Quantum Numbers and Rules

Transcription

1 OpenStax-CNX module: m Quantum Numbers and Rules OpenStax College Tis work is produced by OpenStax-CNX and licensed under te Creative Commons Attribution License 3.0 Abstract Dene quantum number. Calculate angle of angular momentum vector wit an axis. Dene spin quantum number. Pysical caracteristics tat are quantizedsuc as energy, carge, and angular momentumare of suc importance tat names and symbols are given to tem. Te values of quantized entities are expressed in terms of quantum numbers, and te rules governing tem are of te utmost importance in determining wat nature is and does. Tis section covers some of te more important quantum numbers and rulesall of wic apply in cemistry, material science, and far beyond te realm of atomic pysics, were tey were rst discovered. Once again, we see ow pysics makes discoveries wic enable oter elds to grow. Te energy states of bound systems are quantized, because te particle wavelengt can t into te bounds of te system in only certain ways. Tis was elaborated for te ydrogen atom, for wic te allowed energies are expressed as E n 1/n 2, were n = 1, 2, 3,... We dene n to be te principal quantum number tat labels te basic states of a system. Te lowest-energy state as n = 1, te rst excited state as n = 2, and so on. Tus te allowed values for te principal quantum number are n = 1, 2, 3,... (1) Tis is more tan just a numbering sceme, since te energy of te system, suc as te ydrogen atom, can be expressed as some function of n, as can oter caracteristics (suc as te orbital radii of te ydrogen atom). Te fact tat te magnitude of angular momentum is quantized was rst recognized by Bor in relation to te ydrogen atom; it is now known to be true in general. Wit te development of quantum mecanics, it was found tat te magnitude of angular momentum L can ave only te values L = l (l + 1) (l = 0, 1, 2,..., n 1), (2) were l is dened to be te angular momentum quantum number. Te rule for l in atoms is given in te parenteses. Given n, te value of l can be any integer from zero up to n 1. For example, if n = 4, ten l can be 0, 1, 2, or 3. Note tat for n = 1, l can only be zero. Tis means tat te ground-state angular momentum for ydrogen is actually zero, not / as Bor proposed. Te picture of circular orbits is not valid, because tere would be angular momentum for any circular orbit. A more valid picture is te cloud of probability sown for te ground state of ydrogen in. Te electron actually spends time in and near te nucleus. Version 1.10: Sep 6, :39 pm ttp://creativecommons.org/licenses/by/3.0/

2 OpenStax-CNX module: m Te reason te electron does not remain in te nucleus is related to Heisenberg's uncertainty principle te electron's energy would ave to be muc too large to be conned to te small space of te nucleus. Now te rst excited state of ydrogen as n = 2, so tat l can be eiter 0 or 1, according to te rule in L = l (l + 1). Similarly, for n = 3, l can be 0, 1, or 2. It is often most convenient to state te value of l, a simple integer, rater tan calculating te value of L from L = l (l + 1). For example, for l = 2, we see tat L = 2 (2 + 1) = 6 = = J s. (3) It is muc simpler to state l = 2. As recognized in te Zeeman eect, te direction of angular momentum is quantized. We now know tis is true in all circumstances. It is found tat te component of angular momentum along one direction in space, usually called te z-axis, can ave only certain values of L z. Te direction in space must be related to someting pysical, suc as te direction of te magnetic eld at tat location. Tis is an aspect of relativity. Direction as no meaning if tere is noting tat varies wit direction, as does magnetic force. Te allowed values of L z are L z = m l (m l = l, l + 1,..., 1, 0, 1,... l 1, l), (4) were L z is te z-component of te angular momentum and m l is te angular momentum projection quantum number. Te rule in parenteses for te values of m l is tat it can range from l to l in steps of one. For example, if l = 2, ten m l can ave te ve values 2, 1, 0, 1, and 2. Eac m l corresponds to a dierent energy in te presence of a magnetic eld, so tat tey are related to te splitting of spectral lines into discrete parts, as discussed in te preceding section. If te z-component of angular momentum can ave only certain values, ten te angular momentum can ave only certain directions, as illustrated in Figure 1.

3 OpenStax-CNX module: m Figure 1: Te component of a given angular momentum along te z-axis (dened by te direction of a magnetic eld) can ave only certain values; tese are sown ere for l = 1, for wic m l = 1, 0, and+1. Te direction of L is quantized in te sense tat it can ave only certain angles relative to te z-axis. Example 1: Wat Are te Allowed Directions? Calculate te angles tat te angular momentum vector L can make wit te z-axis for l = 1, as illustrated in Figure 1. Strategy Figure 1 represents te vectors L and L z as usual, wit arrows proportional to teir magnitudes and pointing in te correct directions. L and L z form a rigt triangle, wit L being te ypotenuse and L z te adjacent side. Tis means tat te ratio of L z to L is te cosine of te angle of interest. We can nd L and L z using L = l (l + 1) and L z = m. Solution We are given l = 1, so tat m l can be +1, 0, or 1. Tus L as te value given by L = l (l + 1). l (l + 1) 2 L = = (5)

4 OpenStax-CNX module: m L z can ave tree values, given by L z = m l. L z = m l = {, m l = +1 0, m l = 0, m l = 1 As can be seen in Figure 1, cos θ = L z /L, and so for m l = +1, we ave (6) Tus, cos θ 1 = L Z L = 2 = 1 2 = (7) Similarly, for m l = 0, we nd cos θ 2 = 0; tus, θ 1 = cos = (8) And for m l = 1, θ 2 = cos 1 0 = (9) so tat cos θ 3 = L Z L = 2 = 1 2 = 0.707, (10) θ 3 = cos 1 ( 0.707) = (11) Discussion Te angles are consistent wit te gure. Only te angle relative to te z-axis is quantized. L can point in any direction as long as it makes te proper angle wit te z-axis. Tus te angular momentum vectors lie on cones as illustrated. Tis beavior is not observed on te large scale. To see ow te correspondence principle olds ere, consider tat te smallest angle (θ 1 in te example) is for te maximum value of m l = 0, namely m l = l. For tat smallest angle, cos θ = L z L = l, (12) l (l + 1) wic approaces 1 as l becomes very large. If cos θ = 1, ten θ = 0. Furtermore, for large l, tere are many values of m l, so tat all angles become possible as l gets very large. 1 Intrinsic Spin Angular Momentum Is Quantized in Magnitude and Direction Tere are two more quantum numbers of immediate concern. Bot were rst discovered for electrons in conjunction wit ne structure in atomic spectra. It is now well establised tat electrons and oter fundamental particles ave intrinsic spin, rougly analogous to a planet spinning on its axis. Tis spin is a fundamental caracteristic of particles, and only one magnitude of intrinsic spin is allowed for a given type of particle. Intrinsic angular momentum is quantized independently of orbital angular momentum. Additionally, te direction of te spin is also quantized. It as been found tat te magnitude of te intrinsic (internal) spin angular momentum, S, of an electron is given by S = s (s + 1) (s = 1/2 for electrons), (13)

5 OpenStax-CNX module: m were s is dened to be te spin quantum number. Tis is very similar to te quantization of L given in L = l (l + 1), except tat te only value allowed for s for electrons is 1/2. Te direction of intrinsic spin is quantized, just as is te direction of orbital angular momentum. Te direction of spin angular momentum along one direction in space, again called te z-axis, can ave only te values ( S z = m s m s = 1 ) 2, +1 (14) 2 for electrons. S z is te z-component of spin angular momentum and m s is te spin projection quantum number. For electrons, s can only be 1/2, and m s can be eiter +1/2 or 1/2. Spin projection m s =+1/2 is referred to as spin up, wereas m s = 1/2 is called spin down. Tese are illustrated in. : In later capters, we will see tat intrinsic spin is a caracteristic of all subatomic particles. For some particles s is alf-integral, wereas for oters s is integraltere are crucial dierences between alf-integral spin particles and integral spin particles. Protons and neutrons, like electrons, ave s = 1/2, wereas potons ave s = 1, and oter particles called pions ave s = 0, and so on. To summarize, te state of a system, suc as te precise nature of an electron in an atom, is determined by its particular quantum numbers. Tese are expressed in te form (n, l, m l, m s ) see Table 1: Atomic Quantum NumbersFor electrons in atoms, te principal quantum number can ave te values n = 1, 2, 3,... Once n is known, te values of te angular momentum quantum number are limited to l = 1, 2, 3,..., n 1. For a given value of l, te angular momentum projection quantum number can ave only te values m l = l, l + 1,..., 1, 0, 1,..., l 1, l. Electron spin is independent of n, l, and m l, always aving s = 1/2. Te spin projection quantum number can ave two values, m s = 1/2 or 1/2. Atomic Quantum Numbers Name Symbol Allowed values Principal quantum number n 1, 2, 3,... Angular momentum l 0, 1, 2,...n 1 Angular momentum projection m l l, l +1,..., 1, 0, 1,..., l 1, l (or 0, ±1, ±2,... continued on next page

6 OpenStax-CNX module: m Spin 6 s 1/2 ( electrons ) Spin projection m s 1/2, + 1/2 Table 1 Figure 2 sows several ydrogen states corresponding to dierent sets of quantum numbers. Note tat tese clouds of probability are te locations of electrons as determined by making repeated measurements eac measurement nds te electron in a denite location, wit a greater cance of nding te electron in some places rater tan oters. Wit repeated measurements, te pattern of probability sown in te gure emerges. Te clouds of probability do not look like nor do tey correspond to classical orbits. Te uncertainty principle actually prevents us and nature from knowing ow te electron gets from one place to anoter, and so an orbit really does not exist as suc. Nature on a small scale is again muc dierent from tat on te large scale. 7 Te spin quantum number s is usually not stated, since it is always 1/2 for electrons

7 OpenStax-CNX module: m Figure 2: Probability clouds for te electron in te ground state and several excited states of ydrogen. Te nature of tese states is determined by teir sets of quantum numbers, ere given as (n, l, m l ). Te ground state is (0, 0, 0); one of te possibilities for te second excited state is (3, 2, 1). Te probability of nding te electron is indicated by te sade of color; te darker te coloring te greater te cance of nding te electron.

8 OpenStax-CNX module: m We will see tat te quantum numbers discussed in tis section are valid for a broad range of particles and oter systems, suc as nuclei. Some quantum numbers, suc as intrinsic spin, are related to fundamental classications of subatomic particles, and tey obey laws tat will give us furter insigt into te substructure of matter and its interactions. : Te classic Stern-Gerlac Experiment sows tat atoms ave a property called spin. Spin is a kind of intrinsic angular momentum, wic as no classical counterpart. Wen te z-component of te spin is measured, one always gets one of two values: spin up or spin down. Figure 3: Stern-Gerlac Experiment 8 2 Section Summary Quantum numbers are used to express te allowed values of quantized entities. Te principal quantum number n labels te basic states of a system and is given by n = 1, 2, 3,... (15) Te magnitude of angular momentum is given by L = l (l + 1) (l = 0, 1, 2,..., n 1), (16) were l is te angular momentum quantum number. Te direction of angular momentum is quantized, in tat its component along an axis dened by a magnetic eld, called te z-axis is given by L z = m l (m l = l, l + 1,..., 1, 0, 1,... l 1, l), (17) were L z is te z-component of te angular momentum and m l is te angular momentum projection quantum number. Similarly, te electron's intrinsic spin angular momentum S is given by S = s (s + 1) (s = 1/2 for electrons), (18) s is dened to be te spin quantum number. Finally, te direction of te electron's spin along te z-axis is given by S z = m s ( m s = 1 ) 2, +1, (19) 2 were S z is te z-component of spin angular momentum and m s is te spin projection quantum number. Spin projection m s =+1/2 is referred to as spin up, wereas m s = 1/2 is called spin down. Table 1: Atomic Quantum Numbers summarizes te atomic quantum numbers and teir allowed values. 8 ttp://cnx.org/content/m42614/latest/stern-gerlac_en.jar

9 OpenStax-CNX module: m Conceptual Questions Exercise 1 Dene te quantum numbers n, l, m l, s, and m s. Exercise 2 For a given value of n, wat are te allowed values of l? Exercise 3 For a given value of l, wat are te allowed values of m l? Wat are te allowed values of m l for a given value of n? Give an example in eac case. Exercise 4 List all te possible values of s and m s for an electron. Are tere particles for wic tese values are dierent? Te same? 4 Problem Exercises Exercise 5 (Solution on p. 10.) If an atom as an electron in te n = 5 state wit m l = 3, wat are te possible values of l? Exercise 6 An atom as an electron wit m l = 2. Wat is te smallest value of n for tis electron? Exercise 7 (Solution on p. 10.) Wat are te possible values of m l for an electron in te n = 4 state? Exercise 8 Wat, if any, constraints does a value of m l = 1 place on te oter quantum numbers for an electron in an atom? Exercise 9 (Solution on p. 10.) (a) Calculate te magnitude of te angular momentum for an l = 1 electron. (b) Compare your answer to te value Bor proposed for te n = 1 state. Exercise 10 (a) Wat is te magnitude of te angular momentum for an l = 1 electron? (b) Calculate te magnitude of te electron's spin angular momentum. (c) Wat is te ratio of tese angular momenta? Exercise 11 (Solution on p. 10.) Repeat Exercise for l = 3. Exercise 12 (a) How many angles can L make wit te z-axis for an l = 2 electron? (b) Calculate te value of te smallest angle. Exercise 13 (Solution on p. 10.) Wat angles can te spin S of an electron make wit te z-axis?

10 OpenStax-CNX module: m Solutions to Exercises in tis Module Solution to Exercise (p. 9) l = 4, 3 are possible since l < n and m l l. Solution to Exercise (p. 9) n = 4 l = 3, 2, 1, 0 m l = ±3, ± 2, ± 1, 0 are possible. Solution to Exercise (p. 9) (a) J s (b) J s Solution to Exercise (p. 9) (a) J s (b) s = J s (c) L S = 12 = 4 3/4 Solution to Exercise (p. 9) θ = 54.7º, 125.3º Glossary Denition 1: quantum numbers te values of quantized entities, suc as energy and angular momentum Denition 2: angular momentum quantum number a quantum number associated wit te angular momentum of electrons Denition 3: spin quantum number te quantum number tat parameterizes te intrinsic angular momentum (or spin angular momentum, or simply spin) of a given particle Denition 4: spin projection quantum number quantum number tat can be used to calculate te intrinsic electron angular momentum along te z-axis Denition 5: z-component of spin angular momentum component of intrinsic electron spin along te z-axis Denition 6: magnitude of te intrinsic (internal) spin angular momentum given by S = s (s + 1) Denition 7: z-component of te angular momentum component of orbital angular momentum of electron along te z-axis