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.
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...
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} )...
I've already found the turning points, in the case of the left turning point, the local minimum of the potential, ##\delta_{min}=1.11977## when evaluating for an arbitrary value of current ##J=0.9I_C##. The left turning point is therefore ##\delta_r=2.48243##.
I know the Bohr-Sommerfeld...
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) =...
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"?
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?
What is the nonperturbative approach to quantum mechanics as opposed to perturbative one? When does the latter method fail and one has to apply nonperturbative approach? Please keep your discussion confined within non-relativistic quantum mechanics.
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...
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}...
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...
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...
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...
I would like to understand how to find wave functions using WKB.
Homework Statement
Given an electron, say, in the nuclear potential
$$U(r)=\begin{cases}
& -U_{0} \;\;\;\;\;\;\text{ if } r < r_{0} \\
& k/r \;\;\;\;\;\;\;\;\text{ if } r > r_{0}
\end{cases}$$
With the barrier region given...
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...
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...
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...
i think every system is accurately described by Schrodinger equation.
so what is the point of using old quantum mechanics methods?
with Schrodinger equation, at least numerically, you can solve the eigenvalues and eigenvectors readily and accurately. So what is the point of using...
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...
Hi i have the WKB approx of:
u_{+} = \sqrt{1-\frac{bm}{f}}e^{i\int f_{k} dt} + \sqrt{1+\frac{bm}{f}}e^{-i\int f_{k} dt}
to the differential equation:
\frac{d^{2}u_{+}}{dt^2} + [f_{k}^{2} + i(\frac{d(bm)}{dt})]u_{+} =0
This equation can be written as:
\frac{d^{2}u_{+}}{dN^2} + [p^2 - i...
It is not from my howework(due I'm not in the undergrad now), but it seems to be a very easy question I have to know answer to, but I fail to do so.
Homework Statement
I have to go from classical to quantum Hamiltonian via WKB method (and both to solve Schroedinger equation)
It looks...
Homework Statement
I am given the modified Poschl-Teller potential:
V(x)=-\frac{U_0}{\cosh^2(\alpha x)}. I have to make WKB approximation of bound energy states and compare it to exact solution (analytical and numerical).
Homework Equations
Exact solution is given...
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...
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...
Dear All,
I have recently read about WKB approximation and about perturbation theory.
Both methods are applicable in the range of slowly varying potentials. What I have not understood is which is the range of applicability of one of the method compared with the other one. More...
Hi,
Why is it that the WKB approximation produces the correct eigenvalues for the Harmonic Oscillator problem, but the wrong wavefunctions, whereas for the square well, we get correct wavefunctions and the wrong eigenvalues?
I've been trying to dig through the approximations we make in...
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}}...
WKB theory?? Pls help
Homework Statement
WKB theory and WKB approximation. Are they same??
Homework Equations
I have to do a presentation about WKB theory in Physics (Quantum mechanics) but I don't know how to start from the WKB theory and come to WKB approximation. i.e. I don't find a...
Hi,
This is just a quick question -- I'm puzzled by the way this answer sheet represents the potential function.
The question asks us to determine the energy eigenvalues of the bound states of a well where the potential drops abruptly from zero to a depth Vo at x=0, and then increases...
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
If in 1-D the WKB wave and energy quantization are:
\Psi (x) = e^{iS(x)/\hbar} and \oint_C dq p =2\pi (n+1/2) \hbar
My question is what happens with more than one dimension ?? (many body system or 3-D system), what happens with QFT ?? i know that as an analogy you could always put the...
for a Hamiltonian H=H_0 + \epsilon V(x)
my question is (for small epsilon) can WKB and perturbative approach give very different solutions ?? to energies eigenvalues and so on the index '0' means that is the Hamiltonian of a free particle.
problem arises perhaps in calculation of...
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) +...
If we have a 1-dimensional problem so for big n "Energies" can be found in the form:
2 \int_{a}^{b}dx \sqrt (E_{n} - V(x) ) = (n+1/2) \hbar
where "a" and "b" are the turning points, then could we writte the equation for energies (where a>c>b using Mean-value theorem for integrals )...
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)}...
let be e an small parameter e<<<1 then if we want to find a solution to the equation:
e\ddot x + f(t)x=0
then we could write a solution to it in the form:
x(t)=exp(i \int dt f(t)^{1/2}/e)[a_{0}(t)+ea_{1}(t)+e^{2}a_{2}(t)+...]
My question is if we could apply Borel resummation (or...
let,s suppose we have a particle with mass m\rightarrow\infty then my question is if would be fair to make the WKB approach by setting the solution of the Schroedinguer equation as \phi=e^{iS/\hbar} wiht S hte classical action satisfying the equation:
(dS/sx)^{2}+2m(V(x)-E_{n})=0 with E_n...
Consider the potential
V(x) = \beta x for x \geq\ 0
V(x) = 0 for x < 0.
Find the exact and WKB wavefunction for the situation where a particle has
E = 10 in units where \beta = \hbar = m = 1.
Any suggestions guys?
James