What is Delta function potential: Definition and 32 Discussions
In quantum mechanics the delta potential is a potential well mathematically described by the Dirac delta function - a generalized function. Qualitatively, it corresponds to a potential which is zero everywhere, except at a single point, where it takes an infinite value. This can be used to simulate situations where a particle is free to move in two regions of space with a barrier between the two regions. For example, an electron can move almost freely in a conducting material, but if two conducting surfaces are put close together, the interface between them acts as a barrier for the electron that can be approximated by a delta potential.
The delta potential well is a limiting case of the finite potential well, which is obtained if one maintains the product of the width of the well and the potential constant while decreasing the well's width and increasing the potential.
This article, for simplicity, only considers a one-dimensional potential well, but analysis could be expanded to more dimensions.
Hi
Let's consider a potential of the form
The Schrodinger equation reads
as shown in the book 'Introduction to Quantum mechanis' by D.J. Griffiths, Chaper 2 Section 5, the solution of the equation yields (only for bound state, which means E<0):
My question:
if i have one particle and i apply...
is it correct that the continuum states will be free particle states? and the probability will be |< Ψf | ΨB>|^2 . Where Ψf is the wave function for free particle and ΨB is the wave function for the bound state when the depth is B.
Homework Statement
Consider a particle of mass m moving in a one-dimensional double well potential
$$V(x) = -g\delta(x-a)-g\delta(x+a), g > 0$$
This is an attractive potential with ##\delta##-function dips at x=##\pm a##.
In the limit of large ##\lambda##, find a approximate formula for the...
I'm working on an assignment where I'm required to numerically find the energy of a delta-potential's bound state. To do this, we've converted the time-independent schrödinger equation to an eigenvalue problem with E the eigen value, ψ the eigen vector and H a matrix as follows:
with ##t =...
Homework Statement
Using the equations given, show that the wave function for a particle in the periodic delta function potential can be written in the form
##\psi (x) = C[\sin(kx) + e^{-iKa}\sin k(a-x)], \quad 0 \leq x \leq a##
Homework Equations
Given equations:
##\psi (x) =A\sin(kx) +...
In Griffith's "Introduction to Quantum Mechanics, second edition" he states: For the delta-function potential, when considering the scattered states (with E > 0), we have the general solutions for the time-independent Schrodinger equation: $$\psi(x) = Ae^{ikx} + Be^{-ikx}~~~~\text{for }x<0$$ and...
Hi, in the book 'Introduction to Quantum Mechanics' by Griffiths, on page 71 in the section 'The Delta-Function Potential' he states that the general solution to time independent Schrodinger Equation is $$\psi(x) = Ae^{-\kappa x} + B e^{\kappa x}$$
he then notes that the first term blows up as...
I'm working on a problem out of Griffith's Intro to QM 2nd Ed. and it's asking to find the bound states for for the potential ##V(x)=-\alpha[\delta(x+a)+\delta(x-a)]## This is what I'm doing so far:
$$
\mbox{for $x\lt-a$:}\hspace{1cm}\psi=Ae^{\kappa a}\\
\mbox{for $-a\lt x\lt...
I'm reading through Griffiths Intro to QM 2nd Ed. and when it comes to bound/scattering states (2.5) they say:
##E<0 \implies## bound state
##E>0 \implies## scattering state
Why doesn't this change depending on whether you have a positive or negative delta-function potential?
Homework Statement
Background: The problem is to find the uncertainty relationship for the wave equation for a delta function potential barrier where ##V(x)=\alpha\delta(x)##.
Check the uncertainty principle for the wave function in Equation 2.129 Hint: Calculating ##\left< p^2 \right> ##...
Homework Statement
Consider the TISE for a particle of mass m moving along the x-axis and interacting an attractive delta function potential at origin:
Part(a): What is the difference between a bound state particle and a free particle?
Part(b): Show ##\psi _{(x)} = exp (-|k|x)## is a...
What is the physical meaning to a bound state with negative energy? As I understand it, this is the case with the delta function potential, which admits only one bound state with a negative energy.
If the potential function is identically zero throughout (except at the delta function peak)...
In the double delta function potential well, where one delta function ( -αδ(x) ) is at -a and one at +a, if the energy is less than zero, there can be either one or two bound states, depending on the magnitude of α...if α is large enough, there can be two bound states, but if α is small, there...
Homework Statement
Consider the double Dirac delta function V(x) = -α(δ(x+a) + δ(x-a)). Using this potential, find the (normalized) wave functions, sketch them, and determine the # of bound states.
Homework Equations
Time-Independent Schrodinger's Equation: Eψ = (-h^2)/2m (∂^2/∂x^2)ψ +...
Consider a one-dimensional system described by a particle of mass m in the presence
of a pair of delta function wells of strength Wo > 0 located at x = L, i.e.
V(x) = -Wo (x + L) - Wo(x - L) This is a rough but illuminating toy model of an electron in the presence of two positive.
charges...
Homework Statement
I need to give scattering amplitude f(θ) in Born approximation to the first order in the case of delta function scattering potential δ(r). The problem is in spherical coordinate and I'll give major equation concerned.Homework Equations
The equation for scattering amplitude is...
V(x) = |g| (δ(x+L)+δ(x-L)
Consider scattering from a repulsive twin-delta function potential.
Calculate R and T.
I'm mostly confused about computing the T coefficients for multiple barriers. Would I compute the T coefficient for the barrier at x = -L and at x = L seperately? Then...
Homework Statement
An ideal particle of energy E is incident upon a rectangular barrier of width 2a and height V_{0}. Imagine adjusting the barrier width and height so that it approaches V(x)=\alpha \delta(x). What is the relationship between V0, alpha and a?
Homework Equations
The...
Hello Again!
My question:
Find the bound energy spectrum of the potential that contains two delta-function
wells: V(x) = -V_{0}\delta(x-\frac{a}{2}) -V_{0}\delta(x+\frac{a}{2}) under the assumption that the wells are located very far away from each other. Find and plot the associated stationary...
Homework Statement
Pro #2 if you click on this link.
http://s1104.photobucket.com/albums/h332/richard78931/?action=view¤t=hw4.jpg
Homework Equations , The Attempt at a Solution
Click here
http://s1104.photobucket.com/albums/h332/richard78931/?action=view¤t=2a.jpg...
First of all, let me copy the standard solution from Griffiths, section 2.5, just for the sake of clarity.
PotentialV(x) = - \alpha \delta (x)
The bound state eigenfunction:
\psi (x) = \left\{ \begin{array}{l}
B{e^{\kappa x}}{\rm{ (}}x \le 0{\rm{)}} \\
B{e^{ - \kappa x}}{\rm{...
I am trying to integrate a charge density over a volume in order to obtain a total charge, but there is a delta function involved and I am not entirely sure how to treat it.
\rho = q* \delta (\textbf{r})- \frac {q\mu^{2} Exp(- \mu r)} {4 \pi r}
Q = \int \rho (\textbf{r})d^{3}r...
So let's say we have a particle in the delta function potential, V = - \alpha \delta(x). I calculated that the reflection coefficient (scattering state) is
R = \frac{1}{1 + (2 \hbar^2 E/m\alpha^2)}
Now, clearly, the term 2 \hbar^2 E/m\alpha^2 is very small, as \hbar^2 has an order of magnitude...
Homework Statement
Find the bound state energy for a particle in a Dirac delta function potential.
Homework Equations
\newcommand{\pd}[3]{ \frac{ \partial^{#3}{#1} }{ \partial {#2}^{#3} } } - \frac{\hbar^2}{2 m} \ \pd{\psi}{x}{2} - \alpha \delta (x) \psi (x) = E\psi (x)
where \alpha >...
Homework Statement
write the radial equation for a particle with mass m and angular momentum l=0 which is under the influence of the following potential:
V(r)=-a*delta(r-R)
a,R>0
write all the conditions for the solution of the problem.Homework Equations
Schroedinger's equation:
Hu=Eu...
Homework Statement
Consider the double delts-function potential
V(x)=-\alpha[\delta(x+a)+\delta(x-a)]
How many bound states does this possess? Find the allowed energies for
\alpha=\frac{\hbar^{2}}{ma^{2}}and\alpha=\frac{\hbar^{2}}{4ma^{2}}Homework Equations
The Attempt at a Solution
I divided...
Alright, I'm in my first QM course right now, and one of the topics we've looked at is solving the one-dimensional time-independent Schrodinger equation for various potentials, such as the harmonic oscillators, infinite and finite square wells, free particles, and last, but not least, the dirac...
Homework Statement
Why does it make sense that a negative delta function potential represents a highly localized attractive force and a positive delta function potential represents a highly localized repulsive force?
How do you explain that using
-dV/dx = f(x)
?
I guess I am confused about...
Homework Statement
How many stationary states exist for this potential? What are the allowed energies if the strength of the well, \alpha= \hbar^2/ma and \hbar^2/4ma where a= the position of the well(one at a, one at -a) Homework Equations
V(x) = -\alpha(\delta(x+a) +\delta(x-a))
E_{one...
I have a time dependent wavefunction for inside a delta function potential well: V(x) = -a delta(x).
It reads
Psi(x,t) = (sqrt(m*a)/hbar) * exp(-m*a*abs(x)/hbar^2) * exp(-iEt/hbar)
I'm supposed to stick this back into the time dependent Schrodinger Equation and solve for E.
Taking my...
So I read that the delta function potential well has one and only one bound state. This seems to give a precise momentum and position as the bound state has a definite energy and the particle must be in the well. This seems to be a violation of the HUP. Is the physical impossibility of...