Double delta-function potential

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In summary, The conversation is about problem 2.27 in Griffiths, which involves determining the number of bound states, finding their associated energies, and sketching the wave functions for two delta potential wells at +a and -a. The solution involves solving the Schrodinger equation in three regions and paying attention to boundary conditions. The problem was challenging and confusing, but the main difficulty was solving for the energy, which required numerical solving of two equations for even and odd wave functions.
  • #1
TheFerruccio
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My session expired while typing this post, so this is my second attempt at typing it. I *always* forget to paste into notepad before submitting these darned things.

Homework Statement



Problem 2.27, Griffiths.

Given two delta potential wells at +a and -a, determine the number of bound states, find their associated energies, sketch the wave functions.

Homework Equations



[itex]V(x) = -\alpha\left[\delta\left(x+a\right)+\delta\left(x-a\right)\right][/itex]


The Attempt at a Solution



This is the first problem in the book where I really do not know where to begin. I know that the answer has to be in some form of exp(kt) where k is sqrt(-2*m*E)\hbar.

I vaguely understand the book's process for the logic behind constructing the wave function for a single delta potential well, but clearly not well enough.
 
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  • #2
Start by solving the Schrodinger equation in the three regions x<-a, -a≤x≤a, and x>a.
 
  • #3
Thanks for the assistance. All I had to do was solve for a 0 potential and pay attention to the boundary conditions (continuous function, discontinuous derivative by a specific value).

The hard part, after that, was not getting lost in the algebra. I still don't quite understand how to solve for the energy, though. I ended up with two equations that needed numerical solving, one for the even wave function, and one for the odd wave function.

Overall, this problem was a real curve ball, and the specific questions this problem asked were far more confusing to me than any of the others.
 

1. What is a double delta-function potential?

A double delta-function potential is a mathematical model used in quantum mechanics to describe a system with two point-like potentials at specific locations. It is often used as a simplified model for studying interactions between particles in a one-dimensional system.

2. How is a double delta-function potential represented mathematically?

A double delta-function potential is represented by the function V(x) = A[δ(x - a) + δ(x - b)], where A is a constant representing the strength of the potential and a and b are the locations of the two delta-functions.

3. What are the properties of a double delta-function potential?

A double delta-function potential is symmetric around the midpoint between the two delta-functions, and the potential energy is zero everywhere except at the two delta-function locations. It is also an infinite potential barrier, meaning that particles cannot penetrate the potential barrier.

4. What is the significance of a double delta-function potential in quantum mechanics?

A double delta-function potential is a simple model that can be used to understand more complex systems with multiple potentials. It is also used to study scattering and tunneling phenomena in quantum mechanics.

5. How does a double delta-function potential affect the behavior of particles?

A double delta-function potential can cause particles to reflect or transmit depending on their energy. If a particle's energy is lower than the potential energy, it will be completely reflected by the potential barrier. However, if the particle's energy is higher than the potential energy, it may tunnel through the barrier and continue on its path.

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