= 襯ん , ア霽 0 ん D QUANTUM THEORY 襯 = 士 1 , 士 2 , 襯ん -7b ( 7 ー 14 ) Hence the component 0f angular momentum 0f the orbit along the 名 axis can assume only the quantized values which are integral multiples 0f ん / 2T. The quantum number is called the 襯 - netic 0 れ襯襯 6 催 , because it serves t0 distinguish the various slightly separated levels int0 which the field-free energy levels are split upon the application 0f a magnetic field t0 the atom. This quantum number is closely conne cted with the orientation 0f the old-quantum-theory orbit ⅲ space' a question discussed ⅲ Section 7d. The second integral is easily discussed by the introduction 0f the angle x and its conjugate momentum Px = 2 , the t0tal angular momentum 0f the system , bY means 0f the relation' given ⅲ Equation 1-41 , Section pxdx = d 十ル d . ln this way we obtain the equation $pxdx = んん , ( 7 ー 15 ) ( 7 ー 16 ) ⅲ which px is a constant 0f the motion and ん is the sum 0f が and 襯 . This integrates at once t0 = たん , 2 or た = 1 , 2 , ( 7 ー 17 ) Hence the total angular momentum 0f the orbit was restricted by the 01d quantum theory t0 values which are integral mul- tiples 0f the quantum unit 0f angular momentum ん / 2 The quantum number ん is called the 0 襯ん記 9 社 0 れ襯忸わな . To evaluate the first integral it is convenient t0 transform it ⅲ the following way, involving the introduction 0f the angle X and the variable れ = 1 を with the use 0f Equation 7 ー 6 : 1 du 2 dr 2 dx = ・ ル d = 市 r2 dx 2 dx From Equation 7 ー 10 we find 0 Ⅱ differentiation dx. を COS (X ー xo) dx ( 7 一 18 ) ( 7 ー 19 )

I -1 司 Ⅳルア 0 Ⅳ ' S EQUA TIONS 0 MO ア 70 Ⅳ 13 px and about different fixed axes, one 0f which, px, relates tO the axis normal t0 the plane 0f the motion. This is pxdx = 十鮖 d ( 1 ー 41 ) an equation easily derived by considering Figure 1 ー 3. The sides of the small triangle have the lengths r sin +dp, rdx, and rdB. Since they form a right triangle, these distances are conne cted by the relation r2(dx)2 = ド sin2 癶 ) 2 十ド ( ) 2 , which gives, 0 Ⅱ introduction of the angular velocities , 物 , and and multiplication by 襯 / 襯 , 襯ドえ dx = 襯ド sin2 d 十襯ド Bd Equation 171 follows from this and the definitions of px, , and . Conservation of angular momentum may be applied to more general systems than the one described here. lt is at once evident that we have not used the special form of the potential- energy expression except for the fact that it is independent of direction, since this function enters into the equation only. Therefore the above results are true for a particle moving ⅲ any spherically symmetric potential field. Furthermore, we can extend the theorem to a collection of point particles interacting with each other ⅲ any desired way but influenced by external forces only through a spherically symmetric potential function. If we describe such a system by using the polar coordinates 0f each particle, the Lagrangan function iS = 叫十第十 sin2 鬢 ) ー ( 1 ー 42 ) イ = 1 lnstead 0f , , coordinates 嘶 。 , we now introduce new angular , K given by the linear equations = 夜十 610 十 92 = 十 62 爆十・ = 伐十し調十・ The vames given the constants 61 , 十ん IK , 十ん的 ・十ん謎 . ( 173 ) んル are unimportant SO 10 Ⅱ g they make the above set Of equations mutually independ-

344 T 〃刃 STRUCTURE OF SIMPLE MOLECULES [XII-43a lt may be mentioned that the accuracy of the energy calculation is greater than appears from the values quoted for の。 , inasmuch the energy Of the electrons ⅲ the field 0f the two nuclei (differing from ル 8 by the term e2 / の at rAB = 1.500 is calculated to be 2 ー仏ー 18.1 v. e. , and the error of 1.5 v. e. is thus only a few per cent 0f the t0tal electronic interaction energy. lt is interesting and clarifying for this system also ()s for the hydrogen molecule-ion) t0 discuss the enei ・ gy function for a hypothetical case. Let us suppose that the wave function for the system were VI = 術 & ( 1 ) 1 & ( 2 ) a10 Ⅱ e. The energy of the system would then be HI 1, which is shown as curve Ⅳⅲ Figure -22 0 FIG. 43 ー 2. 43- ー 2. lt is seen that this curve gives only a small attraction between the two atoms, with a bond energy at equilibrium only a few per cent Of the observed value. The wave function 協 & differs from this function in the interchange 0f the coordinates Of the electrons, and we conse— 2 4 5 —Energy curves for the molecule is in the maln reso— of the bond ⅲ the hydrogen 5 6 quently say that the energy hydrogen molecule ()n units e2 / 200 ). nance or interchange energy. So far we have not taken into consideration the spins of the electrons. 0 Ⅱ d0ing this we find, exactly as for the helium atom, that in order tO make the complete wave functions anti- symmetric in the electrons, as required by Pauli's principle, the orbital wave functions must be multiplied by suitably chosen spin functions, becoming and 1 ( 1 ) 0 ②ー 0 ( 1 ) 伐② } 2 ・伐 ( 1 ) 夜 ( 2 ) , { 伐 ( 1 ) 0 ②十 1 ) ② } , の・ ( 1 ) 0 ② . 1 2 There are hence three repulsive states for one attractive state S ; the chance is } 4 that two normal hydrogen atoms

IV-12a] ル刃 EQ UA TION 0 は S YSTEM OF PARTICLES 85 the one-dimensional system, the only changes being those conse- quent tO the increase in the number Of dimensions 0f configura- tion space. A detailed account 0f the postulates made regarding the wave equation and its solutions for a general system 0f point particles is given ⅲ the following sections, together with a dis- cussion Of various simple systems for illustration. 12a. The Wave Equation lncluding the Time. —Let us con- sider a system consisting 0f point particles 0f masses 襯 1 , ? 〃Ⅳ moving in three-dimensional space under the influence 0f forces expressed by the potential function V ' ( ェ 1 ・ 2 Ⅳ being the 3 Cartesian coordinates ・名Ⅳ , の , 0f the ル particles. The potential function , representing the interaction 0f the particles with one another or with an external field or both, may b e a function 0f the 3 coordinates alone or may depend on the time also. The former case, with ・ 2 ) , corresponds tO conservative system. Our main interest lies in systems 0f this type, and we shall S00 Ⅱ restrict our discussion tO them. We assume with Schrödinger that the wave equation for this 2 十 system iS ん 2 1 ö2 ö2 重ö2 んöV 8 2 十 工挈 öz? 十平 = ( 12 ー 1 ) This equation is 0ften written as ー▽ : 平十平 イ = 1 2 2T 乞 ⅲ which ▽挈 is the ん 0 厄 ce 0 催襯 or ん 0 厄 c れ for the ith particle. 1 ln Cartesian coordinates, it is given bY the expression 2 十 The wave function 平 = 平 ( ェ 1 ・ 2 十一一 öz? ・ 2 Ⅳ , の is a function 0f the 3 coordinates 0f the system and the time. lt will be noted that the Schrödinger time equation for this general system is formally related t0 the classical ene.rgy equation ⅲ the same way as for the one-dimensional system 0f the preced- 1 The symbol △ is sometimes used in place 0f V2. The symbol ▽ 2 commonly read as del sq 社 0 d ・