Question

A one-dimensional square well of infinite depth and 1 Å width contains 3 electrons. The potential well is described by V = 0 for 0 ≤ 2 ≤ 1 Å and V = +\(\infty\) for x < 0 and x > 1 Å. For a temperature of T = 0 K , the average energy of the 3 electrons is E = 12.4 eV in the approximation that one neglects the Coulomb interaction between electrons. In the same 1 approximation and for T = 0 K, what is the average energy for 4 electrons in this potential well?

Answer 1

For a one-dimensional potential well the energy levels are given by

E_n = E₁n²

where E₁ is the ground state energy and n = 1,2,. . . . Pauli抯 exclusion principle and the lowest energy principle require that two of the three electrons are in the energy level E₁ and the third one is in the energy level E₂. Thus 12 .4 x 3 = 2E₁ + 4E₁, giving E₁ = 6.2 eV. For the case of four electrons, two are in E₁ and the other two in E₂, and so the average energy is