Delta function potential Definition and 32 Threads

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.

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  1. E

    I Energies of bound state for delta function potential

    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...
  2. M

    A sudden change in the depth of delta function potential well

    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.
  3. D

    Energy Difference with a Two Delta Function Potential

    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...
  4. M

    Numerically find the energy of the delta-well's bound state

    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 =...
  5. 1

    Wavefunction in a delta potential well

    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) +...
  6. Danny Boy

    I Why is reflection coefficient defined this way

    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...
  7. Danny Boy

    I Delta fuction potential general solution

    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...
  8. Logan Rudd

    Determining bound states for delta function potential

    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...
  9. Logan Rudd

    What are the Bound States for a Sum of Two Negative Delta-Function Potentials?

    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?
  10. kq6up

    Delta Function Potential Barrier

    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> ##...
  11. U

    Delta function potential; Schrodinger Equation

    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...
  12. S

    Bound state of the delta function potential.

    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)...
  13. A

    Double delta function potential: two bound states vs one ?

    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...
  14. P

    The Double Dirac Delta Function Potential wave functions

    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 +...
  15. B

    Double Delta function potential well

    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...
  16. L

    Calculate scattering amplitude by delta function potential

    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...
  17. J

    Transmission Coefficient of a double delta function potential

    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...
  18. J

    Double Delta Function Potential

    I have V (x) = \sqrt{((h-bar ^{2})V_{0})/2m} [\delta(x-a)+ \delta(x+a)] How do I find R and T? Under what condition is there resonant transmission?
  19. A

    A problem with a Dirac delta function potential

    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...
  20. D

    Double Delta Function Potential Well

    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...
  21. R

    QM Infinite square well with delta function potential in middle

    Homework Statement Pro #2 if you click on this link. http://s1104.photobucket.com/albums/h332/richard78931/?action=view&current=hw4.jpg Homework Equations , The Attempt at a Solution Click here http://s1104.photobucket.com/albums/h332/richard78931/?action=view&current=2a.jpg...
  22. K

    A seeming contrdiction in deriving wave function for delta function potential

    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{...
  23. R

    What is the treatment of a delta function potential in charge integration?

    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...
  24. D

    Is the Reflection Coefficient for a Delta Function Potential Always Close to 1?

    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...
  25. B

    Bound state for a Dirac delta function potential

    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 >...
  26. M

    How Does a Delta Function Potential Affect a Quantum Particle's Radial Equation?

    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...
  27. R

    How Many Bound States Exist in a Double Delta Function Potential?

    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...
  28. M

    Dirac Delta Function Potential (One Dimension)

    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...
  29. E

    Delta function potential problem

    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...
  30. G01

    How Many Stationary States Exist for a Double Delta Function Potential?

    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...
  31. S

    Delta function potential and Schrodinger Equation

    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...
  32. A

    Can Bound States with Exact Energies Violate the Uncertainty Principle?

    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...
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