How to calculate the exact wavefunction of two electrons in a 1-d infinite well?

[peter308]

as title,
the electron's interaction is coulomb force.
1.is it unsolvable?(exact solution)
2.will computer simulation be the only way to work it out?

thanks a lot,dude

 

[turin]

Try a coordinate transformation. This is a 2-D problem particle-in-a-box problem with a twist: another potential. I am trying the coordinate transformation:

η = x - y, ξ = x + y.

EDIT:
Sorry, I typed that without much explanation. I will now reveal the secret meaning of those variables. I intended x to be the position of one of the electrons and y to be the position of the other. This would give a 2-D box problem with a ridge of infinite potential running along one of the diagonals that tapers off at the opposing corners.

 

[peter308]

the way u solve the problem will encounter some difficulties,one is the bondary conditions(infinite well!),as u transfrom the coordiates,
the boundary will transfrom,either .and i believe this will result in the conditon that u can't separate your variables!by the way,i am grateful y give me replies!

 

[Dr Transport]

How is the problem stated orginally, is it "two non-interacting particles" in a 1-d well or something else. The wavefunction of two non-interacting particles is the product of the wavefunctions taking into account odd and even symmetry.

If it is truly two interacting electrons, the problem is more difficult and depends on the boundary conditions with other things involved. The "1-d hydrogen problem" is difficult enough, two mutually repelling particles is difficult at best. you then have to take into account spin etc...the fact that electrons are fermions cannot be neglected.

Please state the problem exactly from the text ot professors notes and I can give you some more ideas...

dt

 

[peter308]

the two electrons are fermions(electons),and they interact through coulomb force. i got this problem from my prof,saying this work could have some connections to entanglement and quantum dots.i wonder if we could throw the spin part away,just concentrate on the space part first!or have we consider it both at the same time?i have asked several profs about this problem,they all pointed out the main obstacle is the non-seperable variables!maybe we need some more advanced mathematical framework?likely the differential geometry
or...etc.

by the way,since i am a foreign guy,some corrections or instructions in my grammer will be highly wellcomed,that will make progress in my composition a lot!

best wishes

 

[Ackbar]

The electronic repulsion term (couloumbic force) is quite challenging. In fact, that term is one of the reasons that we are unable to solve the helium atom exactly. For this reason, I am not sure that your problem can be solved exactly.

I suppose your hamiltonian should be...
-h^2/2m*(d/dx1+d/dx2) + e^2/(4*pi*epsilon0*|r1-r2|)
In which case, how do you separate the last term? Like I said earlier, this is challenging. I would suggest you attempt approximation techniques. This is not a hard differential equation to solve numerically, but it is analytically.

Good luck.

 

[vanesch]

I don't know how to solve this problem just like that, but I'd suggest the following approach. Because we know that the two electrons are confined to a finite interval, we know that their respective hilbert spaces in the space representation take on the following form:

psi_n(x) = N sin(n pi x / L)

The product hilbert space then has a basis psi_{n,m}(x,y) = psi_n(x) cross psi_m(y)
which we can represent by the dirac ket |n,m>.

If we ignore the fermionic part, in that basis, we can work out the hamiltonian from the coulomb interaction:

E_{k,l} + E_{m,n} + <k,l | 1/|x-y| | m,n>

the last term is an integral which I think can be worked out, and the first two E-terms are the free energy terms for the wave functions psi_n etc...

Now the final problem - which I don't know how to tackle - is how to diagonalise this infinite matrix.

If we want to introduce the fermionic character of the electrons, the modification is essentially that we should only consider anti-symmetric combinations (a redefinition of |k,l> as a function of the sine functions).

cheers,
Patrick.

 

[turin]

Originally posted by peter308
the way u solve the problem will encounter some difficulties,one is the bondary conditions(infinite well!),as u transfrom the coordiates,
the boundary will transfrom,either .and i believe this will result in the conditon that u can't separate your variables!

I totally agree. That's what happens when I take a semester break from school. I hadn't solved it when I posted. I decided on a slightly different transformation, but I haven't worked through it yet. It makes the kinetric energt pretty messy. I tried to attach an image of the basic idea. It at least allows you to separate the potential (including the boundaries of the well). In equation form, the transformation that I have in mind is:

r = x - y
&theta; = arctan((a - y)/x)

These coordinates are not orthogonal, so, I'm a little discouraged. Notice that this is not the conventional r and &theta;.