Multiple Electrons in Rectangular Traps (PROBLEMS)
To prepare for our discussion of multiple electrons in atoms. let us discuss two electrons confined to the rectangular traps . We shall again use the quantum numbers we found for those traps when only one electron was confined. However. here we shall also include the spin angular momenta of the two electrons. To do this. we assume that the traps are located in a uniform magnetic field. Then according to Eq. 41-12. an electron can be either spin up with m, = !or spin down with m, = +]. (We shall assume that the magnetic field is very weak so that we can neglect the potential energies of the electrons due to the field.) As we confine the two electrons to one of the traps. we must keep the Pauli exclusion principle in mind; that is. the electrons cannot have the same set of values for their quantum numbers.
Seven electrons are confined to the square corral of Sample Problem 40-5, where the corral is a two-dimensional infinite potential well with widths L, = L; = L . Assume that the electrons do not electrically interact with one another. (a) What is the electron configuration for the ground state of the system of seven electrons?
SOLUTION: We can determine the electron configuration of the system by placing the seven electrons in the corral one by one. to build up the system. One Key Idea here is that because we assume the electrons do not electrically interact with one another, we can use the energy-level diagram for a single trapped electron in order to keep track of how ‘we place the seven electrons in the corral. That one-electron energy-level diagram is given in and partially reproduced here as Recall that the levels are labeled as for their associated energy. For example. the lowest level is for energy £1.1. where quantum number nx is I and quantum number n,. is I. A second Key idea here is that the trapped electrons must obey the Pauli exclusion principle; that is. no two electrons can have the same set of values for their quantum numbers nx• ny, and ms’ The first electron goes into energy level £1.1 and can have ms = !or ms = -to We arbitrarily choose the latter and draw down arrow (to represent spin down) on the £1.1 level in . The second electron also goes into the £1.1 level but must have ms = +! so that one of its quantum numbers differs from those of the first electron. We represent this second electron with an up arrow (for spin up) on the £1.1 level in l3b. Another Key Idea now comes into play: The level for energy £1.1 is fully occupied, and thus the third electron can not have that energy. Therefore, the third electron goes into the next higher level, which is-for the equal energies £2.1 and £1.2 (the level is degenerate). This third electron can have quantum numbers nx and ny of either 1 and 2 or 2 and I. respectively. It can also have a quantum number iris of either +! or -!. Let us arbitrarily assign it the quantum numbers nx = 2, ny = I, and ms = -!. We then represent it with a down mow on the level for 13c. You can show that the next three electrons can also go into where Below is the energy of the level where Ihejump begins and EIUShis the energy of the level where the jump ends. 3. The Pauli exclusion principle must still apply; in particular, an
3. The Pauli exclusion principle must still apply; in particular, an electron cannot jump to a level that is fully occupied. us consider the three jumps shown in Fig. 41-13e; all are allowed by the Pauli exclusion principle because they are jumps to empty or partially occupied states. In one of those possible jumps.an electron jumps from the EI•I level to the partially occupied E2;l level. The change in the energy is
(We shall assume that the spin orientation of the electron making the jump can change as needed.) In another of the possible jumps in Fig. 41-131′. an electron jumps from the degenerate level of E2•1 and EI.2 to the partially occupied E2•2 level. The change in the energy is
level jumps to the unoccupied. degenerate level of EIJ and E3•1• The change in energy is
The energy Ere of the first excited state of the system is then
We can represent this energy and the energy Ear for the ground state of the system on an energy-level diagram for the system. as shown in .