286 T ″刃 ROTA ー 0 Ⅳ AND IB A770 Ⅳ 0 MO ん刃 CU ん刃 S [X-37a (after Secs. 24 , 26a , etc. ) , this set 0f equations possesses a solution = 0 0 Ⅱ ly when the other than the trivial one 1 = 2 corresponding determinantal equation (the secular equation 0f perturbation and variation problems) is satisfied. This equa- tion IS = 0. ( 37 ー 17 ) 3 れ 3 れ 6 37 一 16 and solve for the ratiosl 0f the 's. If we put Having found one Of these r00tS' we can substitute it ⅲ Equation satisfy Equation 37 ー 17. (Some 0f these roots may be equal. ) equations 0f motion only when 入 has one 0f the 3 れ values which ln Other words, Equation 37 ー 15 can represent a solution 0f the = Bk1Q?' and introduce the extra condition ( 37 ー 18 ) ( 37 ー 19 ) ⅲ which the subscript specifies which root 0f the secular equation has been used' then we can determine the values of the お ' s , Q? being left arbitrary. By this procedure we have obtained 3 れ particular solutions of the equations 0f motion' one for each root 0f the secular equation. A general solution may be obtained by adding all of these together, a process which yields the equations Q? ・ sin 十の ). 3 れ ( 37 ー 20 ) This solution 0f the equations 0f motion contains 6 れ arbitrary constants, the 0 襯紐 s and the phases の , which ⅲ any particular case are determined from a knowledge of the initial positions and velocities 0f the nuclei. We have thus solved the classical problöm 0f determining the positions Of the nuclei as a function 0f the time' glven any set of initial conditions. Let us now discuss the nature 0f the 1 These equations are homogeneous' SO that 0 Ⅱ ly the ratios Of the can be determined. The extra condition 37 ー 19 on the Bu's then allows them tO be completely determined.

I-2a] EQUA TIONS 0 0770 ル / Ⅳ AM / TO ⅣーⅣ 0 社 15 restricted t0 be a function 0f the coordinates 0 Ⅱ ly , can be written öアöL た = 1 , 2 , 3 れ . ( 2 ー 1 ) Angular momenta can likewise be expressed in thiS manner. Thus, for one particle ⅲ a spherically symmetric potential field, the angular momentum about the 名 axis was defined in Section le by the expression = mp = 襯ド sin2 物 . ( 2 ) Reference t0 Equation 1 ー 31 , which gives the expression for the kinetic energy ⅲ polar coordinates shows that öT öL ( 2 ー 3 ) Likewise, ⅲ the case 0f a number 0f particles, the angular momentum comugate t0 the coordinate 伐 is öT öL ( 27 ) as shown by the discussion 0f Equation 176. BY extending this to 0ther coordinate systems, the generalized 襯 0 襯 e れ m ん conjugate t0 the coordinate is defined as öL ん = 1 , 2 , 3 れ . ( 2 ー 5 ) The form taken by Lagrange's equations ( q. 1 ー 29 ) when the definition Of 2 ん is introduced is öL ö(lk ん = 1 , 2 , 3 れ , ( 2 ) so that Equations 5 and 2 ー 6 form a set 0f 6 れ first-order dif- ferential equations equivalent tO the 3 れ second-order equations of Equatio n 1 ー 29. öL being ⅲ general a function 0f both the q's and ä'S' the ödlk definition 0f ん gwen bY Equation 2 ー 5 provides 3 れ relations between the variables , , and ん , permitting the elimination of the 3 れ velocities , SO that the system can now be described ⅲ terms Of the 3 れ coordinates 9 ん and the 3 れ conjugate momenta

20 SUR 刃 Y OF CLASSICAL MECHANICS [I-2d lt is noticed that the last six 0f these equations ( 2 ー 26 , 27 ) are identical with the equations which define the momenta involved. An inspection Of Equations 2 ー 25 indicates that they are closely related to Equations 1 ー 33 , 1 ー 34 , and 1 ー 35. If ⅲ these equations 襯 is replaced by and if ⅲ Equation 1 ー 35 4 ド襯鴫 r is replaced by öV ーー , we obtain just the equations which result from substituting 0r for , , の their expressions ⅲ terms 0f テ , and のⅲ Equation 2 ー 25. The first three, 2 ー 24 , show that the center 0f gravity of the system moves with a constant velocitY' while the next three are the equations Of motion 0f a particle 0f mass 料 bound t0 a fixed center by a forc e whose potential-energy function is ). This problem illustrates the fact that ⅲ most actual problems the Lagrangian equations are reached ⅲ the process 0f solution of the equations 0f motion in the Hamiltonian form. The great value of the Hamiltonian equations lies ⅲ their particular suita- bility for general considerations , such as , for exampl% Liouville 's theorem ⅲ statistical mechanics, the rules 0f quantization ⅲ the 01d quantum theory, and the formulation 0f the Schrödinger wave equation. This usefulness is in part due t0 the symmetrical or conjugate form 0f the equations in P and Prob1em 2 ー 1. Discuss the motion 0f a charged particle ⅲ a uniform electric field. ProbIem 2 ー 2. S01ve the equation 0f motion for a charged particle 0 「 mass 襯 constrained tO move on the 工 axis in a uniform electric field (the potential energy due t0 the field being ¯eFC' where e is the electric charge constant) and connected tO the origin bY a spring Of force constant ん . ー e at the ongin, Obtain an expression for the Assuming a fixed charge average electric moment Of the system as a function Of the quantities 襯 , ん , and the energy 0f the system. See Equation 3 ー 5. Prob1em 2 ー 3. Derive an expression for the kinetic energy 0f a particle ⅲ terms Of cylindrical coordinates and then treat the equations Of motion for a cylindrically symmetrical potential function. ProbIem 2 ー 4. Using spherical polar coordinates, solve the equations 0f motion for a free particle and discuss the results. ProbIem 2 ー 5. Obtain the solution for x ⅲ Section ld. ProbIem 2 ー 6. By eliminating the time in the result 0f Problem 2 ー 5 and the equation for アⅲ section ld, shOW that the orbit Of the particle is an ellipse. ProbIem 2 ー 7. prove the identity of the motion 0f the plane isotropic harmomc oscillator found by solution ⅲ Cartesian and polar coordinates. ProbIem 2 ー 8. Show ow to obtain an immediate integral 0f one equation Of motion, the Lagrangian function does not involve the corresponding

CHAPTER IV THE WAVE EQUATION FO A SYSTEM 0 POINT PARTICLES IN THREE DIMENSIONS 12. THE WAVE EQUATION FOR A SYSTEM OF POINT PARTICLES The Schrödinger equation for a system of ル interacting point particles in three-dimensional space is closely similar t0 that for the simple one-dimensional system treated ⅲ the preceding chapter. The time equation is a partial differential equation ⅲ 3 ル十 1 independent variables (the 3 Cartesian coordinates, say, 0f the particles, and the time) instead 0f only two inde- pendent variables, and the wave function is a function 0f these 3 ル十 1 variables. The same substitution as that used for the simpler system leads t0 the separation 0f the time equation intO an equation involving the time alone and an amplitude equation involving the 3 coordinates. The eauation involving the time alone is found t0 be the same as for the simpler system, so that the time dependency 0f the wave functions for the stationary states of a general system 0f point particles is the same as for the one-dimensional system. The amplitude equation, however, instead of being a t0tal differential equation ⅲ one independent variable, is a partial differential equation ⅲ 3 independent variables, the 3 coordinates. lt is convenient t0 say that this iS an equation in a 3N-dimensional configuration meaning by this that solutions are to be found for all values 0f the 3 ー to 十 . The 2 Ⅳ from Cartesian coordinates 工 1 amplitude function, dependent on these 3 eoordinates, is said t0 be a function ⅲ configuration space. A や 0 ⅲ t ⅲ configuration space corresponds t0 a definite value 0f each 0f the 3 coordi- ・ 2 Ⅳ , and hence t0 definite positions 0f the ル particles nates 工 1 ⅲ ordinary space, that is, t0 a definite configuration 0f the system. The wave equation, the auxiliary conditions imposed on the wave functions, and the physical interpretation 0f the wave functions for the general system are closely similar t0 those for 84

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 .

X -37 可 T ″刃 / B / 0 Ⅳ 0 PO ん Y TO 既 / C 」一 0 ん ECU. ん刃 s 285 coordinates undergoes harmonic oscillation, the frequency being determined by the constant いた . Now it is always possible by a simple transformation of variables tO change the equations of motion from the form 37 ー 8 to the form 37 ー 9 ; that is, to eliminate the cross-products from the potential energy and at the same time retain the form 37 ー 3 for the kinetic energy. Let us call these new coordinates QI(I = , 3 れ ). ln terms of them the kinetic and the potential energy would be written and = 入・ Q and the solutions of the equations of motion would be ( 37 ー 11 ) ( 37 ー 12 ) = Q?sin ( 、 / 十の ) , I = 1 , 2 , , 3 れ . ( 37 ー 13 ) lnstead of finding the equations of transformation from the q's tO the Q's by the consideration of the kinetic and potential energy functions, we shall make use of the equations of motion. ln case that all of the amplitude constants QIO are zero except one, Q?, say, then QI will vary with the time in accordance with Equation 37 ー 13 , and, inasmuch as the q's are related to the 3 れ Q's by the linear relation ( 37 ー 14 ) each 0f the q's will vary with the time ⅲ the same way, namely, = いⅲ ( ~ 爪こ十 61 ) , ん = 1 , 2 , , 3 ル ( 37 ー 15 ) ln these equations スん represents the product おた IQ % and 入 the quantity 入 1 , inasmuch as we selected QI as the excited coordinate ; the new symbols are introduced for generality. On substituting these expressions in the equations 0f motion 37 ー 8 , we obtain the set 0f equations 3 れ ー、十い、 = 0 , = 1 先 = 1 , 2 , , 3 れ . ( 37 ー 16 ) This is a set of 3 れ simultaneous linear homogeneous equations 洫 the 3 れ unknown quantities スた . As we know well by this time

296 ~ 刃 TU 記 BÄ TION T ″刃 0 一Ⅳ 0 一Ⅳ G T 社アー霽 [XI-39a ö重 0 2T を 2 The first and last terms ⅲ this equation cancel ()y Eqs. 39 ー 1 and 39 ー 2 ) , leaving 2 乞 If we now multiply by and integrate over configuration space, noting that all terms 0 Ⅱ the left vanish except that for れ = 襯 because 0f the orthogonality properties 0f the wave functions' we obtain ( 3 6 ) This is a set of simultaneous differential equations ⅲ the functions 0 。 ( の , by means 0f which these functions can be evaluated ⅲ particular cases. 39a. A Simp1e Examp1e. —As an illustration 0f the use 0f the set of equ ations 39 ー 6 , -- - Ⅱ s... -. 、四旦 si は . - ー . ↓ hat a. t. - ー the -- ー ti 埠 0 ー上ー〒一 0 ・・ we knoy-that a, system ⅲ .. Vhi. ch weare interested is ⅲ a particulgr stationary sta our knowledge perhaps having been obtained y a measurement Of the energy 0f the system. The wave func- tion representing the system is then 平尸 , ⅲ which has a particular value. If a small perturbation 〃′ acts 0 Ⅱ the system for a short time , 〃′ being independent Of こ during this period, we may solve the equations 39 ー 6 by neglecting all terms on the right side except that with れ = ; that is, by assuming that only the term ⅲの ( の need be retained 0 Ⅱ the right side 0f these equations. lt is first necessary for us t0 discuss the equation for の itself. This equation ( 刊 q. 39 ー 6 with = I and = 0 for れ l) ・ s れ = 0 d の ( の 2 ーの ( のゎ 7 ⅲ which 〃 工 ? * 〃 ! 市 , which can be integrated at once t0 ー 2 i を / ん の ( の ( 3 7 )

111-91 ア〃お SC ″ÖD / Ⅳ G 刃 WA EQ UA 770 Ⅳ 51 the hydrogen atom ⅲ an electric field (Stark effect). For the last problem he developed his perturbation theory, and for the discussion of dispersion he also developed the theory of a perturbation varying with the time. His methods were rapidly adopted by other investigators, and applied with such success that there is hardly a field of physics or chemistry that has remained untouched by Schrödinger's work. Schrödinger's system of dynamics differs from that of Newton, Lagrange, and Hamilton ⅲ its aim as well as its method. lnstead of attempting to find equations, such as Newton's equations, which enable a prediction to be made of the exact positions and velocities 0f the particles of a system ⅲ a given state of motion, he devised a method of calculating a function of the coordinates of the system and the time (and not the momenta or velocities) , with the aid of which, ⅲ accordance with the interpretation developed by Born, 1 probable values 0 the coordinates and of other dynamical quantities can be predicted for the system. lt was later recognized that the acceptance of dynamical equa- tions 0f this type involves the renunciation of the hope of describ- ing ⅲ exact detail the behavior 0f a system. The degree of accuracy with which the behavior of a system can be discussed by quantum-mechanical methods forms the subj ect of 〃 e な e れ一 印 ' s れ ce び principle,2 to which we shall recur in Chapter XV. The Schrödinger wave equation and its auxiliary postulates enable us to determine certain functions 平 of the coordinates of a system and the time. These functions are called the Sc ん市れ 0 な じ砒℃ / 社れ c 0 れ s or ro し 0 い I 寘リ 0 ルを襯 de / れ厩あれ s. The square 0f the absolute value of a given wave function is interpreted as a 、 0 I リ市 s なんれ / れ c 0 れ for the coordinates of the system ⅲ the state represented by this wave function, as will be discussed in Section 100. The wave equation has been given this name because it is a differential equation of the second order ⅲ the coordinates of the system, somewhat similar to the wave equation 0f classical theory. The similarity is not close, however, and we shall not utilize the analogy ⅲ our exposition. Besides yielding the probability amplitude or wave function 平 , the Schrödinger equation provides a method 0f calculating values 1 M. BORN, Z. /. P んい . 37 , 863 ; 38 , 803 ( 1926 ). 2 r. HEISENBERG, Z. /. んい . 43 , 172 ( 1927 ).

I -1 可 Ⅳ刃ルア 0 Ⅳ ' S EQUA / 0 Ⅳ S 0 外 0 ア 70 Ⅳ 7 This gves the relation between any Cartesian component of velocity and the time derivatives of the new coordinates. Similar relations, Of course, hold for 広 and for any particle. The quantities の , by analogy with , are called generalized 怩ん c s , even though they do not necessarily have the dimensions of length divided by time (for example, の may be an angle). Since partial derivatives transform in just the same manner, we have öV ööェ 1 öVöY1 ö(li öVözn özn ö(li öVöCi öVöYi öVöZi öqi = 1 = ( 1 ー 20 ) Since Qiis glven by an expression ⅲ terms of and の which is analogous t0 that for the force Xi ⅲ terms of and , it is called ln exactly similar fashion, we have a generalized force. öT öTö±i öTöÜi öTöii ö4i öÜi ö4i ( 1 ー 21 ) を = 1 tion of Equation 1 ー 50 by ーー , of 1 ー 56 by 、一ご , etc. , gives tions 1 ー 5 , using the methods of the previous section. MuItiplica- proof we shall apply a transformation of coordinates to Equa- they are valid for any choice of coordinate system. For this ton's equations are written ⅲ the form given by Equation 1 一 7 gian FO て m. ・一、 Ve are now ⅲ a position to show that when New- lc. The lnvariance of the Equations of Motion ⅲ the Lagran- öCli öェ 1 d öT öqj ö上 1 öェ 2 d öT öqj 襯ö上 2 öェれイöT öV öェ 1 ö工 1 ö(li öV öェ 2 ö工 2 öqj ööェれ öれöqi ( 1 ー 22 ) with similar equations ⅲ and 名 . Adding all of these together gves

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・