Multiple Electrons in Rectangular Traps
To prepare for our discussion of multiple electrons in atoms. let us discuss two electrons confined to the rectangular traps of Chapter 40. 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.
1. One-dimensional trap. In the one-dimensional trap of . filling an electron wave to the trap’s width L requires the single quantum number II. Therefore. any electron confined to the trap must have a certain value of n. and its quantum
number nls can be either +} or -!. Th e two electrons could have different values of II. or they could have the same value of II if one of them is spin up and the other is spin down.
2. Rectangular corral. In the rectangular corral of . fitting an electron wave to the corral’s widths L, and L,. requires the two quantum numbers II, and n,.. Thus. any electron confined to the trap must have certain values for those two quantum numbers. and its quantum number 11I., can be either +~or -4-so now there are three quantum numbers. According to the Pauli exclusion principle. two electrons confined to the trap must have different values for at least one of those three quantum numbers.
3. Rectangular box. In the rectangular box of . fitting an electron wave to the box’s widths Lx. Lv. and L; requires the three quantum numbers II. and n; Thus, any electron confined to the trap must have certain values for these three quantum numbers, and its quantum number m, can be either +tor -tso. now there are four quantum numbers. According to the Pauli exclusion principle, two electrons confined to !he trap must have different values for at least
one of those four quantum numbers. Suppose we add more than two electrons, one by one, to a rectangular trap in
the preceding list. The first electrons naturally go into the lowest ‘possible energy level-they are said to occupy that level/However, eventually the Pauli exclusion principle disallows any more electrons from occupying that lowest energy level, and the next electron must occupy the next higher level. When an energy level cannot be occupied by more electrons because of the Pauli exclusion principle, we say that level is full or fully occupied. In contrast, a level that is not occupied by any electrons is empty or unoccupied. For intermediate situations, the level is partially occupied. The electron con inauguration of a system of trapped electrons is a listing or drawing of the energy levels the electrons occupy, or the set of the quantum numbers of the electrons.
Finding the Total Energy
We shall later want to find the energy of a system of two or’ more electrons confined to a rectangular trap. That is, we shall want to find the total energy for any configuration of the trapped electrons. For simplicity, we shall assume that the electrons do no electrically interact with one another; that is, we shall neglect the electric potential energies of pairs of electrons. In that case, we can calculate the total energy for any electron configuration Iby calculating the energy of each electron as we did in Chapter 40, and then summing those energies. (In Sample Problem 41-3 we do so for seven electrons confined to a rectangular corral.) . A good way to organize the enegy values of a given system of electrons is with an energy-level diagram for the system, just as we did for a single electron in the traps ofChapter 40, The lowest level, with energy Err’ corresponds to the ground state of the system. The next higher level, with energy Ere, corresponds to the first excited state of the system. The next level, with energy Ese, corresponds to the second excited state of the system. And so on.