Quantum Physics

28.7 The Spin Magnetic Quantum Number 915Solution(a) Determine the number of states with a unique set ofvalues for and m in the hydrogen atom for n 2.Determine the different possible values of for n 2: 0 n 1, so, for n 2, 0 1 and 0 or 1Find the different possible values of m for 0: m , so 0 m 0 implies m 0List the distinct pairs of (, m ) for 0: There is only one: (, m ) (0, 0).Find the different possible values of m for 1: m , so 1 m 1 implies m 1, 0, or 1List the distinct pairs of (, m ) for 1: There are three: (, m ) ( 1, 1), (1, 0), and (1, 1).Sum the results for 0 and 1: Number of states 1 3 4(b) Calculate the energies of these states.The common energy of all of the states can be foundwith Equation 28.13:E n 13.6 eV13.6 eVn 2 : E 2 2 2 3.40 eVRemarks While these states normally have the same energy, applying a magnetic field will result in their takingslightly different energies centered around the energy corresponding to n 2. As seen in the next section, there arein fact twice as many states, corresponding to a new quantum number called spin.Exercise 28.3(a) Determine the number of states with a unique pair of values for and m in the n 3 level of hydrogen.(b) Determine the energies of those states.Answers (a) 9 (b) E 3 1.51 eV28.7 THE SPIN MAGNETIC QUANTUM NUMBERAs we’ll see in this section, there actually are eight states corresponding to n 2for hydrogen, not four as given in Example 28.3. This happens because anotherquantum number, m s , the spin magnetic quantum number, has to be introduced toexplain the splitting of each level into two.The need for this new quantum number first came about because of an unusualfeature in the spectra of certain gases, such as sodium vapor. Close examination ofone of the prominent lines of sodium shows that it is, in fact, two very closelyspaced lines. The wavelengths of these lines occur in the yellow region of the spectrum,at 589.0 nm and 589.6 nm. In 1925, when this doublet was first noticed,atomic theory couldn’t explain it. To resolve the dilemma, Samuel Goudsmit andGeorge Uhlenbeck, following a suggestion by the Austrian physicist WolfgangPauli, proposed the introduction of a fourth quantum number to describe atomicenergy levels, called the spin quantum number.In order to describe the spin quantum number, it’s convenient (but technically incorrect)to think of the electron as spinning on its axis as it orbits the nucleus, just asthe Earth spins on its axis as it orbits the Sun. Strangely, there are only two ways inwhich the electron can spin as it orbits the nucleus, as shown in Figure 28.11. If thedirection of spin is as shown in Figure 28.11a, the electron is said to have “spin up.” Ifthe direction of spin is reversed, as in Figure 28.11b, the electron is said to have “spindown.” The energy of the electron is slightly different for the two spin directions, andthis energy difference accounts for the sodium doublet. The quantum numbers associatedwith electron spin are m s for the spin-up state and m s 1 22 for the spin-1down state. As we’ll see in Example 28.5, this new quantum number doubles thenumber of allowed states specified by the quantum numbers n, , and m .Nucleus(a)Nucleus(b)Spin upSpin downFigure 28.11 As an electron movesin its orbit about the nucleus, its spincan be either (a) up or (b) down.