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Προσεγγιστική Μέθοδος WKB

- Μία διαδικασία.

## ΕτυμολογίαEdit

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## ΕισαγωγήEdit

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 slowly changing.

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 Edit

This method is named after physicists Wentzel, Kramers, and 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, which includes the Schrödinger equation.

Even though the Schrödinger equation was developed two years later, 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 R. B. Dingle.[1]

Earlier references to the method are:

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.[2][3]

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.

## WKB methodEdit

Generally, WKB theory 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.

$\epsilon \frac{d^ny}{dx^n} + a(x)\frac{d^{n-1}y}{dx^{n-1}} + \cdots + k(x)\frac{dy}{dx} + m(x)y= 0,$

assume a solution of the form of an asymptotic series expansion

$y(x) \sim \exp\left[\frac{1}{\delta}\sum_{n=0}^{\infty}\delta^nS_n(x)\right]$

in the limit δ → 0.

The asymptotic scaling of δ in terms of 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 Sn(x) in the expansion.

WKB theory is a special case of multiple scale analysis.[4][5][6]

## An exampleEdit

This example comes from the text of Bender and Orszag.[6] Consider the second-order homogeneous linear differential equation

$\epsilon^2 \frac{d^2 y}{dx^2} = Q(x) y,$

where $Q(x) \neq 0$. Substituting

$y(x) = \exp\left[\frac{1}{\delta}\sum_{n=0}^\infty \delta^nS_n(x)\right]$

results in the equation

$\epsilon^2\left[\frac{1}{\delta^2}\left(\sum_{n=0}^\infty \delta^nS_n'\right)^2 + \frac{1}{\delta}\sum_{n=0}^{\infty}\delta^nS_n''\right] = Q(x).$

To leading order (assuming, for the moment, the series will be asymptotically consistent), the above can be approximated as

$\frac{\epsilon^2}{\delta^2}S_0'^2 + \frac{2\epsilon^2}{\delta}S_0'S_1' + \frac{\epsilon^2}{\delta}S_0'' = Q(x).$

In the limit δ → 0, the dominant balance is given by

$\frac{\epsilon^2}{\delta^2}S_0'^2 \sim Q(x).$

So δ is proportional to ε. Setting them equal and comparing powers yields

$\epsilon^0: \quad S_0'^2 = Q(x),$

which can be recognized as the Eikonal equation, with solution

$S_0(x) = \pm \int_{x_0}^x \sqrt{Q(t)}\,dt.$

Considering first-order powers of ε fixes

$\epsilon^1: \quad 2S_0'S_1' + S_0'' = 0.$

This is the unidimensional transport equation, having the solution

$S_1(x) = -\frac{1}{4}\ln Q(x) + k_1,$

where k1 is an arbitrary constant.

We now have a pair of approximations to the system (a pair, because S0 can take two signs); the first-order WKB-approximation will be a linear combination of the two:

$y(x) \approx c_1Q^{-\frac{1}{4}}(x)\exp\left[\frac{1}{\epsilon}\int_{x_0}^x\sqrt{Q(t)}\,dt\right] + c_2Q^{-\frac{1}{4}}(x)\exp\left[-\frac{1}{\epsilon}\int_{x_0}^x\sqrt{Q(t)}\,dt\right].$

Higher-order terms can be obtained by looking at equations for higher powers of δ.

Explicitly,

$2S_0'S_n' + S''_{n-1} + \sum_{j=1}^{n-1}S'_jS'_{n-j} = 0$

for n ≥ 2.

## ΥποσημειώσειςEdit

1. R.B. Dingle, Asymptotic Expansions: Their Derivation and Interpretation (Academic Press, 1973).
2. Adrian E. Gill (1982). Atmosphere-ocean dynamics. Academic Press. σελ. 297. ISBN 978-0-12-283522-3.
3. Renato Spigler & Marco Vianello (1998). A Survey on the Liouville–Green (WKB) approximation for linear difference equations of the second order. Advances in difference equations: proceedings of the Second International Conference on Difference Equations : Veszprém, Hungary, August 7–11, 1995. CRC Press. σελ. 567. ISBN 978-90-5699-521-8.
4. Filippi, Paul (1999). Acoustics: basic physics, theory and methods. Academic Press. σελ. 171. ISBN 978-0-12-256190-0.
5. Kevorkian, J.; Cole, J. D. (1996). Multiple scale and singular perturbation methods. Springer. ISBN 0-387-94202-5.
6. 6,0 6,1 Bender, C.M.; Orszag, S.A. (1999). Advanced mathematical methods for scientists and engineers. Springer. σελ. 549–568. ISBN 0-387-98931-5.

## ΙστογραφίαEdit

Κίνδυνοι Χρήσης

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σε αυτήν την εγκυκλοπαίδεια
ωστόσο, παρακαλούμε να λάβετε σοβαρά υπ' όψη ότι
η "Sciencepedia" δεν μπορεί να εγγυηθεί, από καμιά άποψη,
την εγκυρότητα των πληροφοριών που περιλαμβάνει.

"Οι πληροφορίες αυτές μπορεί πρόσφατα
να έχουν αλλοιωθεί, βανδαλισθεί ή μεταβληθεί από κάποιο άτομο,
η άποψη του οποίου δεν συνάδει με το "επίπεδο γνώσης"
του ιδιαίτερου γνωστικού τομέα που σας ενδιαφέρει."

Πρέπει να λάβετε υπ' όψη ότι
όλα τα άρθρα μπορεί να είναι ακριβή, γενικώς,
και για μακρά χρονική περίοδο,
αλλά να υποστούν κάποιο βανδαλισμό ή ακατάλληλη επεξεργασία,
ελάχιστο χρονικό διάστημα, πριν τα δείτε.

Επίσης,
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αλλά και κάθε διαδικτυακού ιστότοπου (ή αλλιώς site)),
αν και άκρως απαραίτητοι,
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(λόγω της ρευστής φύσης του Web),
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σε παραπλανητικό, κακόβουλο ή άσεμνο περιεχόμενο.
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