QM - delta function potential

In summary, the conversation discussed the radial equation for a particle with zero angular momentum under the influence of a potential. The equation is derived from Schroedinger's equation and has conditions such as continuity, square integrability, and a jump in the first derivative at r=R. Another condition is that u(0)=0, which ensures a bounded wave function.
  • #1

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
Hamiltonian: H=p/2m +V = pr/2m+L^2/2mr^2+V(r)

The Attempt at a Solution



since the angular momentum is zero, the radial equation appears as:
(-hbar/2m)(d^2u/dr^2)-a*delta(r-R)u=Eu
the conditions I can think of are:
1) continuity of u
2) u(infinity)= 0 (for u to be square integrable)
3) from integration of Schroedinger's equation on the interval [R-epsilon, R+epsilon] the jump in the first derivative of u at r=R should be -2mau(R)/hbar^2

but there is another condition according to the answers, that is, u(0)=0.
where does this condition come from?
 
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  • #3
you mean why u(0) = 0 ?

This is due to that [tex] \Psi (r) = \frac{u(r)}{r} [/tex] so [tex] u(r) [/tex] must go to zero faster than r, in order to have a bounded wave function [tex] \Psi (r) [/tex].
 

1. What is the delta function potential in quantum mechanics?

The delta function potential, also known as the Dirac delta potential, is a concept in quantum mechanics that describes a point-like potential energy function. It is represented by the Dirac delta function, which is a mathematical function that is infinite at one point and zero everywhere else. In quantum mechanics, the delta function potential is often used to model interactions between particles at a specific point in space.

2. How is the delta function potential used in quantum mechanics?

The delta function potential is used in quantum mechanics to model point-like interactions between particles. It is often used in the context of scattering problems, where particles interact at a specific point in space. The delta function potential can also be used to model interactions between particles and a potential barrier, such as a potential well or potential barrier.

3. What are the properties of the delta function potential?

The delta function potential has several important properties in quantum mechanics. Firstly, it is a point-like potential, meaning that it only exists at a single point in space. Secondly, it is infinite at this point and zero everywhere else. Lastly, the delta function potential is a non-analytic function, meaning it cannot be expressed as a power series.

4. How does the delta function potential affect the wave function of a particle?

The delta function potential affects the wave function of a particle by causing a discontinuity in the derivative of the wave function at the point of interaction with the potential. This results in a change in the phase of the wave function, which can lead to scattering or other effects depending on the specific problem being studied.

5. What are some real-life applications of the delta function potential?

The delta function potential has many real-life applications in fields such as atomic and molecular physics, nuclear physics, and solid-state physics. It is commonly used to model interactions between particles in these systems, as well as in the study of quantum mechanical phenomena such as tunneling and resonance. The delta function potential is also used in engineering and applied mathematics, such as in the study of electric fields and potential barriers in electronic devices.

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