How to Solve the Schrodinger Equation with Delta Function Potential?

[QualTime]

Homework Statement

Consider an electron subject to the following 1-D potential:
<br /> U(x) = -U_0 \left( \delta(x+a) + \delta(x-a) \right)<br />
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 Schrödinger 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&#039;(a+\varepsilon)-\psi&#039;(a-\varepsilon) should be derived by integrating Schrödinger 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.