Fine: Chapter 5, Problem 28

Problem   On page 218: List all four quantum numbers for each of the following:

1. A lone electron in a ground-state hydrogen atom
2. A 3p electron

Subject   Quantum numbers; Electron Configuration, Fine: pages 191-201.

Solution   Basically, the solution for the two cases is the same. We start by examining the collective part. At the end we will solve the two separate cases. We can collect the necessary information from the text (see pages 193-196), from figures 5.11 and 5.12, and tables 5.5 and 5.6.

The principle quantum number, n, depends on which shell the electron is in. The angular momentum quantum number, l, ranges between zero (0) and l = n - 1, so it has n possible values. It corresponds to the subshell of the electron. The magnetic quantum number m_l ranges from -l to +l, and therefore has 2l+ 1 ( = 2n - 1) values. Finally, there is only one electron (and no magnetic field) so for both electrons, the spin quantum number m_s = -1/2, +1/2. We can not determine m_s for single electrons without additional information.

1. For hydrogen in the ground state, the principle quantum number is n = 1 (as stated in the text on page 193). This automatically gives us the value for the angular momentum quantum number as l = 0, which in turn means that m_l = 0. The spin quantum number m_s can be +1/2 or - 1/2: this is not specified.
2. The 3 from 3p means that n = 3. The p subshell tells us that l = 1. Actually, from the provided information, we can only say that m_l is either +1,0,  or  -1, but without a (magnetic) field, we can not tell which of these it is. As mentioned above m_s can be either +1/2 or - 1/2.