Question

For two dimensional particle in a box, how many electrons can we get at most in the first energy level and why?

Answer 1

Befor proceeding i will tell you my approach

1. first i calculated ground state energy for a particle in a box

2. Then we will calculate accordingly

A quantum particle of mass in a two-dimensional square box by a potential energy that is zero if and and infinite otherwise. Inside the box, the energy is entirely kinetic because , so the classical energy is

where and are the two components of the particle's momentum. We see that the energy naturally is expressible as a sum of kinetic energies associated with motion in the and directions:

Because the energy is a simple sum, the solutions of the Schrödinger equation can be expressed as simple products of the solutions of the one-dimensional Schrödinger equation for this problem. Note that it is only when the energy is expressible in this way that simple product solutions are rigorously correct. Thus, the wave function is of the form

which satisfies the boundary conditions at and , namely and . In order to satisfy the remaining boundary conditions and , we have two conditions:

The first one can be satisfied if

independent of , while the second can be satisfied if

independent of . These are the same conditions that we encountered for the one-dimensional box, hence we already know the function in each case can be zero in many places. In fact, these two conditions are satisfied if

which yield the allowed values of and as

We need two different integers and because the conditions are completely independent and can be satisfied by any two different (or similar) values of these integers. The allowed values of the total energy are now given by

Note that the allowed energies now depend on two integers and rather than one. These arise from the two independent boundary conditions in the and directions. As in the one-dimensional box, the values of and are both restricted to the natural numbers Note, therefore, that the ground state energy is

=  

Here energy depends on other parameters like Length so no of electrons can't be determined exactly without knowing what exactly value of L is.