What is Wkb approximation: Definition and 23 Discussions

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.

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  1. P

    I Error of the WKB approximation

    hello everyone I tell you a little about my situation. I already found the approximate wavefunctions for the schrodinger equation with the potential ##V(x) = x^2##, likewise, energy, etc. I have the approximate WKB solution and also the exact numeric solution. What I need to do is to calculate...
  2. Samama Fahim

    Limits of Integration in the Transmission Coefficient

    Initially '0' is the upper limit and ##a = \frac{Ze^2}{E}## is the lower limit. With change of variable ##x = \frac{Er}{Ze^2}##, for ##r=0##, ##x=0##, and for ##r=\frac{Ze^2}{E}##, ##x=1##, so 1 should be the lower limit. However, he takes 1 as the upper limit, and without a minus sign. Why is...
  3. C

    I Difficulty in understanding step in Deriving WKB approximation

    In Zettili book, it is given that ## \nabla^2 \psi \left( \vec{r} \right) + \dfrac{1}{\hbar ^2} p^2 \left( \vec{r} \right) \psi ( \vec{r} ) =0 ## where ## \hbar## is very small and ##p## is classical momentum. Now they assumed the ansatz that ## \psi ( \vec{r} ) = A ( \vec{r} ) e^{i S( \vec{r} )...
  4. C

    I Quantization of Quasiperiodic Orbits in the Bohr-Sommerfeld Model

    Recall that in the semi-classical Bohr-Sommerfeld quantization scheme from the early days of quantum mechanics, bound orbits were quantized according to the value of the action integral around a single loop of a closed path. Clearly this only makes sense if the orbits in question permit closed...
  5. B

    WKB method transmission coefficient

    Homework Statement: The Task is to calculate the Transmission coefficient with the WKB Approximation of following potential: V(x) = V_0(1-(x/a)²) |x|<a ; V(x) = 0 otherwise Homework Equations: ln|T|² = -2 ∫ p(x) dx I have inserted the potential in the equation for p(x) and recieved p(x) =...
  6. Q

    Asymptotic behavior of Airy functions in the WKB method

    If it is the asymptotic behavior of the Airy's function what it's used instead of the function itself: Does it mean that the wkb method is only valid for potentials where the regions where ##E<V## and ##E>V## are "wide"?
  7. Wrichik Basu

    B "Slowly varying" potential for WKB approximation

    In order to use WKB approximation, the potential has to be "slowly varying". I learned the method from this video: But the Professor hasn't mentioned in detail what the measure of "slowly varying" is. What is the limit beyond which we cannot use the WKB method accurately?
  8. deep838

    I Interpreting "momentum" in WKB approximation

    According to WKB approximation, the wave function \psi (x) \propto \frac{1}{\sqrt{p(x)}} This implies that the probability of finding a particle in between x and x+dx is inversely proportional to the momentum of the particle in the given potential. According to the book, R. Shankar, this is...
  9. DeldotB

    WKB Approximation with V-Shaped well

    Homework Statement Good day all! I'm studying for finals and i'd like to know how to do this problem (its not homework): "Using the WKB method, find the bound state energies E_n of a particle of mass m in a V-shaped potential well: V(x)= \begin{Bmatrix} -V_0 (1- \begin{vmatrix}...
  10. Mrinmoy Naskar

    A WKB Approximation: Suggestions for Starting Points

    Any good book for starting wkb approximation except Griffth... Please suggest some...
  11. C

    Understanding WKB Approximation for E-V(x)

    Consider E>V(x). WKB states the wavefunction will remain sinusoidal with a slow variation of wavelength $ \lambda $ and amplitude given that V(x) varies slowly. From the equation \begin{equation} k(x)=\frac{\sqrt{2m(E-V(x))}}{\hbar} \end{equation}, I can see that the k(x) is directly...
  12. P

    Trying to understand the WKB approximation

    I'm trying to understand why the WKB approximation doesn't seem to work in the following case. Suppose you have a particle of mass ##m## in a potential ##V(x)=q m\cos(2mx/\hbar)##, where ##q\ll 1##. Consider then the stationary solution with energy ##E=m/2##. The Schroedinger equation is then...
  13. S

    Quantum Wells: WKB approximation

    1. Consider a quantum well described by the potential v(x)=kx^{2} for \left|x\right|<a and v(x)=ka^{2} for \left|x\right|>a. Given a^{2}\sqrt{km}/\hbar =2, show that the well has 3 bound states and calculate the ratios between the energies and ka^{2}. You may use the standard integral...
  14. A

    How to apply the WKB approximation in this case?

    Homework Statement I'm trying to learn how to apply the WKB approximation. Given the following problem: An electron, say, in the nuclear potential $$U(r)=\begin{cases} & -U_{0} \;\;\;\;\;\;\text{ if } r < r_{0} \\ & k/r \;\;\;\;\;\;\;\;\text{ if } r > r_{0} \end{cases}$$ 1. What is the...
  15. G

    Book suggestions for WKB approximation and Perturbations in Cosmology

    Hi everyone, I was wondering if you guys could suggest me some good books in cosmology with finely explained WKB method and Perturbations especially in Structure formation area. I have "The early universe" by Klob and Turner and "Cosmology" by Weinberg , but they seem unpalatable at first...
  16. H

    How to solve for allowed energies with the WKB approximation?

    Hello, I'm trying to solve for the allowed energies with the WKB approximation of the Schrodinger equation, using the Morse potential. So I have (as per equation 35 at http://hitoshi.berkeley.edu/221a/WKB.pdf), \int_a^b \sqrt{2m(E-V(x))}dx=\left(n+\frac{1}{2}\right)\pi\hbar However, how do I...
  17. M

    Proof of vailidity of WKB approximation Please help

    Hi, I have been looking for rigorous mathematical conditions for when the WKB approximation may be applied. Here is my understanding of the topic. We start with the most general form that the wavefunction could take, i.e. exp[if(x)/h] , Where "i" stands for square root of -1, f(x) is...
  18. M

    Problem with integration for WKB approximation

    problem with integration for WKB approximation in MATLAB hi all, i have been having troubles with getting MATLAB to solve the following problem (the language is not the MATLAB one, the functions are not solvable by the symbolic integration and i was trying to get one of the quad functions to...
  19. F

    Estimating Electron Lifetime in 1D Quantum Well with WKB Approximation

    Homework Statement Using the WKB approximation, estimate the lifetime of an electron in the ground state of a 1D quantum well with 10 nm width, surrounded from both sides by 0.3 eV high and 8nm wide barriers. Homework Equations Hint: Estimate the tunneling probability and find the...
  20. maverick280857

    Connection Formulas in the WKB approximation

    Hello friends, I've been reading Schiff's book on QM (3rd Edition), esp the section on the WKB approximation. (This isn't homework.) I have a few questions: What is the physical significance of the arrow on the connection formulas, like \frac{1}{2}\frac{1}{\sqrt{\kappa}} e^{-\zeta_{2}}...
  21. I

    Understanding the Limitations of WKB Approximation in One-Dimensional Problems

    Hi all I have a question about WKB approximation Why is it that WKB method can be applied only to problems that are one dimensional or those which can be reduced to forms that are one dimensional ones? any help is deeply appreciated
  22. Repetit

    Deriviation of WKB approximation

    Hey! In deriving the WKB approximation the wave function is written as \psi \left( x \right) = exp\left[ i S\left( x \right) \right ] Now, in some of the deriviations I've seen, the function S(x) is expanded as a power series in \hbar as S(x) = S_0(x) + \hbar S_1(x) +...
  23. J

    Questions on WKB approximation

    The description in P.252in liboff's quantum mechanics, I cannot not figure out the continuity and continue in first order derivative of the wave function \varphi_I = \frac{1}{\sqrt{\kappa}} \exp {(\int_{x_1}^{x} \kappa dx)}...