GENERALIZED FOURIER SERIES 113 pointwise convergence but not vice versa. We conclude this section by stating a theorem from Courant and Hilbert (p. 427, vol. I). The expansion theorem: Any piecewise continuous function defined in the fundamental interval [a, b] with a square integrable first derivative (i.e., sufficiently smooth) could be expanded in an eigenfunction series: m=O which converges absolutely and uniformly in all subintervals free of points of discontinuity. At the points of discontinuity this series rep resents (as in the Fourier series) the arithmetic mean of the right- and the left-hand limits. In this theorem the function F (x) does not have to satisfy the boundary conditions. This theorem also implies convergence in the mean and point- wise convergence. That the derivative is square integrable means that the integral of the square of the derivative is finite for all the subintervals of the fundamental domain [a, b] in which the function is continuous. 8.5 GENERALIZED FOURIER SERIES Series expansion of a sufficiently smooth F (z) in terms of the eigenfunction set {urn (3)) can now be written as 00 F (x) = C amurn (XI, (8.34) which is called the generalized Fourier series of F(z). Expansion coefficients, urn, are found using the orthogonality relation of {um (x)} as m=O - - am. Substituting this in Equation (8.34) we get (8.36) 114 STURM-LIOUVILLE THEORY Using the basic definition of the Dirac-delta function, that is, g(x) = 1 g(x’)6(x - x’)dx’, we can now give a formal expression of the completeness of the set {$m (x)} as 00 c u:, (x’) w (x’) urn (x) = s (x - x’) . (8.37) m=O It is needless to say that this is not a proof of completeness. 8.6 TRIGONOMETRIC FOURIER SERIES Trigonometric Fourier series are defined with respect to the eigenvalue pro& lem (8.38) with the operator given as k‘ = @/dx2. This could correspond to a vibrating string. Using the periodic boundary conditions we find the eigenfunctions as u, = cosnx, n = 0,1,2, urn = sinmx, m = 1,2, .__ . Orthogonality of the eigenfunctions is expressed as sin mx sin nxdx = A,&,, cos mx cos nxdx = B,S,,, (8.39) (8.41) sin mx cos nxdx = 0, HERMITIAN OPERATORS IN QUANTUM MECHANICS 115 where (8.42) (8.43) 7i n#O ?r n#O 27r n=O' An= { 0 n=o ' Bn= { Now the trigonometric Fourier series of any sufficiently well-behaved function becomes 00 a0 2 f(z) = - + C [a,cosnz + b, sinnz] , n= 1 (8.44) where the expansion coefficients are given as 1" a, = ; 1 f (t) cos ntdt , n = 0, 1 ,2, . . . (8.45) and (8.46) Example 8.1. Trigonometric Fourier series: Trigonometric Fourier se 1" bn = ; 1, f (t)sinntdt, n = 1,2 , . ries of a square wave can now be written as 2d O0 sin (2n + 1) 3: f (z) = ,c (2n + 1) ' n=O where we have substituted the coefficients a, = 0 d n = even n?r n = odd b, = - (1 -cosn?r) = (8.48) (8.49) 8.7 HERMITIAN OPERATORS IN QUANTUM MECHANICS In quantum mechanics the state of a system is completely described by a complex valued function, @(z), in terms of the real variable z. Observable 116 STURM-LIOUVKLE THEORY quantities are represented by differential operators (not necessarily second order) acting on the wave functions. These operators are usually obtained from their classical expressions by replacing position, momentum, and energy with their operator counterparts as (8.50) a at E + ih- For example, the angular momentum operator is obtained from its classical expression L = 7 x 9 as + i L =-ih(T+xv+). Similarly, the Hamiltonian operator is obtained from its classical expression H = p2/2m + V(X) as 1 2m H = v2 + V(Z). The observable value of a physical property is given by the expectation value of the corresponding operator L as (L) = /@*L@dx. (8.51) Because (L) corresponds to a measurable quantity it has to be real; hence observable properties in quantum mechanics are represented by Hermitian operators. For the real Sturm-Liouville operators Hermitian property [Eq. (8.20)] was defined with respect to the eigenfunctions u and v, which sat- isfy the boundary conditions (8.13) and (8.15). To accommodate complex operators in quantum mechanics we modify this definition as / 9;L@adz = (L@1)*92dx, (8.52) J where 9land 92 do not have to be the eigenfunctions of the operator L. The fact that Hermitian operators have real expectation values can be seen from = /(L@)*@dx = (L)* (8.53) HERMITIAN OPERATORS /N QUANTUM MECHANICS 117 A Hermitian Sturm-Liouville operator must be second order. However, in quantum mechanics order of the Hermitian operators is not restricted. Remember that the momentum operator is first order, but it is Hermitian because of the presence of a in its definition: a ax rm (p) = / 9*(-itz-)9dZ =Irm ax = itz 9'91:00 - 1, **(itz-)*dx ax = s_, **(-ili-)*dx. dX -rm 00 a (-&-9)**dX a 00 a 00 (8.54) (8.55) (8.56) (8.57) In proving that the momentum operator is Hermitian we have imposed the boundary condition that 9 is sufficiently smooth and vanishes at large dis- tances. A general boundary condition that all wave functions must satisfy is that they have to be square integrable, and thus normalizable. Space of all square integrable functions actually forms an infinite dimensional vector space called L2 or the Hilbert space. Functions in this space can be expanded as general- ized Fourier series in terms of the complete and orthonormal set of eigenfunc- tions, {urn (z)}, of a Hermitian operator. Eigenfunctions satisfy the eigenvalue equation Lum(z> = Amum(z), (8.58) where A, represents the eigenvalues. In other words, {urn(.)} spans the infinite dimensional vector space of square integrable functions. The inner product (analog of dot product) in Hilbert space is defined as (8.59) which has the following properties: (91, a92) = 491, 92), ("*1,@2) = "*(91,*2), (91, *2)* = (*2, *l), (91 f *2>*3) = (91, 93) f (*2, 93), where (Y is a complex number. The inner product also satisfies the triangle inequality: (8.60) 118 STURM-LIOUVILLE THEORY and the Schwartz inequality: 1911 I%l L I(~l,*Z)l. (8.62) An important consequence of the Schwartz inequality is that convergence of (@I, 92) follows from the convergence of (@I, 91) and (92, @2). Problems 8.1 Show that the Laguerre equation d2Y dY x- + (1 -x) - +ny = 0 dx2 dx can be brought into the self-adjoint form by multiplying it with e-" . 8.2 Write the Chebyshev equation (1 - X2)Tl(X) - XTL(X) +n2Tn(x) = 0 in the self-adjoint form. 8.3 Find the weight function for the associated Laguerre equation 8Y dY 5- + (k+ 1 - x) - +ny = 0. dx2 dx 8.4 A function y(x) is to be a finite solution of the differential equation (2 + 5~ - x') dx 4~( 1 - X) in the entire interval x E [O, 11. )a Show that this condition can only be satisfied for certain values of X and write the solutions explicitly for the lowest three values of A. b) Find the weight function ~(x). c) Show that the solution set {yx(x)} is orthogonal with respect to the w(x) found above. 8.5 Show that the Legendre equation can be written as d dx -[(l - x">4] + l(Z+ 1)9 = 0. 8.6 For the Sturm-Liouville equation with the boundary conditions Y(0) = 0 Y( ) - Y'(4 = 0, PROBLEMS 119 find the eigenvalues and the eigenfunctions. 8.7 tem Find the eigenvalues and the eigenfunctions of the Sturm-Liouville sys- Y(0) = 0, y( 1) = 0. Hint: 73-y the substitution x = tant. 8.8 Show that the Hermite equation can be written as 8.9 Given the Sturm-Liouville equation If yn(x) and y,(x) are two orthogonal solutions and satisfy the appropriate boundary conditions, then show that &(x) and yA(x) are orthogonal with the weight function p(x). 8.10 as Show that the Bessel equation can be written in the self-adjoint form d n2 dx X -[xJ;] + (x - -)Jn = 0. 8.11 Find the trigonometric Fourier expansion of f(x)=7r -7r<X<O =x O<X<T. 8.12 Show that the angular momentum operator + L =-ih(TixT) and its square are Hermitian. 8.13 show that they have the same eigenfunctions. a) What are their eigenvalues? b) Write the L, and L, operators in spherical polar coordinates. Write the operators t2, and L, in spherical polar coordinates and 120 STURM-LIOUVILLE THEORY 8.14 For a Sturm-Liouville operator let u(z) be a nontrivial solution satisfying Xu = 0 with the boundary condition at z = a, and let V(X) be another nontrivial solution satisfying Lu = 0 with the boundary condition at x = b. Show that the Wronskian is equal to A/p(z), where A is a constant. 8.15 For the inner product defined as (*I, *2) = J *;(x)*2(X)dX7 prove the following properties, where a is a complex number: 8.16 Prove the triangle inequality: and the Schwartz inequality: 1*i1 I*2\ 2 l(*i7*2)l. 8.17 Show that the differential equation Y” +Pl(X)Y’ + [132(x) + Wx)ly(x) = 0 can be put into self-adjoint form as 9 ST URM- LIO U VILLE SYSTEMS and the FACTORIZATION METHOD The factorization method allows us to replace a Sturm-Liouville equation, which is a second-order differential equation, with a pair of first-order differ- ential equations. For a large class of problems satisfying certain boundary conditions the method immediately yields the eigenvalues and allows us to write the ladder or the step-up/-down operators for the problem. These op erators are then used to construct the eigenfunctions from a base function. Once the base function is normalized, the manufactured eigenfunctions are also normalized and satisfy the same boundary conditions as the base func- tion. First we introduce the method of factorization and its features in terms of five basic theorems. Next, we show how eigenvalues and eigenfunctions are obtained and introduce six basic types of factorization. In fact, factor- ization of a given second-order differential equation is reduced to identifying the type it belongs to. To demonstrate the usage of the method we discuss the associated Legendre equation and spherical harmonics in detail. We also discuss the radial part of Schradinger’s equation for the hydrogen-like atoms, Gegenbauer polynomials, the problem of the symmetric top, Bessel functions, and the harmonic oscillator problem via the factorization method. Further details and an extensive table of differential equations that can be solved by this technique can be found in Infeld and Hull (1951), where this method was introduced for the first time. 121 122 STURM-LIOUVILLE SYSTEMS AND THE FACTORIZATION METHOD 9.1 ANOTHER FORM FOR THE STURM-LIOUVILLE EQUATION The Sturm-Liouville equation is usually written in the first canonical form as d -& [P(X)F] + q(z)*(z) + AW(X)*(Z) = 0, 2 E [alp], (9.1) where p(z) is different from zero in the open interval (a, p); however, it could have zeroes at the end points of the interval. We also impose the boundary conditions and where eigenvalue. Solutions also satisfy the orthogonality relation and Ik are any two solutions corresponding to the same or different If p(z) and w(.) are never negative and w(~)/p(~) exists everywhere in (a, p), using the transformations and (9.5) we can cast the Sturm-Liouville equation into another form, also known as the second canonical form 2m 2.'") + {A + T(Z, m)}yy(z) = 0, where 2 r(z,m)=-+- + :6[:2 :El 2 dpdw ld2w + (9.7) m and X are two constant parameters that usually enter into our equations through the process of separation of variables. Their values are restricted by the boundary conditions and in most cases take discrete (real) values like [...]... ( z , m )and the p ( m ) functions corresponding t o a given r ( z ,m) are known For m > 0, depending on whether p ( m ) is a n increasing or a decreasing function, there are two cases 9.4.1 Case I ( m > 0 and p ( m ) is an increasing function) In this case, from Theorem IV there is a maximum value for m, m = 0 , 1 , 2 , ,1, and the eigenvalues (9 .54 ) are given as x = x1 = p(2 + 1) (9 .55 ) Since there... say mmax 1 This determines X as = X = p(l + 1) = ( 1 + $ 1 (9.141) 2 On the other hand, for m < 0 we could write 1 4 4 = (Iml+ $2 (9.142) Again from the conclusions of Theorem IV there exists a minimum value, mmin, thus determining X as X = mmin (9.143) To find mminwe equate the two expressions [Eqs (9.141) and (9.143)] for X t o obtain (9.144) mmin -1 = (9.146) Since m changes by integer amounts, we... Equation (9.83) and integrating gives us k o ( z )= cacota(z + p ) + sin+ d + p ) ’ (9.89) where p and d are integration constants With these Ico and Icl functions in Equation (9. 75) and the p ( m ) given in Equation (9. 85) we obtain r ( z , m ) from Equation (9.71) or (9.72) as +,m) = - + a 2 ( m c)(m + c + 1)+ d2 + 2ad(m + c + a) cos a ( z + p ) sin2a ( z + p ) (9.90) We now obtain our first factorization... defined as (9.163) Using Equation (9.162) we write (9.164) Using (9.1 65) 142 STURM-LIOUVILLE SYSTEMS AND THE FACTORIZATION METHOD and Equation (9.160) we could also write (9.166) L - - Cancelling J(l a e c i 4 5 J(l m e - + m)(l - m + 1) [-$-mcotO + m)(2- rn + 1) J c l + m)(l- m + 1) I F; ~" (o) [ - A -S )H( 61 ( ) mcot r P m on both sides and noting that (9.167) and using Equation (9.164) we finally... elab- orate this case in the next chapter in Problem 11.11 9.6.2 Construction of the Eigenfunctions Since mmax 1, there are no states with m > 1 Thus = (9. 152 ) (9. 153 ) This gives = (1+;) Hence the state with mmax 1 is determined as = yi, (8) = N(sin Q)(’++) (9. 155 ) N is a normalization constant t o be determined from (9. 156 ) (9. 157 ) which g i v s N = (-1)L [+ (22 l)!] 221+11!2 (9. 158 ) The factor of (-l)’,... lower than 1 = m, using the definition of 0- ( z , l ) , we can use Equation (9.66) to find yx' z ) as ( Je=-mJ (9.206) sinh z cosh z dz (9.207) Vd-(z, = " 1 coshm z ' (9.208) where N' is a normalization constant in the z-space Using the transformation given in Equation (9.194) and, since 1 = m, we write VA-( z ) as V E ( ~ )= ~ ' ( 6= Nsin'O ) (9.209) From Equations (9.162) and (9. 155 ) we note that for... (9.178) and 9.6.4 Interpretation of the & Operators In quantum mechanics the angular momentum operator (we set ti = 1) is given as L = - 7 3 4x -+ (9.179) We write this in spherical polar coordinates to get e , A r A ee 0 a i - _- a ar , rd8 rsinQd4 (9.180) 144 STURM-LIOUVILLE SYSTEMS AND THE FACTORIZATION METHOD (9.181) g The basis vectors g and g+ in spherical polar coordinates are written in terms... coordinates as of e0 = A + (cos 6 sin 4)Gy - (sin Q)Gz, (cos 6 cos c))& e+ = -(sinB)P, A + (cos4)Sy (9.182) (9.183) Thus the angular momentum operator in Cartesian coordinates becomes + (9.184) L = LxPx LySy L,G, + + a icotQcosq5-++sinq534) It is now clearly seen that + L+ = L, ZL,, L- = L, - ZL,, and Lz -i- (9.1 85) a 34 Also note that t2 =L:+L;+L: 1 = - (L+L- + L- L+) + L: 2 (9.186) From the definition... SYSTEMS AND THE FACTORIZATION METHOD This determines p ( m ) as p ( m )= p ( 0 ) + a2(m2+ 2mc) for a +o (9. 85) and p ( m ) = p ( 0 ) - 2mb for a = 0 (9.86) In these equations we could take p ( 0 ) = 0 without any loss of generality Using these results we now obtain the following categories: A) For a # 0 Equation (9.82) gives (9.87) Ic* = a c o t a ( z + p ) (9.88) Substituting this into Equation (9.83) and. .. SYSTEMS AND THE FACTORIZATION METHOD Theorem III: If (9.32) Ja exists and if p(m) is an increasing function of m ( r n > 0), then (9.33) also exists If p(m) is a decreasing function of m (and m > 0), then lbdz [o-( v 4 Y i W l 2 (9.34) also exists O+(z,m+l)yE(z) and O - ( z , m ) y T ( z ) alsosatisfy thesame (z) boundary condition as Proof: We take Y2 = YEk) (9. 35) and Y1 = Y y k ) (9.36) in Theorem . m=O which converges absolutely and uniformly in all subintervals free of points of discontinuity. At the points of discontinuity this series rep resents (as in the Fourier series) the arithmetic. square integrable means that the integral of the square of the derivative is finite for all the subintervals of the fundamental domain [a, b] in which the function is continuous. 8 .5 GENERALIZED. 00 a 00 (8 .54 ) (8 .55 ) (8 .56 ) (8 .57 ) In proving that the momentum operator is Hermitian we have imposed the boundary condition that 9 is sufficiently smooth and vanishes at large