# Two particle in a square potential well?

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1. Dec 18, 2016

### H Psi equal E Psi

Hi guys!

I'm struggling with the following problem:

Consider two distinguishable (not interacting) particles in a quadratic 2 dimensional potential well. So

$V(x,y)=\left\{\begin{matrix} 0,\quad\quad-\frac { L }{ 2 } \le \quad x\quad \le \quad \frac { L }{ 2 } \quad and\quad -\frac { L }{ 2 } \le \quad y\quad \le \quad \frac { L }{ 2 } \\ \infty ,\quad \quad \quad rest \end{matrix}\right.$

I am now asked to find the normalized wave function in the ground state for two particles within the given potential. I tried to solve the schroedinger equation by means of the method of separation of variables:

$\psi ({ x }_{ 1 },{ x }_{ 2 },{ y }_{ 1 }{ y }_{ 2 })=\alpha ({ x }_{ 1 })\beta ({ x }_{ 2 })\delta ({ y }_{ 1 })\varepsilon ({ y }_{ 2 })$

This was harder then i thought so i didn't quiet got an sensible answer...

The second part of the exercise is to replace the two distinguishable particle with two spin ½ (Not interacting) electrons. Now pauli's principle has to be taken in to account. Since i didn't manged to find radial function I am not able to construct the anti symmetric wave function for the electrons:

$\psi ({ x }_{ 1,2 },{ y }_{ 1,2 })=\phi ({ x }_{ 1,2 },{ y }_{ 1,2 })\cdot \frac { 1 }{ \sqrt { 2 } } ((\left| \left \uparrow \downarrow \right> \right) -(\left| \left \downarrow \uparrow \right> \right))$

While $\phi (x,y)$ is the searched function (has to be symmetric).

Sorry for my English ( not my mother tongue )

Thanks and Cheers!

Last edited: Dec 18, 2016
2. Dec 18, 2016

### Staff: Mentor

I moved the thread to our homework section.

If the particles are distinguishable and not interacting, you can solve for their wave functions individually and then combine them.
Why do you want to find a radial function (a function of r?)?

Finding an antisymmetric wave function works the same as for every problem, you just have to find the ground state for a single electron.