Quantum Physics

914 Chapter 28 Atomic PhysicsTABLE 28.2Three Quantum Numbers for the Hydrogen AtomNumber ofQuantumAllowedNumber Name Allowed Values StatesN Principal quantum number 1, 2, 3, . . . Any numberOrbital quantum number 0, 1, 2, . . . , n 1 nm Orbital magnetic quantum , 1, . . . , 2 1number 0, . . . , 1, According to quantum mechanics, the energies of the allowed states are in exactagreement with the values obtained by the Bohr theory (Eq. 28.12) when theallowed energies depend only on the principal quantum number n.In addition to the principal quantum number, two other quantum numbersemerged from the solution of the wave equation: and m . The quantum number is called the orbital quantum number, and m is called the orbital magnetic quantumnumber. As pointed out in Section 28.4, these quantum numbers had alreadyappeared in empirical modifications made to the Bohr theory. The significance ofquantum mechanics is that those numbers and the restrictions placed on their valuesarose directly from mathematics and not from any ad hoc assumptions tomake the theory consistent with experimental observation. Because we will need tomake use of the various quantum numbers in the sections that follow, the allowedranges of their values are repeated:The value of n can range from 1 to in integer steps.The value of can range from 0 to n 1 in integer steps.The value of m can range from to in integer steps.From these rules, it can be seen that for a given value of n, there are n possible valuesof , while for a given value of there are 2 1 possible values of m . For example,if n 1, there is only 1 value of , 0. Because 2 1 2 0 1 1,there is only one value of m , which is m 0. If n 2, the value of may be 0 or 1;if 0, then m 0, but if 1, then m may be 1, 0, or 1. Table 28.2 summarizesthe rules for determining the allowed values of and m for a given value of n.States that violate the rules given in Table 28.2 cannot exist. For instance, onestate that cannot exist is the 2d state, which would have n 2 and 2. This stateis not allowed because the highest allowed value of is n 1, or 1 in this case.Thus, for n 2, 2s and 2p are allowed states, but 2d, 2f, . . . are not. For n 3, theallowed states are 3s, 3p, and 3d.In general, for a given value of n 1 there are n 2 states with distinct pairs of valuesof and m .Quick Quiz 28.3When the principal quantum number is n 5, how many different values of (a) and (b) m are possible? (c) How many states have distinct pairs of values of and m ?EXAMPLE 28.3GoalThe n 2 Level of HydrogenCount states and determine energy based on atomic energy level.Problem (a) Determine the number of states with a unique set of values for and m in the hydrogen atom for n 2.(b) Calculate the energies of these states.Strategy This is a matter of counting, following the quantum rules for n, , and m . “Unique” means that no otherquantum state has the same pair of numbers for and m the energies are all the same because all states have thesame principal quantum number, n 2.