# Principles of Quantum Mechanics as Applied to Chemistry and Chemical Physics

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it satisfies two criteria A^(\u1421 \u1422) A^\u1421 A^\u1422 (3:2a) A^(c\u142) cA^\u142 (3:2b) where c is any complex constant. In the three examples given above, the operators are linear. Some nonlinear operators are \u2018exp\u2019 (take the exponential of) and [ ]2 (take the square of), since 65 ex y exe y 6 ex e y e cx 6 cex [x y]2 x2 2xy y2 6 x2 y2 [c(x y)]2 6 c[x y]2 The operator C^ is the sum of the operators A^ and B^ if C^\u142 (A^ B^)\u142 A^\u142 B^\u142 The operator C^ is the product of the operators A^ and B^ if C^\u142 A^B^\u142 A^(B^\u142) where first B^ operates on \u142 and then A^ operates on the resulting function. Operators obey the associative law of multiplication, namely A^(B^C^) (A^B^)C^ Operators may be combined. Thus, the square A^2 of an operator A^ is just the product A^A^ A^2\u142 A^A^\u142 A^(A^\u142) Similar definitions apply to higher powers of A^. As another example, the differential equation d2 y dx2 k2 y 0 may be written as (D^2x k2)y 0, where the operator (D^2x k2) is the sum of the two product operators D^2x and k 2. In multiplication, the order of A^ and B^ is important because A^B^\u142 is not necessarily equal to B^A^\u142. For example, if A^ x and B^ D^x, then we have A^B^\u142 xD^x\u142 x(d\u142=dx) while, on the other hand, B^A^\u142 D^x(x\u142) \u142 x(d\u142=dx). The commutator of A^ and B^, written as [A^, B^], is an operator defined as [A^, B^] A^B^ÿ B^A^ (3:3) from which it follows that [A^, B^] ÿ[B^, A^]. If A^B^\u142 B^A^\u142, then we have A^B^ B^A^ and [A^, B^] 0; in this case we say that A^ and B^ commute. By expansion of each side of the following expressions, we can readily prove the relationships [A^, B^C^] [A^, B^]C^ B^[A^, C^] (3:4a) [A^B^, C^] [A^, C^]B^ A^[B^, C^] (3:4b) The operator A^ is the reciprocal of B^ if A^B^ B^A^ 1, where 1 may be regarded as the unit operator, i.e., \u2018multiply by unity\u2019. We may write A^ B^ÿ1 66 General principles of quantum theory and B^ A^ÿ1. If the operator A^ possesses a reciprocal, it is non-singular, in which case the expression ö A^\u142 may be solved for \u142, giving \u142 A^ÿ1ö. If A^ possesses no reciprocal, it is singular and the expression ö A^\u142 may not be inverted. 3.2 Eigenfunctions and eigenvalues Consider a finite set of functions f i and the relationship c1 f1 c2 f 2 \ufffd \ufffd \ufffd cn f n 0 where c1, c2, . . . are complex constants. If an equation of this form exists, then the functions are linearly dependent. However, if no such relationship exists, except for the trivial one with c1 c2 \ufffd \ufffd \ufffd cn 0, then the functions are linearly independent. This definition can be extended to include an infinite set of functions. In general, the function ö obtained by the application of the operator A^ on an arbitrary function \u142, as expressed in equation (3.1), is linearly independent of \u142. However, for some particular function \u1421, it is possible that A^\u1421 Æ1\u1421 where Æ1 is a complex number. In such a case \u1421 is said to be an eigenfunction of A^ and Æ1 is the corresponding eigenvalue. For a given operator A^, many eigenfunctions may exist, so that A^\u142i Æi\u142i (3:5) where \u142i are the eigenfunctions, which may even be infinite in number, and Æi are the corresponding eigenvalues. Each eigenfunction of A^ is unique, that is to say, is linearly independent of the other eigenfunctions. Sometimes two or more eigenfunctions have the same eigenvalue. In that situation the eigenvalue is said to be degenerate. When two, three, . . . , n eigenfunctions have the same eigenvalue, the eigenvalue is doubly, triply, . . . , n-fold degenerate. When an eigenvalue corresponds only to a single eigenfunc- tion, the eigenvalue is non-degenerate. A simple example of an eigenvalue equation involves the operator D^x mentioned in Section 3.1. When D^x operates on e kx, the result is D^xe kx d dx ekx kekx Thus, the exponentials ekx are eigenfunctions of D^x with corresponding eigenvalues k. Since both the real part and the imaginary part of k can have any values from ÿ1 to 1, there are an infinite number of eigenfunctions and these eigenfunctions form a continuum of functions. 3.2 Eigenfunctions and eigenvalues 67 Another example is the operator D^2x acting on either sin nx or cos nx, where n is a positive integer (n > 1), for which we obtain D^2x sin nx ÿn2 sin nx D^2x cos nx ÿn2 cos nx The functions sin nx and cos nx are eigenfunctions of D^2x with eigenvalues ÿn2. Although there are an infinite number of eigenfunctions in this example, these eigenfunctions form a discrete, rather than a continuous, set. In order that the eigenfunctions \u142i have physical significance in their application to quantum theory, they are chosen from a special class of func- tions, namely, those which are continuous, have continuous derivatives, are single-valued, and are square integrable. We refer to functions with these properties as well-behaved functions. Throughout this book we implicitly assume that all functions are well-behaved. Scalar product and orthogonality The scalar product of two functions \u142(x) and ö(x) is defined as 1 ÿ1 ö\ufffd(x)\u142(x) dx For functions of the three cartesian coordinates x, y, z, the scalar product of \u142(x, y, z) and ö(x, y, z) is 1 ÿ1 ö\ufffd(x, y, z)\u142(x, y, z) dx dy dz For the functions \u142(r, \u141, j) and ö(r, \u141, j) of the spherical coordinates r, \u141, j, the scalar product is 2ð 0 ð 0 1 0 ö\ufffd(r, \u141, j)\u142(r, \u141, j)r2 sin \u141 dr d\u141 dj In order to express equations in general terms, we adopt the notation dô to indicate integration over the full range of all the coordinates of the system being considered and write the scalar product in the form ö\ufffd\u142 dô For further convenience we also introduce a notation devised by Dirac and write the scalar product of \u142 and ö as hö j\u142i, so that hö j\u142i \ufffd ö\ufffd\u142 dô The significance of this notation is discussed in Section 3.6. From the definition of the scalar product and of the notation hö j\u142i, we note that 68 General principles of quantum theory hö j\u142i\ufffd h\u142 jöi hö j c\u142i chö j\u142i hcö j\u142i c\ufffdhö j\u142i where c is an arbitrary complex constant. Since the integral h\u142 j\u142i\ufffd equals h\u142 j\u142i, the scalar product h\u142 j\u142i is real. If the scalar product of \u142 and ö vanishes, i.e., if hö j\u142i 0, then \u142 and ö are said to be orthogonal. If the eigenfunctions \u142i of an operator A^ obey the expressions h\u142 j j\u142ii 0 all i, j with i 6 j the functions \u142i form an orthogonal set. Furthermore, if the scalar product of \u142i with itself is unity, the function \u142i is said to be normalized. A set of functions which are both orthogonal to one another and normalized are said to be orthonormal h\u142 j j\u142ii äij (3:6) where äij is the Kronecker delta function, äij 1, i j 0, i 6 j (3:7) 3.3 Hermitian operators The linear operator A^ is hermitian with respect to the set of functions \u142i of the variables q1, q2, . . . if it possesses the property that \u142\ufffdj A^\u142i dô \u142i(A^\u142 j) \ufffd dô (3:8) The integration is over the entire range of all the variables. The differential dô has the form dô w(q1, q2, . . .) dq1 dq2 . . . where w(q1, q2, . . .) is a weighting function that depends on the choice of the coordinates q1, q2, . . . For cartesian coordinates the weighting function w(x, y, z) equals unity; for spherical coordinates, w(r, \u141, j) equals r2 sin\u141. Special variables introduced to simplify specific problems have their own weighting functions, which may differ from unity (see for example Section 6.3). Equation (3.8) may also be expressed in Dirac notation h\u142 j j A^\u142ii hA^\u142 j j\u142ii (3:9) in which the brackets indicate integration over all the variables using their weighting function. 3.3 Hermitian operators 69 For illustration, we consider some examples involving only one variable, namely, the cartesian