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Appendix a Mathematical Operations Appendix A Mathematical Operations This appendix presents a number of mathematical techniques and equations for the convenience of the reader. Although we have attempted to sum­ marize accurately some of the most useful relations, we make no attempt at rigor. A bibliography is included at the end of this appendix. A-1 COMPLEX NUMBERS A complex quantity may be represented as follows: u = x + iy = re+i<l> (A-I) where i 2 = -1, x, y, r, and cp are real numbers and ei<l> = cos cp + i sin cpo One refers to x and y as the real and the imaginary components, respectively of u, whereas r is the absolute magnitude of u, that is, r = lui. cp is called the phase angle. The complex conjugate of u, viz., u*, is obtained by changing the sign of i wherever it appears; that is, u* = x - iy. The relation between complex numbers and their conjugates is clarified by representing them as points in the "complex plane" (Argand diagram, see Fig. A-I). The abscissa is chosen to represent the real axis (x), and the ordinate the imaginary axis 391 392 ELECTRON SPIN RESONANCE Imaginary axis u iy ~E2---------t-- Real axis u' Fig. A-l Representation of a point u and of its complex conjugate u* in complex space (Argand diagram). (y). Note that the real component of u is equal to one-half the sum of u and u*; the product of u and its complex conjugate is the square of the absolute magnitude, that is, (A-2) A-2 OPERATOR ALGEBRA A-2a. Properties of operators An operator A is a symbolic instruction to carry out a stipulated mathematical operation upon some function which is called an operand. Unless its form is explicitly indicated, an operator will be designated by a circumflex. One of the simplest operators is a constant multiplier; for example, ka = ka. An operator n is said to be linear if the result of operation upon a sum of functions is the same as that obtained by operating on each function separately; i.e., if na = {3, then (A-3) Also, if e is a constant, n(ca) = ena = e{3 (A-4) If ai = !(qi) , then a/aqi is a linear operator. An example of a nonlinear operator is "Y". The reader will be familiar with such operators as the summation operator ~ n 2: ai == a1 + a2 + a3 + . + an (A-5) i=1 Its use permits a concise representation of a series. Frequently one wishes to summarize a set of equations with constant coefficients such as MATHEMATICAL OPERATIONS 393 t/h = C11 c/>1 + c 12c/>2 + c 13c/>3 + t/12 = C21 c/>1 + C22c/>2 + C23c/>3 + t/13 = C31 c/>1 + C32c/>2 + C33 c/>3 + + C3nc/>n (A-6) The sum of the t/1j can be represented by a double summation, viz., L t/1j = L L Cjkc/>k (A-7) j j k Here one encounters a juxtaposition of two operators, which in the general case are represented as Aii. It is understood that AB implies operation first with B and then with A. Interchange of order of the operators may give a different result; for example, but If AB = BA, then A and B are said to be commuting operators. The dif­ ference A B - BA is called the commutator of A and B. It is represented by the symbol [A ,B] == (AB - BA). The magnitude of the commutator of two operators is of profound significance in quantum-mechanical systems. The commutators of angular-momentum operators will be treated in Appendix B. An operator fl is said to be hermitian if it obeys the following relation: (A-8) A useful aspect of the hermitian property is that (with care!) one may operate "backwards," i.e., to the left, when the operator occurs between two operands, as in Eq. (A-8). An example of operation to the left is found in Sec. B-4. Hermitian operators have the important property that if the result Table A-l Classical and quantum-mechanical dynamical variables Dynamical variable Classical quantity Quantum-mechanical operator Position q q Time t dq , ... d Linear momentum pq= m dt Pq=-/l, dq Pd> = r x p:j: Pd> = r x P Angular momentum P",= (xPy - ypx) pd>z=-ili (x :y - y :J = -iii a~ § 2 2 2 1f Kinetic energy T= Po" :!f=-!L.=--, P -h a associated with 2m 2m 2m aq2 coordinate q Potential energy V(q) V(q) :j: The evaluation of vector products is described in Sec. A-4. § The angle </> measures rotation about the z axis. 1f This form of the kinetic-energy hamiltonian is valid only for cartesian coordinates. 394 ELECTRON SPIN RESONANCE of operating upon a function is the function itself multiplied by a constant, one is assured that the constant is real. (See Sec. A-2b.) Some of the most important operators of quantum mechanics are those associated with observable properties of a physical system, i.e., the "dy­ namical variables." A few important linear operators are listed in Table A-I. Some of these operators are identical with the variable itself, whereas others involve derivatives. A-2b. Eigenvalues and eigenfunctions If the result of the application of an operator A to a function I/In is Al/ln = A.nl/ln (A-9) where A.n is a constant, then I/In is said to be an eigenfunction of A with eigenvalue A.n- (The set of functions I/In is often called a "basis set. ") The spin functions I/Ia and 1/1/3 introduced in Sec. 1-5 are examples of eigenfunc­ tions, in this case of the operator Sz; that is, Szl/la = +tl/la (A-lOa) Szl/l/3 = -tl/l/3 (A-lOb) (Angular momentum operator expressions are considered in detail in Sec. B-4.) A given set of eigenfunctions I/In may simultaneously be eigenfunctions of several operators. Operators having the same set of eigenfunctions have the very useful property that the operators must commute. In the case of the particle in a ring considered in Sec. 1-3, the wave functions 1/1 are eigen­ functions both of the angular-momentum operator PcP and of the hamiltonian operator:fr. The eigenvalue equations are (A-II) and Schrodinger equation (A-12) Table A-I gives PcP = -ih d/d~, where ~ measures the angular position ofthe particle. The kinetic energy of a classical particle having an angular mo­ mentum PcP and moment of inertia 1 is 2 W=PcP (A-13) 21 The quantum-mechanical operator for a system with V = 0 is ~ _ PcP2 _ (-ih)2 d 2 _ -h2 d2 flt - 2i - 2I d~2 - 2T d~2 (A-14) Substitution of:fr from Eq. (A-14) into (A-12) gives MATHEMATICAL OPERATIONS 395 (A-I5) Rearranging, 2 d 1/1 = -21WI/1 = -M2./. (A-I6) d1J2 1i2 'I' Here the constant 21W/1i2 has been set equal to M2. Two solutions of (A-16) are (A-17a) and 1/12 = A e-iMe/> (A-I7b) as is evident by substitution. From the requirement that the functions 1/1 be normalized, i.e., that J:7T 1/1*1/1 d1J = 1 (A-IS) one finds that A = (27T)-!. Hence 1/11 = (27T) -!eiMe/> (A-I9a) and 1/12 = (27T)-!e-iM e/> (A-I9b) Insertion of 1/11 into Eq. (A-I2) gives -1i2 d 2 (_1_ iMe/» _M21i2 (_1_ iMe/» 21 d1J2 Y2ii e - 21 Y2ii e (A-20) Hence, the eigenvalue W of the operator it, corresponding to the eigenfunc­ tion 1/11, is M21i2/2/. Use of 1/12 gives an identical energy value. Operation by Pe/> on 1/11 and 1/12 leads to the following equations: -iii ~ (_1_ eiMe/» = Mli (_1_ eiMe/» (A-2Ia) d1J Y2ii Y2ii -iii ~ (vk e-iMe/» = - Mli ( vk e-iMe/» (A-2Ib) Hence, the eigenvalues of Pe/>' corresponding to the eigenfunctions 1/1] and 1/12, are +MIi and -MIi, respectively. The wave functions can be eliminated from Eq. (A-20) or (A-2I) by multiplication on the left by the corresponding complex conjugate function 1/1* followed by integration. This yields expressions for the energy and for the angular momentum of a particle moving in a circle. [See Eq. (1-21) and Prob. A-3.] 396 ELECTRON SPIN RESONANCE A-3 DETERMINANTS A determinant is a scalar quantity which represents a linear combination of products of terms. It may be represented by a square array, for example (A-22a) Determinants are denoted in this book by boldface type enclosed by vertical lines. More generally a determinant of order k is represented as (A-22b) A determinant may be expanded by the "method of minors." The minor of any element aij is the determinant remaining after the row and column con­ taining the element au are removed. The expansion is carried out by mul­ tiplying the elements of a specific row or column by their corresponding minors as follows: IAkl = 2: (-l)(i+j) au IA"-llu (A-23) iorj Here IA"-llu is the minor corresponding to the element aij' For a determinant of order 3 this expansion may be carried out as follows: all a l2 a21 a22 a31 a32 = alla22a33 - alla23a32 - al2a2la33 + al2a23a31 + a l 3a 21 a32 - al3a22a31 (A-24a) Here the elements of the first row have been used. One could equally well have used the elements of any other row or column. The method of minors is a valuable technique for stepwise reduction of the order of a determinant.
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