Question

Use classical Coulomb potential energy expressions in finding the lowest potential energy of two electrons that occupy four quantum dots in a quantum automata cell. Show the energy of the electrons is lower when the diagonal configuration is occupied than when the electrons occupy dots that are adjacent. The dimensions of the 2-D box is a×a×a×a.

Answer 1

The lowest potential energy of two electrons in a quantum automata cell with four quantum dots is achieved when the electrons occupy the diagonal configuration.

Step-by-step explanation:

In a 2-D box with dimensions a×a×a×a, the classical Coulomb potential energy between two point charges (electrons in this case) is given by Coulomb's law:
`\(U = (k \cdot q_1 \cdot q_2)/(r)\)`, where (k) is Coulomb's constant,
`\(q_1\) and \(q_2\)` are the charges, and (r) is the separation between the charges. For electrons,
`\(q_1\)` and
`\(q_2\)` are both negative, indicating an attractive force.

Now, considering two electrons in adjacent quantum dots, the separation (r) is larger than in the diagonal configuration. As a result, the potential energy is higher when the electrons occupy adjacent dots compared to the diagonal configuration. This can be expressed as
`\(U_(adjacent) > U_(diagonal)\).`

The diagonal configuration allows the electrons to be closer together, leading to a lower potential energy. Electrons naturally arrange themselves in configurations that minimize their potential energy. Therefore, the diagonal arrangement is favored, demonstrating that the lowest potential energy of two electrons is achieved when they occupy the diagonal configuration in the quantum automata cell with four quantum dots.