Six electrons are in a two-dimensional square potential energy well with sides of length L.The potential energy is infinite at the sides and zero inside.The single-particle energies are given by where n_x and n_y are integers.If a seventh electron is added to the system when it is in its ground state the least energy the additional electron can have is: A)2 (h²/8mL²) B)5 (h²/8mL²) C)8 (h²/8mL²) D)10 (h²/8mL²) E)13 (h²/8mL²)

Question

Quizplus · Accepted Answer

The first six electrons fill up the lowest energy states, with each state having an energy given by the equation 11eab079_97d0_dbec_a51f_75c4197b74fe_TB6585_11. Since there are two possible states for each value of n_x and n_y, the first six electrons will fill up all of the states with energies up to 2(11eab079_97d0_dbec_a51f_75c4197b74fe_TB6585_11) = 22(h/(8mL^2)). To add a seventh electron, we must add it to a state with energy greater than 22(h/(8mL^2)), which means it must have n_x=2 or n_y=2 (since these are the next highest energy states). The energy of the lowest energy state with n_x=2 or n_y=2 is given by the equation 11eab079_97d0_dbec_a51f_75c4197b74fe_TB6585_11 with n_x=1 and n_y=1 (since these are the lowest possible values for the other coordinate). Plugging in these values, we get: E = 2(11eab079_97d0_dbec_a51f_75c4197b74fe_TB6585_11) - (11eab079_97d0_dbec_a51f_75c4197b74fe_TB6585_11) = 8(h/(8mL^2)) Therefore, the least energy the additional electron can have is 8(h/(8mL^2)).

Six Electrons Are in a Two-Dimensional Square Potential Energy Well