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

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.

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

lll-llc] ″ RMONIC OSC ーんんア 0 7 Ⅳ WA 刃 MECHANICS れ ! 十 2 ー 2 79 Since this equation is true for all values 0f s, the coeffcients 朝 individual powers Of S must vanish, giving as the recursion formula for the Hermite polynomials the expression ( 11 ー 15 ) 〃。 + 1 ( の一 2 ″れ ( 9 十 2 ? 卍← 1 ( 9 Similarly, by differentiation with respect tO ミ , we derive the equation öS ー 2sS, which gives, ⅲ just the same manner as above' the equation = 2 れ″← 1 ( の , or ( 11 ー 16 ) involving the first derivatives 0f the Hermite polynomials. This can be further differentiated with respect t0 ミ t0 obtain expressions involving higher derivatives. Equations 11 ー 15 and 11 ー 16 lead t0 the differential equation for ″れ ( , for from 11 ー 16 we obtain ″ : ( の = 2 れ″に I(E ) = 4 れ ( れ一 1 ) ″← 2 ( の , while Equation 11 ー 15 may be rewritten as ″。 ( 一 2 ミ″← 1 ( 十 2 ( れ一 1 ) 〃← 2 ( 9 = 0 , ( 11 ー 18 ) which becomes, with the use 0f Equations 11 ー 16 and 11 ー 17 , ( 11 ー 17 ) 2 ミ or 〃 : ( めー 2 韓 : ( の十 2 れ〃。 (E ) ( 11 ー 19 ) This is just the equation, 11 ー 6 , which we obtained from the har- monic oscillator problem, if we put 2 れⅲ place 0f 1 , as required by Equation 11 ー 8. Since for each integral value 0f れ this equation has 0 Ⅱ IY one solution with the proper behavior at

VI-241 FIRS T-ORDER P 刃 T UR BA T70 Ⅳア霽 0 169 all the , ド s ・ If we now multiply both sides of this equation by : and integrate over configuration space, we obtain the result ル〃′ ). ( 24 ー 12 ) The left side of this equation is zero because 協 and , are orthogonal if ん / 雇 and ルール is zero if ん = ん′ . Ⅱ we introduce the symbols = ル蹼ル , we may express Equation 24 ー 12 ⅲ the form 国ー△川 D = 0 , and = 1 , 2 , 3 , ( 24 ー 13 ) ( 24 ー 14 ) , 伐 . ( 24 ー 15 ) ThiS 沁 a system of 夜 homogeneous linear simultaneous equations in the 伐 unknown quantities い , “ . Written out in full, these equations are ( 丑気 △ 11 ル : D 新十 ( 〃 : 2 △ 21 ル気 ) 新十 ( 42 △。 1 D 新十田 22 △ 12 ル ) 2 十 ・十 △ 22 ル ) 2 十 ( 。ー△ 2 。瑕 D 。 = 0 , △。 2 ル ) 2 十 ( 24 ー 16 ) Such a set Of equations can be solved only for the ratios of the K'S ; i. e. , any one ん may be chosen and all of the others expressed ⅲ terms 0f it. For an arbitrary value of , however, the set 0f equations may have Ⅱ 0 solution except the trivial one Ku, ー lt is only for certain values Of 幵物 that the set of equations has non-trivial solutions ; the condition that must be satisfied if such a set Of homogeneous linear equations is to have non-zero solutions is that the determinant of the coeffcients of the unknown quantities vanish ; that is, that

VII -27d 】 0 THER APPROXIMA TE ME THODS 203 The differential equation 27 ー 39 is the relation between the curvature at a point and the function ん 2 ( ルー均 at that 2 point, SO that we may approximate to the differential equation by the set 0f equations 27 ー 40 , there being one such equation for each point . The more closely we space the points 明 , the more accurately d0 Equations 27 ー 40 correspond to Equation 27 ー 39. Just as the lowest energy 幵 ' of the differential equation can be obtained by minimizing the energy integral E = 工の * ″のア with respect t0 the function の , keeping 工の * ゆ 1 , so the lowest value of ル giving a solution of Equations 27 ー 40 may be obtained by minimizing the quadratic form ( 27 ー 43 ) ⅲ which の 1 , の 2 , are numbers Which are varied until 刃 is a minimum. (Just as の must obey the boundary conditions 0f Section 9c, so the numbers ゆィ must likewise approxl- mate a curve WhiCh iS a satisfactory wave function. ) A convenient methodl has been devised for carrymg out this minimization. A set 0f trial values 0f の一 is chosen and the value 0f 刃 is calculated from them. The true solutions , t0 which the values 0f will converge as we carry out the variation, satisfy Equations 27 ー 40. Transposing one 0f these glves ー 1 十 + 1 2 ー。 2 ん 2 { ルー } ( 27 ー 44 ) If the +i's we choose are near enough tO the true values , then it can be shownl that, by putting 転ー 1 and + 1 ⅲ place 0f ー 1 and + 1 and 刃ⅲ place 0f ルⅲ Equation 27 ー 44 , the resulting expression glves an improved value の : for のわ namely, ー 1 十転 + 1 2 ー。 2 ん 2 囮ー取 } ( 27 ー 45 ) ln this way a new set の 1 , ゆ 2 , ・ can be built up from the initial set の 1 , ゆ 2 , the new set giving a 1 G. E. KIMBALL and G. Ⅱ . SHORTLEY, P ん . 北 e 既 45 , 815 ( 1934 ).

VII -27a 】 OT ″霽れス pp OX ー町は TE METHODS 193 or an arbitrary choice of the functions 。 ( の Equation 27 ー 5 represents an infinite number Of equations ⅱ 1 an infinite number Of unknown coemcients …・ Under these circumstances ques- tions Of convergence arise which are not always easily answered. ln special cases, however, only れ finite number of functions F 鼠ェ ) will be needed to represent a given function 協 ( ェ ) ; ⅲ these cases we know that the set of simultaneous homogeneous linear equations 27 ー 5 has a non-trivial solution only when the deter- minant Of the coeffcients 0f the , 、 's vanishes ; that is, when the condition 〃 11 ール〃 12 is satisfied. We shall assume that ⅲ the infinite case the mathe- matical questions 0f convergence have been settled, and that Equation 27 ー 7 , involving a convergent infinite determinant, 沁 applicable. Our problem is now in principle solved : We need only to eval- uate the roots 0f Equation 27 ー 7 to obtain the allowed energy values for the original wave equation, and substitute them ⅲ the set 0f equations 27 ー 5 to evaluate the coeffcients れ and obtain the wave functions. The relation of this treatment to the perturbation theory of Chapter VI can be seen from the following arguments. Ⅱ the functions 。 ( ェ ) were the true solutions 鼠ェ ) of the wave equation 27 ー 1 , the determinantal equation 27 ー 7 would have the form 0 ル 2 ール 0 0 0 ″ 22 ール ″ 23 = 0 ( 27 ー 7 ) 〃 33 ール 0 0 一 1 ( 27 ー 8 ) with roots ル = ル 1 , Ⅳ = ル 2 , etc. Now, if the functions F の closely approximate the true solutions 。 ( の , the non- diagonal terms ⅲ Equation 27- ー 7 will be small, and an approxi- mation we can neglect them. This gives ル 1 = 〃 11 Ⅳ 2 = 〃 22 , ル 3 = ″ 33 , etc. , 3 ( 27 ー 9 )

X -37 切 7 ' ″刃 / お TION 0 20 ん必 TOMIC 」 0 ん刃 CU. ん刃 S 289 tion 34 ー 4. ln terms Of the Cartesian coordinates previously described (Fig. 37 ー 1 ) , we write By changing the scale of the coordinates as indicated by Equation 37 ー 2 we eliminate the M's, obtaining for the wave equation the 3 れ 1 ö2V 4 ムö〆 2 ( 37 ー 26 ) = 1 expresslon ) : 当十 ( ルー v ル = 0. 3 れ ( 37 ー 27 ) We now introduce the normal coordinates QI. The reader can easily convince himself that an orthogonal transformation will leave the form 0f the first sum in the wave equation unalterod. , so that, using also Equation 37 ー 23 , we obtain the wave equation in the form 3 れ 3 れ ( 37 ー 28 ) This equation, however, is immediately separable into 3 ル one-dimensional equations. We put 協 = 協 1 ( QI ) 協 2 ( Q2 ) ・ and obtain the equations d2 れ 2 十一可ルん一一入ん Q れ dQk2 ( 37 ー 30 ) ( 37 ー 29 ) each 0f which is identical with the equation for the one-dimen- sional harmomc oscillator (Sec. 11 の . The t0tal energy W is the sum Of the energies 幵气 associated with each normal coordi- 3 れ nate ; that is, ( 37 ー 31 ) see that Applyin g this to the problem 0f the polyatomic mole cule, we tum number and the classical frequency Of the oscillator. Section 110 to have the values @ 十 ) ん , where is the quan- The energy levels Of the harmonic oscillator were found ⅲ

273 一 B んス TION 0 DIÄTOMIC 」 0 CU. も & X -35 d2S 1 dS 8 籠ル の一 co 2D C1 D ー C2 S = 0 , , 十ッ・十 02 ん 2 ( 3 ト 31 ) in which The substitutions の = e 222 ② , 2 2d 8 2 (D 十 C2), 02 ん 2 32k2 ( ルー D ー cD 02 ん 2 simplify Equation 35 ー 31 considerably, yielding the equation d2F 6 十 1 ( 3 ト 34 ) ー 1 - 一十 -F = 0 , 2 472 1 ()D ー CI) ー ( 6 十 1 ). ( 35 ー 35 ) 02 ん 2d 2 Equation 3 ト 34 is closely related to the radial equation 18 ー 37 0f the hydrogen atom and may be solved ⅲ exactly the same manner. If this 沁 done, it is found that it is necessary t0 ・ⅲ order t0 obtain a poly- restrict 紗 to the values 0 , 1 , 2 nomial so ⅲ tio Ⅱ .1 Ⅱ we solve for ル by means 0f Equations 35 ー 35 and the definitions of Equations 3 ト 33 , 35 ー 32 , and 35 一 28 , we obtain the equation (D 一 ) 2 ah(D ー 1/2CI) 1 ル = D 十 co ー 2 十一 (D 十 C2) 2 1 02 ん 2 十一 2 8T 2 1 The solutions for integral satisfy the boundary conditions F → 0 ー instead of 「→ 0 (Sec. 3 ). 3 3 are co = 1 十 0 ケ 2 、 ). 3 ケ 0 0 6 4 0r6 ( 3 32 ) 02r : ( 3 33 ) ⅲ which 2

XII-42cJ T 〃〃 D 刃 . 」耄 0 ん ECU. ん刃イ 0 Ⅳ 339 We now consider the set of equations 42 ー 28 for different values 0f ~ as a set 0f simultaneous linear homogeneous equations in the unknown quantities の . ln order that the set may possess a Ⅱ 0 Ⅱ - trivial solution, the determinant formed by the coefhcients of the cz's must vanish. This gwes a determinantal equation involving 入 and from which we determine the relation between them. are interested ⅲ the normal state of the system, with 襯 = 0 and I even. The determinantal equation for this case is 1 2 0 0 3 4 0 21 り 1 0 1 一 1 21 0 臨 0 0 0 ー 20 ー ( 42 ー 29 ) 39 77 0 The only non-vanishing terms are ⅲ the principal diagonal and the immediately adjacent diagonals. As a rough approximation ()o the first degree ⅲ入 ) we can neglect the adjacent diagonals ; the roots of the equation are then = 当 X , = 1 IX ー 6 , 3 ? イ 7 入ー 20 , etc. We are interested ⅲ the first of these. ln order to obtain it more accurately, we solve the equation agam, including the first two non-diagonal terms, and replacing ⅲ the second diagonal term by ! X. This equation, 1 2 3 15 ・ 1 ・ 1 1 り編 has the solution l//3X 十 2135X2 十 8505X3 , ⅲ which powers of 入 higher than the third are neglected. Hyller- aas carried the procedure one step farther, obtaining l//3X 十イ 35X2 十チく 505X3 ー 0.000013 入 4 ー 0.0000028X5. This equation expresses the functional dependency of 0 Ⅱ入 for the normal state, as determined by the equation. The next step is tO introduce this in the ミ equation, eliminating 円 and then to solve this equation tO Obtain the characteristic