Two particle in a square potential well?

In summary, the conversation discusses solving for the normalized wave function in the ground state for two particles in a quadratic 2-dimensional potential well. The method of separation of variables is used, but it becomes more difficult when considering two spin ½ electrons due to the inclusion of Pauli's principle. The goal is to find an anti-symmetric wave function for the electrons, but the speaker was not able to find a sensible answer due to difficulty finding a radial function.
  • #1
H Psi equal E Psi
11
0
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 into 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:
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  • #2
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.
H Psi equal E Psi said:
Since i didn't manged to find radial function
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.
 
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Related to Two particle in a square potential well?

1. What is a square potential well?

A square potential well is a theoretical model used in quantum mechanics to represent the potential energy of a particle in a confined space. It consists of a square-shaped potential energy barrier that confines the particle within a finite region.

2. What are the properties of a particle in a square potential well?

The properties of a particle in a square potential well include discrete energy levels, quantized energy states, and a wave function that describes the probability of finding the particle at a certain position within the well.

3. How does the depth of the potential well affect the particle?

The depth of the potential well determines the energy levels and spacing between them. A deeper well will have a greater number of energy levels and tighter spacing, while a shallower well will have fewer energy levels and wider spacing.

4. What is the significance of the boundaries of the square potential well?

The boundaries of the square potential well represent the edges of the confined space in which the particle can exist. They act as barriers that confine the particle to a finite region and determine the shape and size of the potential well.

5. How does the wave function of a particle in a square potential well change over time?

The wave function of a particle in a square potential well will oscillate between positive and negative values as the particle moves within the well. The frequency and amplitude of these oscillations depend on the energy level of the particle and the depth of the potential well.

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