Double delta function and bound states.

In summary, we are given a potential function with two delta functions and asked to find the number of bound states and the allowed energies for two specific values. We can solve this problem by setting up the wave function in three parts and using the fact that the potential function is symmetric about x = 0 to reduce the number of unknown coefficients. We also need to consider the condition that the wave function must be continuous. Finally, the values of \frac{\hbar^2}{ma} and \frac{\hbar^2}{4ma} are related to the coefficient of the delta functions and may be used to find the energy for specific cases.
  • #1
petera88
5
0

Homework Statement



Given the delta function -α[δ(x+a) + δ(x-a)] where α and a are real positive constants.
How many bound states does it possess? Find allowed energies for [itex]\frac{hbar2}{ma}[/itex] and [itex]\frac{hbar2}{4ma}[/itex] and sketch the wave functions.

Homework Equations



I know there are three parts of the wave function that has the general solution

Aekx + Be-kx

I also know for -inf to -a the term cancels out leaving only Aekx and for x to inf the only term left is some Fe-kx since it can't blow up as x goes to infinity.

The Attempt at a Solution



I set up the three parts of the wave function.

1) Aekx
2) Cekx + De-kx
3) Fe-kx

I also took the derivatives and will use the fact that they will subtract to give me [itex]\frac{-2ma}{hbar2}[/itex]

I know if I solve for k I can get the energy. Honestly I am kind of lost in the understanding of this problem. Especially the [itex]\frac{hbar2}{ma}[/itex] and [itex]\frac{hbar2}{4ma}[/itex] part.

Sorry I don't know why the fraction code isn't working. It should read (hbar^2)/(ma,4ma)
 
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  • #2
Note that your potential function is symmetric about x = 0. So, you can look for solutions where the wave function is even or odd. Thus, you should be able to write the coefficients F and D in terms of A and C for the two cases of even or odd functions. This reduces the number of unknown coefficients.

petera88 said:
I also took the derivatives and will use the fact that they will subtract to give me [itex]\frac{-2ma}{\hbar^2}[/itex]

This isn't quite correct. Did you mean to write ##\alpha## instead of ##a## (where ##\alpha## is the coefficient of the delta functions)? Even so, there is something missing. Can you show more work here?

Don't forget that you also have the condition that the wave function must be continuous.

I know if I solve for k I can get the energy. Honestly I am kind of lost in the understanding of this problem. Especially the [itex]\frac{\hbar^2}{ma}[/itex] and [itex]\frac{\hbar^2}{4ma}[/itex] part.

I'm not sure what is meant here either. Note that [itex]\frac{\hbar^2}{ma}[/itex] has the dimensions of energy times length. This is the same dimensions as ##\alpha##. So, maybe they mean to find the energy for [itex]\alpha = \frac{\hbar^2}{ma}[/itex].
 

1. What is a double delta function?

A double delta function, also known as the Dirac comb, is a mathematical function that is defined as an infinite sum of delta functions spaced equally apart. It is often used as a model for periodic phenomena in physics and engineering.

2. How is a double delta function related to bound states?

In quantum mechanics, bound states are states in which particles are confined to a certain region of space. The energy levels of bound states can be represented by a double delta function, as it is a sum of discrete energy levels that are spaced equally apart.

3. What is the significance of the spacing between the delta functions in a double delta function?

The spacing between the delta functions in a double delta function represents the distance between energy levels in a bound state system. This distance, also known as the energy spacing, is determined by the properties of the system and can provide valuable information about its behavior.

4. Can a double delta function be used to describe any type of bound state?

No, a double delta function can only be used to describe bound states in systems with discrete energy levels, such as a particle in a box or a hydrogen atom. It cannot be used to describe systems with continuous energy spectra, such as a free particle.

5. How does the amplitude of a double delta function affect the bound states it describes?

The amplitude of a double delta function determines the strength of the interaction between particles in a bound state system. A larger amplitude indicates a stronger interaction and can lead to different energy levels and behaviors of the system.

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