How to Solve the Schrodinger Equation with Delta Function Potential?

[QualTime]

Homework Statement

Consider an electron subject to the following 1-D potential:
$$U(x) = -U_0 \left( \delta(x+a) + \delta(x-a) \right)$$
where U_0 and a are positive reals.

(a) Find the ground state of the system, its normalized spatial wavefunction and the parameter κ related to the ground state energy.

(b) Write the transcendental equation satisfied by κ.
Ok, so I'm a bit confused first by what κ is here. This was from an old qual and I don't quite remember the exact phrasing, and it is possible that my memory has failed me and that k (the wavevector?) was meant here instead of kappa. In any event, I don't quite see what kind of transcendental equation it is supposed to satisfy, exept perhaps by matching the three spatial wavefunction and their first derivative at x=-a and x=a?

How would go about solving the Schrodinger equation with a potential given with delta functions? Any help would be appreciated.

 

[ShayanJ]

You just solve the equation for all places except x=a and x=-a.
Then you should impose boundary conditions.
The first one is that the wave-functions should be equal at x=a and x=-a.The second is the discontinuity of the first spatial derivative of the wave-function and the particular value of $\psi'(a+\varepsilon)-\psi'(a-\varepsilon)$ should be derived by integrating Schrodinger equation with delta potentials with respect to position from $a-\varepsilon$ to $a+\varepsilon$. These should be done for x=-a too.

 

[QualTime]

Thanks. I found out it was a problem from Griffith and was able to fill in the details.

One more question if I may: I remember the last part was asking to consider the case where a second electron was being subjected to the same potential, in addition to the resulting Coulomb repulsion. The question asked to discuss how a 2 electron molecule may form and how spin matters.

With the result from the previous part I can show that a bound state exist provided certain restriction on U_0 and m. Would the spin be relevant because of exchange force / symmetrization of the wavefunction (I'm assuming they would be indistinguishable so one would need an antisymmetric wf)? Always been a bit shaky on this so any hint is welcome.

 

[ShayanJ]

Whether you consider spin or not, the wave function of a number of fermions should be antisymmetric under the exchange of particles. Let's analyse the two cases.
First,without spin: The two particle antisymmetric wave function in this case is $\frac{1}{\sqrt{2}}(\phi(x_1)\psi(x_2)-\phi(x_2)\psi(x_1))$ where $\phi$ and $\psi$ are any two normalized states for the specific problem.
Second,with spin: in this case the wave function has also a spin part. So the spatial and spin part combined make the wave function of the system and so this combination should be antisymmetric under the exchange of particles. So if one part is antisymmetric, the other should be symmetric.
This and this may help too.

 

[paranormal]

I can provide some guidance on how to approach this problem. The first step would be to recognize that the potential given is a double square well, meaning there are two potential wells with a barrier in between. This type of potential can be solved using the scattering method or the transfer matrix method.

To find the ground state, we need to solve the Schrodinger equation for the given potential and find the corresponding energy eigenvalue. This can be done by setting up the Schrodinger equation and applying the boundary conditions at x=-a and x=a. The normalized spatial wavefunction can then be found by solving the differential equation and applying normalization conditions.

As for κ, it is possible that it refers to the wavevector, which is related to the energy eigenvalue. The transcendental equation satisfied by κ would then be related to the energy eigenvalue equation.

To solve the Schrodinger equation with delta functions, we can use the delta function potential as a perturbation and apply the perturbation theory. This will give us an expression for the energy eigenvalue in terms of the perturbation parameter (in this case, the strength of the delta function potential).

I hope this helps in approaching the problem. It is always important to carefully read and understand the given information and use appropriate mathematical and physical concepts to solve the problem. If you need further assistance, I would suggest seeking help from a colleague or a professor.