16 SUR / 刃 0 CLASSICAL MECHANICS に -2c た . HamiIton ⅲ 1834 showed that the equations of motion can ⅲ this way be thrown into an especially simple form, involving a funétion 〃 of the Pk's and qk's called the 〃 0 襯を〃 0 れ 0 れ . 2b. The Hamiltonian Function and Equations. —For con- servative systems.. 1 we shall show that the function 〃 is the total energy (kinetic plus potential) 0f the system, expressed ⅲ terms of the Pk's and ' s. ln order to have a definition which holds for more general systems, we introduce ″ by the relation 3 れ ( 2 ー 7 ) definition we obtain for the total differential of ″ the equation the velocities through the use of Equation 2 ー 5. From the a function 0f the coordinates and momenta only, by eliminating Although this definition involves the velocities , ″ may be made 3 れ 3 れ 3 れ 3 れ or, using the expressions for た and た given ⅲ Equations 2 ー 5 and 2 ー 6 (equivalent to Lagrange's equations) , 3 れ dH = ( 2 ー 9 ) whence, if ″ is regarded as a function of the qk's and Pk's, we obtain the equations öH öPk öll öqk ん = 1 , 2 , 3 ル ( 2 ー 1 の These are the e 0 0 れ s 可襯 0 0 れをれ the 〃 0 襯を〃 0 れ血れ or ca れ 0 〃 0 こ form. 2c. The Hami1tonian Function and the Energy ・—Let us con- sider the time dependence of ″ for a conservative system. We have 1 A eonservative system is a system for which does not depend explicitly 0 Ⅱ the time t. We have restricted our discussion to conservative systems by assuming that the potential function does not depend on こ .

104 ル刃 EQUA ア 70 Ⅳ 0 記 SYSTEM OF PARTICLES [IV-16 coordinates the same technique of solution may often be applied. ln order t0 make such a transformation, which may be repre- sented by the transformation equations リ = 〆もら ) , ( 16 ー 1 の ( 1 1 の ( 16 ー lc ) it is necessary tO know what form the Laplace operator ▽ 2 assumes in the new system, since this operator has been defined only in Cartesian coordinates by the expression 2 十 2 十一 - 0 名 2 ( 1 2 ) The process Of transforming these second partial derivatives is a straightforward application 0f the principles 0f the theory 0f partial derivatives and leads t0 the result that the operator ▽ 2 ⅲ the orthogonol coordinate system has the form 1 加öu öu öv % öv ⅲ which 十 2 öu öu 2 2 2 十 十 十 öw 加öw 2 2 öz öu 2 öz öv ( 1 3 ) ( 16 ー 4 ) 十 2 2 十 öv öw öw Equation 16 ー 3 is restricted tO coordinates , ら幟 which are orthogonal, that is, for which the coordinate surfaces represented constant, and = constant by the equations = constant, 2 intersect at right angles. All the common systems are 0f this type. The volume element 市 for a coordinate system of this type is also determined when ? ,. , , and ク” are known. lt is given by the expression 靃 = d 工面 ( 16 ー 5 ) ln Appendix IV, ク。 , , ?. 。 , and ▽ 2 itself are given for a number 0f lmportant coordinate systems.

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 ・

56 T 〃刃 SCHRÖDINGER ルは「刃 EQ UA ア / 0 Ⅳ [III -9b the hydrogen atom are subsequently verified bY experiment• We might then describe the hydrogen atom bY giving its wave equation ; this description would be eomplete. lt is unsatis- factory, however, because it is unwieldy. On observing that there is a formal relation between this wave equation and the classical energy equation for a system 0f two particles 0f different masses and electrical seize 0 Ⅱ providing a' simple, easy, and familiar way 0f describing the system' and we say that the hydrogen atom consists 0f two particles' the electron and proton, which attract each 0ther according t0 Coulomb's inverse-square law. ActuallY we d0 not know that the electron and roto Ⅱ attract each other ⅲ the same way that two macroscopic electrically charged bodies d0' inasmuch as the force between the two particles in a hydrogen atom has never been directly measured. AII that we d0 know is that the wave equation for the hydrogen atom bears a certain formal relation tO the classical dynamical equations for a system Of tW0 particles attracting each Other in this way. Having emphasized the formal nature 0f this correlation and of the usual description 0f wave-mechanical systems in terms 0f classical concepts, let us now P0int out the extreme practical importance 0f this procedure. lt is found that satisfactory wave equations can be formulated for nearly all atomic and molecular systems by accepting the descriptions 0f them developed during the days 0f the classical and 01d quantum theory and translating them into quantum-mechanical language bY the methods discussed above. lndeed, wave¯mechanical expressions for values 0f experiméntally observable properties 0f systems are identical with those given bY the 01d quantum theory' and ⅲ other cases only small changes are necessary. Throughout the following chapters we shall make use 0f such locutions a system Of tWO particles with inverse-square attraction instead Of ' ' O system whose wave equation six nates and a function e2 / 123 ' etC. 9b. The Amp1itude Equation. —ln order to solve Equation 9 ー 1 , let us ()s is usual ⅲ the so ⅲ ti011 0f a partial differential equation of this type) first study the solutions 平 ()f any exist) which can be expressed as the product 0f two functions, one involving the time alone and the 0ther the coordinate alone : ェ , の協@ル ( の・

318 2 刃 7 ' U B T 0 Ⅳ T 刃 0 1 Ⅳ 0 ん一Ⅳ G T ″刃 TIMB I-41b 工 2 ⅲ the description 0f the system ; that , t0 speak Of the motion of the pendulums individually rather than Of the system as a WhOle. 、、一 e can conceive Of a11 arrangement Of levers whereby an indicator ⅲ an adj acent room would register values 0f , and another values Of 小 An observer ⅲ this room would say that the system was composed Of two independent harmonic oscillators with different frequencies and constant amplitudes' and would not mention resonance all. Despite the fact that we are not required tO introduce the concept Of resonance in mechan-ical systems has been found tO be very useful ⅲ the description 0f the motion 0f sys- tems WhiCh are for S01 れ e reason or Other conveniently described as containing interacting harmonic oscillators. lt is found that a similar state 0f affairs exists ⅲ quantum mechanics. Quantum-mechanical systems WhiCh are conveniently considered tO ShOW resonance occur more than resonating classical systems' the resonance phenomenon has come t0 play an especially important part ⅲ the applications 0f quantum mechanics t0 chemistry. 41b. Resonance ⅲ Quantum Mechanics. ¯ln order t0 illus- trate the resonance phenomenon quantum us continue t0 discuss the system 0f interacting harmonic oscil- lators. 1 Using the potential function 0f Equation 41 ー 1 , the wave equation can be at once separated ⅲ the coordinates and and solved ⅲ terms 0f the Hermite functions. The energy levels are given by the expression 2 ー入 , ( 41 ー 6 ) ル平 = ( 墲十驪 ) ん十 x 十 ( れ。十 ) ん which for 入 small reduces t0 the approxlmate expression 8 鴫 ル ( れ十 1 ) 0 十 ( 墲ーれ。 ) ー ん入 ( れ十 1 ) ん入 2 ( 41 ー 7 ) phenomenon, z. 工 p んい . 38 , 411 ( 1926 ) : 41.239 ( 1927 ). 1 This example was used by Heisenberg in his first papers 0 Ⅱ the resonance nates ミ and makes Ⅱ 0 direct reference tO resonance. us This treatment, like the classical treatment using the coordi- equally spaced levels. 41 ー 2 ; for a given value 0f れ there are れ十 1 approximately ⅲ which れ = 墲十 . The energy levels are shown ⅲ Figure

422 GENERAL T 刃 0 Y OF QUÄNTUM MECIIANICS [XV-51b 2 冠Ⅳ 0 of the wave equation for any system correspond t0 a diagonal ene rgy matrix 0 0 0 0 0 ル 2 0 0 0 3 so that, as mentioned ⅲ Section 10C , the system ⅲ a physical condition represented by one Of these wave functions has a fixed value 0f the t0tal energy. ln the case Of a system with one degree Of freedom Ⅱ 0 Other dynamical quaqtity (except functions 0f onlY' such as 〃 2 ) is represented by a diagonal matrix ; with more degrees Of freedom there are other diagonal matrices. For exampl% the surface- harmonic wave functions ( ) 中襯 ( の for the hydrogen atom and Other two-particle systems separated in P01ar coordinates (Secs. 19 , 21 ) make the matrices for the square 0f the total angular momentum and the c omponent Of angular momentu m along the 2 axis diagonal, these dynamical quantities thus having definite values for these wave functions. The properties 0f angular momentum matrices are 52. The dynamical quantities corresponding t0 diagonal matrices relative t0 the stationary-state wave functions 平 0 , 重 1 , are sometimes called co s れな可 the 襯 0 0 れ 0f the system. The corresponding constants 0f the motion 0f a system ⅲ classical mechanics are the constants Of integration 0f the classical equa- tion Of 1 Ⅱ OtiO Ⅱ . Let us now consider a system whose Schrödinger time functions corresponding t0 the stationary states 0f the system are 0 , 平 1 Suppose that we carry out an experiment (the measurement Of the values 0f some dynamical quantities) such as t0 determine the wave function uniquely. Such an expenment iS called a ~ 0 〃 ~ ? れ e0S47 ℃ ~ e measurement for a system with one degree Of such as the one-dimensional harmonic oscillator, might consist ⅲ the accurate measurement 0f the energy ; the result 0f the measure- ment would be one 0f the characteristic energy values ・ル。 ; and the corresponding wave function 。 would then represent the

Ⅱ I -11a1 社ん盟 0 Ⅳ / C OSC / ん TO / WA 刃 MECHANICS 67 Even if the system is ⅲ a stationary state, represented by the wave function 平れ@, の = 協れ ( の e , only an average ん can be predicted for an arbitrary dynamical quantity. The energy 0f the system, corresponding t0 the Hamiltonian function 〃朝ェ , の , has, however, a definite value for a stationary state 0f the system, equal t0 the characteristic value found 0 Ⅱ solution 0f the wave equation, so that the result 0f a measurement of the energy of the system ⅲ a given stationary state can be predicted accurately. TO prove this, we evaluate 〃 ' and ( 〃 ) ,. ″ is given by the integral 十 ( の ん 2 d2 ( の 十以の ( の , 8 ド m 2 the factor involving the time being equal t0 unity. This trans- forms with the use 0f Equation 9 ー 8 int0 or, smce iS a constant and ( 1g7 ) 0 Ⅱ ly because this provides a good illustration Of the methods system we choose the one-dimensional harmonic oscillator' 0f the solution of the Schrödinger wave equation for a dynamical 11a. S01ution of the Wave Equation. —As our first example 11. THE HARMONIC OSCILLATOR IN WAVE MECHANICS stationary states. we shall restrict the discussion mainly tO the properties 0f under given circumstances will be treated. the earlier sections of deciding which wave function t0 associate with a given system chapters, and especially ⅲ Chapter ⅲ which the question oscillator ⅲ this chapter and 0f 0ther systems ⅲ succeeding will be given in connection with the treatment 0f the harmonic Further discussion Of the physical significance 0f wave functions definite value ルれ . argument set forth above, the energy 0f the system has the be equal t0 ( 〃 ) ' , ⅲ consequence 0f which, in accordance with the it is seen that 〃 ' is equal t0 Ⅱ . 尾 have thus shown 〃 ' t0 By a similar procedure, involving repeated use 0f Equation 9 ー 8 ,

I-2dJ EQUA TIONS 0 0 0 Ⅳ / Ⅳ″ス」 / ん 0 Ⅳ / Ⅳ 0 17 dll 襯 3 れ 襯 3 れ 3 れ öL ん薩十第ん öqk öL ( 2 ー 11 ) using the same substitutions for ん and ん (Eqs. 2 ー 5 and 2 ー 6 ) as before. 〃 is hence a constant of the motion, which is called the e れ e of the system. For Newtonian systems, ⅲ which we shall be chiefly interested, the Hamiltonian function is the sum 0f the kinetic energy and the potential energy, ″ = 実十 ( 2 ー 12 ) expressed as a function of the coordinates and momenta. This is proved by considering the expression for for such systems. For any set of coordinat es , will be a homogeneous quadratic function 0f the velocities 1 け = 1 where the at7's may be functions of the coordinates. 3 れ Hence 3 れ so that 3 れ 3 3 れ ( 2 ー 13 ) ( 2 ー 14 ) 0 = 2 イ議 = 1 〃 = 2 一 = ア十に 2d. A GeneraI Examp1e. ー・ The use of the Hamiltonian equa- tions may be illustrated by the example of two point particles With masses ? れ 1 and respectively, moving under the influence 0f a mutual attraction given by the potential energy function ( の , ⅲ which is the distance between the two particles. The hydrogen atom is a special case of such a system, so that the results obtained below will be used ⅲ Chapter II. Ⅱ the coordi- nates of the first particle are 工 1 , リ 1 , 名 1 and those of the second の 2 , 42 , 22 , the Lagrangian function ん is

XI-41b] T ″刃刃ÅSO Ⅳ C 刃 ~ ″刃Ⅳ 0 ノ If 刃Ⅳ 0 Ⅳ 321 gwen by or approximated by linear c ombinations 0f the initially chosen functions, as found by perturbation or variation methods ・ and various points of analogy between this treatment and the classical treatment of the resonating system have been indicated (see also the following section). ln discussing more complicated systems it is often convenient t0 make use of similar methods Of approximate solution 0f the wave equation, involving the forma- tion Of linear combinations of certain initially chosen functions. The custom has arisen of describing this formation of linear combinations 1 Ⅱ certain cases as corresponding tO resonance the system. ln a given stationary state the system is said t0 resonate among the states or structures corresponding tO those mitially chosen wave functions which contribute t0 the wave function for this stationary state, and the difference between the energy 0f the stationary state and the energy corresponding to the initially chosen wave functions is called so れ 0 れ ce energy. 1 lt is evident that any perturbation treatment for a degenerate level ⅲ which the initial wave functions are not the correct zeroth-order wave functions might be described as involving the resonance phenomenon. Whether this description would be applied or not would depend 0 Ⅱ how important the initial wave functions seem tO the investigator, or hOW convenient thiS description iS in hiS discussion. 2 resonance phenomenon, restricted classical mechanics to interacting harmonic oscillators, is 0f much greater importance in quantum mechanics, this being, indeed, one 0f the most striking differences between the 01d and the new mechanics. lt arises, for example, whenever the system under discussion contains tWO or more identical particles, such as tWO electrons or tWO protons ; and it is also convenient t0 make use Of the terminology ⅲ describing the approximate treatment given the structure 0f polyatomic molecules. The significance Of the phenomenon for many-electron atoms has been seen from the discussion 0f the structure Of the helium atom given ⅲ Chapter Vlll ; it was there pointed out (Sec. 29 の that the splitting 0f levels due to the K 1 There is Ⅱ 0 close classical analogue Of resonan ce energy. 2 The same arbitrariness enters in the use Of t,he word resonance in describ— ing classical systems , inasmuch as if the interaction 0f the classical oscillators is mcreased the motion ultimately ceases tO be even approximately repre- sented by the description 0f the first varagraph 0f Section 41 広

II-5cl ア″ E 0 G / Ⅳ 0 をア〃刃 0 ん D QUANTUM T ″刃 0 ん Y 29 of the fine structure of the spectra of hydrogen and ionized helium, their Zeeman and Stark effects, and many other phe- nomena. The first step of their method consists ⅱ 1 solving the classical equations 0f motion ⅲ the HamiItonian form (Sec. 2 ) , therefore making use of the coordinates 1 , ・ , ク 3 れ and the canonically conjugate momenta 1 , , 3 。 as the independent variables. The assumption is then introduced that only those classical orbits are allowed as stationary states for which the following conditions are satisfied : d = 衂ん , ん = 1 , 2 , れた = an integer. ( 5 ー 2 ) 3 れ ; These integrals, which are called ac 0 れこ 7 な , can be calcu- lated only for conditionally periodic systems ; that is, for systems for which coordinates can be found each of which goes through a cycle as a function 0f the time, independently of the others. The definite integral indicated by the symbol 工 is taken over one cycle 0f the motion. Sometimes the coordinates can be chosen ⅲ several different ways, in which case the shapes of the quantized orbits depend on the choice of coordinate systems, but the energy values do not. shall illustrate the application of this postulate to the determination of the energy levels of certain specific problems ⅲ Sections 6 and 7. Selection Ru1es. The Correspondence Princip1e. —The 01 quantum theory did not provide a satisfactory method of cal- culating the intensities 0f spectral lines emitted or absorbed by a system, that is, the probabilities of transition from one sta- tionary state t0 another with the emission or absorption of a phOton. Qualitative information was provided, however, by an auxiliary postulate, known as Bo んド s correspondence れ c ゆ le , which correlated the quantum-theory transition probabilities with the intensity of the light of various frequencies which would have been radiated by the system according to classical electro- magnetic theory. ln particular, if Ⅱ 0 light of frequency cor- responding tO a given transition would have been emitted classically, it was assumed that the transition would not take place. The results Of such considerations were expressed in selection rules. For example, the energy values ん of a harmonic oscillator ( given ⅲ the following section) are such apparently to