\right)} can be expanded in

950-586: 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 fixes ϵ 1 : 2 S 0 ′ S 1 ′ + S 0 ″ = 0. {\displaystyle \epsilon ^{1}:\quad 2S_{0}'S_{1}'+S_{0}''=0.} This has

1000-753: The Chair for Theoretical Physics, at the University of Zurich , when he succeeded Erwin Schrödinger , in 1928, the same year Wolfgang Pauli was appointed to the ETH Zurich . Together, Wentzel and Pauli built the reputation of Zurich as a center for theoretical physics. In 1948, Wentzel took a professorship at the University of Chicago . He retired in 1970 and went to spend his last years in Ascona , Switzerland . In 1926, Wentzel, Hendrik Kramers , and Léon Brillouin independently developed what became known as

1050-430: The ansatz to most basic problems of thermodynamics . Another example of an ansatz is to suppose the solution of a homogeneous linear differential equation to take an exponential form, or a power form in the case of a difference equation . More generally, one can guess a particular solution of a system of equations, and test such an ansatz by directly substituting the solution into the system of equations. In many cases,

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1100-606: The armed forces during World War I . He then resumed his education at Freiburg until 1919, when he went to the University of Greifswald . In 1920, he went to the Ludwig Maximilian University of Munich (LMU) to study under Arnold Sommerfeld . Wentzel was awarded his doctorate in 1921 and completed his Habilitation in 1922. He remained at LMU as a Privatdozent until he was called to the University of Leipzig in 1926 as an extraordinarius professor of mathematical physics. He became ordinarius professor in

1150-405: 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

1200-1839: The denominator that both of these approximate solutions become singular near the classical turning points , where 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 wavefunction can be chosen to be expressed as: Ψ ( x ′ ) ≈ C cos ⁡ ( 1 ℏ ∫ | p ( x ) | d x + α ) | p ( x ) | + D sin ⁡ ( − 1 ℏ ∫ | p ( x ) | d x + α ) | p ( x ) | {\displaystyle \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)|}}}} and for V ( x ) > E {\displaystyle V(x)>E} , 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 {\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

1250-659: 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 \epsilon ^{2}\left[{\frac {1}{\delta ^{2}}}\left(\sum _{n=0}^{\infty }\delta ^{n}S_{n}'\right)^{2}+{\frac {1}{\delta }}\sum _{n=0}^{\infty }\delta ^{n}S_{n}''\right]=Q(x).} To leading order in ϵ (assuming, for

1300-1084: 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 ) in the expansion. WKB theory is a special case of multiple scale analysis . 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 \epsilon ^{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

1350-445: The equations are solved more precisely for the general function of interest, which then constitutes a confirmation of the assumption. In essence, an ansatz makes assumptions about the form of the solution to a problem so as to make the solution easier to find. It has been demonstrated that machine learning techniques can be applied to provide initial estimates similar to those invented by humans and to discover new ones in case no ansatz

1400-477: The exact quantum energy levels. 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 {\displaystyle x=x_{1}} and the second turning point, where potential is increasing over x, occur at x = x 2 {\displaystyle x=x_{2}} . Given that we expect wavefunctions to be of

1450-721: 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} ) \over \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} )-(\nabla S(\mathbf {x} ))^{2}=2m\left(V(\mathbf {x} )-E\right),} Next,

1500-1361: The following form, we can calculate their coefficients by connecting the different regions using Airy and Bairy functions. Ψ V > E ( x ) ≈ A e 2 3 u 3 2 u 4 + B e − 2 3 u 3 2 u 4 Ψ E > V ( x ) ≈ C cos ⁡ ( 2 3 u 3 2 − α ) u 4 + D sin ⁡ ( 2 3 u 3 2 − α ) u 4 {\displaystyle {\begin{aligned}\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{aligned}}} For U 1 < 0 {\displaystyle U_{1}<0} ie. decreasing potential condition or x = x 1 {\displaystyle x=x_{1}} in

1550-753: The following two inequalities are equivalent since the terms in either side are equivalent, as used in the WKB approximation: ℏ ∣ S 0 ″ ( x ) ∣ ≪ ∣ ( S 0 ′ ( x ) ) 2 ∣ 2 ℏ ∣ S 0 ′ S 1 ′ ∣ ≪ ∣ ( p ′ ( x ) ) 2 ∣ {\displaystyle {\begin{aligned}\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{aligned}}} The first inequality can be used to show

WKB approximation - Misplaced Pages Continue

1600-1057: 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}(\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{aligned}}} which can be solved for 1D systems, first equation resulting in: S 0 ( x ) = ± ∫ 2 m ℏ 2 ( E − V ( x ) ) d x = ± 1 ℏ ∫ p ( x ) d x {\displaystyle S_{0}(x)=\pm \int {\sqrt {{\frac {2m}{\hbar ^{2}}}\left(E-V(x)\right)}}\,dx=\pm {\frac {1}{\hbar }}\int p(x)\,dx} and

1650-976: The following: ℏ ∣ S 0 ″ ( x ) ∣ ≪ ∣ ( p ( x ) ) ∣ 2 1 2 ℏ | p ( x ) | | d p 2 d x | ≪ | p ( x ) | 2 λ | d V d x | ≪ | p | 2 m {\displaystyle {\begin{aligned}\hbar \mid S_{0}''(x)\mid \ll \mid (p(x))\mid ^{2}\\{\frac {1}{2}}{\frac {\hbar }{|p(x)|}}\left|{\frac {dp^{2}}{dx}}\right|\ll |p(x)|^{2}\\\lambda \left|{\frac {dV}{dx}}\right|\ll {\frac {|p|^{2}}{m}}\\\end{aligned}}} where | S 0 ′ ( x ) | = | p ( x ) | {\textstyle |S_{0}'(x)|=|p(x)|}

1700-1870: 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: Bi ⁡ ( u ) → − 1 π 1 u 4 sin ⁡ ( 2 3 | u | 3 2 − π 4 ) where, u → − ∞ Bi ⁡ ( u ) → 1 π 1 u 4 e 2 3 u 3 2 where, u → + ∞ {\displaystyle {\begin{aligned}\operatorname {Bi} (u)\rightarrow -{\frac {1}{\sqrt {\pi }}}{\frac {1}{\sqrt[{4}]{u}}}\sin {\left({\frac {2}{3}}|u|^{\frac {3}{2}}-{\frac {\pi }{4}}\right)}\quad {\textrm {where,}}\quad u\rightarrow -\infty \\\operatorname {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{aligned}}} We cannot use Airy function since it gives growing exponential behaviour for negative x. When compared to WKB solutions and matching their behaviours at ± ∞ {\displaystyle \pm \infty } , we conclude: A = − D = N {\displaystyle A=-D=N} , B = C = 0 {\displaystyle B=C=0} and α = π 4 {\displaystyle \alpha ={\frac {\pi }{4}}} . Thus, letting some normalization constant be N {\displaystyle N} ,

1750-579: The limit δ → 0 , the dominant balance is given by ϵ 2 δ 2 S 0 ′ 2 ∼ Q ( x ) . {\displaystyle {\frac {\epsilon ^{2}}{\delta ^{2}}}S_{0}'^{2}\sim Q(x).} So δ is proportional to ϵ . Setting them equal and comparing powers yields ϵ 0 : S 0 ′ 2 = Q ( x ) , {\displaystyle \epsilon ^{0}:\quad S_{0}'^{2}=Q(x),} which can be recognized as

1800-502: 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. Generally, WKB theory

1850-609: 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 {\epsilon ^{2}}{\delta ^{2}}}S_{0}'^{2}+{\frac {2\epsilon ^{2}}{\delta }}S_{0}'S_{1}'+{\frac {\epsilon ^{2}}{\delta }}S_{0}''=Q(x).} In

1900-996: The number n max will be large, and the minimum error of the asymptotic series will be exponentially small. 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).} The wavefunction can be rewritten as

1950-572: The number of oscillations between x 0 {\displaystyle x_{0}} and the closest turning point. If ϵ − 1 Q ( x ) {\displaystyle \epsilon ^{-1}Q(x)} is a slowly changing function, ϵ | d Q d x | ≪ Q 2 , [might be  Q 3 / 2 ?] {\displaystyle \epsilon \left|{\frac {dQ}{dx}}\right|\ll Q^{2},^{{\text{[might be }}Q^{3/2}{\text{?]}}}}

2000-810: The resulting wavefunction in first order WKB approximation is presented as, Ψ ( x ) ≈ C + e + i ℏ ∫ 2 m ( E − V ( x ) ) d x + C − e − i ℏ ∫ 2 m ( E − V ( x ) ) d x 2 m ∣ E − V ( x ) ∣ 4 {\displaystyle \Psi (x)\approx {\frac {C_{+}e^{+{\frac {i}{\hbar }}\int {\sqrt {2m\left(E-V(x)\right)}}\,dx}+C_{-}e^{-{\frac {i}{\hbar }}\int {\sqrt {2m\left(E-V(x)\right)}}\,dx}}{\sqrt[{4}]{2m\mid E-V(x)\mid }}}} In

2050-672: 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,

WKB approximation - Misplaced Pages Continue

2100-803: The semiclassical approximation is used. This means that each function is expanded as a power series in ħ . 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 , we get: ( ∇ S 0 + ℏ ∇ S 1 ) 2 − i ℏ ( ∇ 2 S 0 ) = 2 m ( E − V ( x ) ) {\displaystyle (\nabla S_{0}+\hbar \nabla S_{1})^{2}-i\hbar (\nabla ^{2}S_{0})=2m(E-V(\mathbf {x} ))} which gives

2150-415: 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 is an arbitrary constant. We now have a pair of approximations to the system (a pair, because S 0 can take two signs); the first-order WKB-approximation will be

2200-422: The solution near + ∞ {\displaystyle +\infty } to the classically allowed region will not agree with the function obtained by connecting the solution near − ∞ {\displaystyle -\infty } to the classically allowed region. The requirement that the two functions agree imposes a condition on the energy E , which will give an approximation to

2250-928: The value n max {\displaystyle n_{\max }} and the magnitude of the last term can be estimated as follows: n max ≈ 2 ϵ − 1 | ∫ x 0 x ∗ − Q ( z ) d z | , {\displaystyle n_{\max }\approx 2\epsilon ^{-1}\left|\int _{x_{0}}^{x_{\ast }}{\sqrt {-Q(z)}}\,dz\right|,} δ n max S n max ( x 0 ) ≈ 2 π n max exp ⁡ [ − n max ] , {\displaystyle \delta ^{n_{\max }}S_{n_{\max }}(x_{0})\approx {\sqrt {\frac {2\pi }{n_{\max }}}}\exp[-n_{\max }],} where x 0 {\displaystyle x_{0}}

2300-456: 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 ℏ {\displaystyle \hbar } gets smaller, the height of the wave function at the turning points grows. It also follows from this approximation that: 1 ℏ ∫ p ( x ) d x = U 1 ∫ x −

2350-1182: The wavefunction is given for increasing potential (with x) as: Ψ WKB ( x ) = { − N | p ( x ) | exp ⁡ ( − 1 ℏ ∫ x x 1 | p ( x ) | d x ) if  x < x 1 N | p ( x ) | sin ⁡ ( 1 ℏ ∫ x x 1 | p ( x ) | d x − π 4 ) if  x 2 > x > x 1 {\displaystyle \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}\\{\frac {N}{\sqrt {|p(x)|}}}\sin {({\frac {1}{\hbar }}\int _{x}^{x_{1}}|p(x)|dx-{\frac {\pi }{4}})}&{\text{if }}x_{2}>x>x_{1}\\\end{cases}}} Gregor Wentzel Gregor Wentzel (17 February 1898 – 12 August 1978)

2400-515: The wavelength λ {\textstyle \lambda } , being much smaller than 1 {\textstyle 1} . Similarly it can be shown that λ ( x ) {\textstyle \lambda (x)} also has restrictions based on underlying assumptions for the WKB approximation that: | d λ d x | ≪ 1 {\displaystyle \left|{\frac {d\lambda }{dx}}\right|\ll 1} which implies that

2450-401: 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 years, he contributed to X-ray spectroscopy , but then broadened out to make contributions to quantum mechanics , quantum electrodynamics , superconductivity and meson theory. Gregor Wentzel

2500-555: Was born in Düsseldorf , Germany, as the first of four children of Joseph and Anna Wentzel. He married Anna Lauretta Wielich and his only child, Donat Wentzel , was born in 1934. The family moved to the United States in 1948 until he and Anny returned to Ascona, Switzerland in 1970. Wentzel began his university education in mathematics and physics in 1916, at the University of Freiburg . During 1917 and 1918, he served in

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