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

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coordinate x, for which w(x) 1. An operator that results in multiplying by a real function f (x) is hermitian, since in this case f (x)\ufffd f (x) and equation (3.8) is an identity. Likewise, the momentum operator p^ ("=i)(d=dx), which was introduced in Section 2.3, is hermitian since 1 ÿ1 \u142\ufffdj p^\u142i dx 1 ÿ1 \u142\ufffdj " i d\u142i dx dx " i \u142\ufffdj \u142i \ufffd\ufffd\ufffd\ufffd1 ÿ1 ÿ " i 1 ÿ1 \u142i d\u142\ufffdj dx dx The integrated part is zero if the functions \u142i vanish at infinity, which they must in order to be well-behaved. The remaining integral is \u142i p^\ufffd\u142\ufffdj dx, so that we have 1 ÿ1 \u142\ufffdj p^\u142i dx 1 ÿ1 \u142i( p^\u142 j) \ufffd dx The imaginary unit i contained in the operator p^ is essential for the hermitian character of that operator. The operator D^x d=dx is not hermitian because 1 ÿ1 \u142\ufffdj d\u142i dx dx ÿ 1 ÿ1 \u142i d\u142\ufffdj dx dx (3:10) where again the integrated part vanishes. The negative sign on the right-hand side of equation (3.10) indicates that the operator is not hermitian. The operator D^2x, however, is hermitian. The hermitian character of an operator depends not only on the operator itself, but also on the functions on which it acts and on the range of integration. An operator may be hermitian with respect to one set of functions, but not with respect to another set. It may be hermitian with respect to a set of functions defined over one range of variables, but not with respect to the same set over a different range. For example, the hermiticity of the momentum operator p^ is dependent on the vanishing of the functions \u142i at infinity. The product of two hermitian operators may or may not be hermitian. Consider the product A^B^ where A^ and B^ are separately hermitian with respect to a set of functions \u142i, so that h\u142 j j A^B^\u142ii hA^\u142 j j B^\u142ii hB^A^\u142 j j\u142ii (3:11) where we have assumed that the functions A^\u142i and B^\u142i also lie in the hermitian domain of A^ and B^. The product A^B^ is hermitian if, and only if, A^ and B^ commute. Using the same procedure, one can easily demonstrate that if A^ and B^ do not commute, then the operators (A^B^ B^A^) and i[A^, B^] are hermitian. By setting B^ equal to A^ in the product A^B^ in equation (3.11), we see that the square of a hermitian operator is hermitian. This result can be generalized to 70 General principles of quantum theory any integral power of A^. Since jA^\u142j2 is always positive, the integral hA^\u142 j A^\u142i is positive and consequently h\u142 j A^2\u142i > 0 (3:12) Eigenvalues The eigenvalues of a hermitian operator are real. To prove this statement, we consider the eigenvalue equation A^\u142 Æ\u142 (3:13) where A^ is hermitian, \u142 is an eigenfunction of A^, and Æ is the corresponding eigenvalue. Multiplying by \u142\ufffd and integrating give h\u142 j A^\u142i Æh\u142 j\u142i (3:14) Multiplication of the complex conjugate of equation (3.13) by \u142 and integrat- ing give hA^\u142 j\u142i Æ\ufffdh\u142 j\u142i (3:15) Because A^ is hermitian, the left-hand sides of equations (3.14) and (3.15) are equal, so that (Æÿ Æ\ufffd)h\u142 j\u142i 0 (3:16) Since the integral in equation (3.16) is not equal to zero, we conclude that Æ Æ\ufffd and thus Æ is real. Orthogonality theorem If \u1421 and \u1422 are eigenfunctions of a hermitian operator A^ with different eigenvalues Æ1 and Æ2, then \u1421 and \u1422 are orthogonal. To prove this theorem, we begin with the integral h\u1422 j A^\u1421i Æ1h\u1422 j\u1421i (3:17) Since A^ is hermitian and Æ2 is real, the left-hand side may be written as h\u1422 j A^\u1421i hA^\u1422 j\u1421i Æ2h\u1422 j\u1421i Thus, equation (3.17) becomes (Æ2 ÿ Æ1)h\u1422 j\u1421i 0 Since Æ1 6 Æ2, the functions \u1421 and \u1422 are orthogonal. Since the Dirac notation suppresses the variables involved in the integration, we re-express the orthogonality relation in integral notation \u142\ufffd2 (q1, q2, . . .)\u1421(q1, q2, . . .)w(q1, q2, . . .) dq1 dq2 . . . 0 This expression serves as a reminder that, in general, the eigenfunctions of a 3.3 Hermitian operators 71 hermitian operator involve several variables and that the weighting function must be used. The functions are, therefore, orthogonal with respect to the weighting function w(q1, q2, . . .). If the weighting function is real and positive, then we can define ö1 and ö2 as ö1 w p \u1421, ö2 w p \u1422 The functions ö1 and ö2 are then mutually orthogonal with respect to a weighting function of unity. Moreover, if the operator A^ is hermitian with respect to \u1421 and \u1422 with a weighting function w, then A^ is hermitian with respect to ö1 and ö2 with a weighting function equal to unity. If two or more linearly independent eigenfunctions have the same eigen- value, so that the eigenvalue is degenerate, the orthogonality theorem does not apply. However, it is possible to construct eigenfunctions that are mutually orthogonal. Suppose there are two independent eigenfunctions \u1421 and \u1422 of the operator A^ with the same eigenvalue Æ. Any linear combination c1\u1421 c2\u1422, where c1 and c2 are any pair of complex numbers, is also an eigenfunction of A^ with the same eigenvalue, so that A^(c1\u1421 c2\u1422) c1A^\u1421 c2A^\u1422 Æ(c1\u1421 c2\u1422) From any pair \u1421, \u1422 which initially are not orthogonal, we can construct by selecting appropriate values for c1 and c2 a new pair which are orthogonal. By selecting different sets of values for c1, c2, we may obtain infinitely many new pairs of eigenfunctions which are mutually orthogonal. As an illustration, suppose the members of a set of functions \u1421, \u1422, . . . , \u142n are not orthogonal. We define a new set of functions ö1, ö2, . . . , ön by the relations ö1 \u1421 ö2 aö1 \u1422 ö3 b1ö1 b2ö2 \u1423 .. . If we require that ö2 be orthogonal to ö1 by setting hö1 jö2i 0, then the constant a is given by a ÿh\u1421 j\u1422i=h\u1421 j\u1421i ÿhö1 j\u1422i=hö1 jö1i and ö2 is determined. We next require ö3 to be orthogonal to ö1 and to ö2, which gives 72 General principles of quantum theory b1 ÿhö1 j\u1423i=hö1 jö1i b2 ÿhö2 j\u1423i=hö2 jö2i In general, we have ös \u142s Xsÿ1 i1 ksiöi ksi ÿhöi j\u142si=höi jöii This construction is known as the Schmidt orthogonalization procedure. Since the initial selection for ö1 can be any of the original functions \u142i or any linear combination of them, an infinite number of orthogonal sets öi can be obtained by the Schmidt procedure. We conclude that all eigenfunctions of a hermitian operator are either mutually orthogonal or, if belonging to a degenerate eigenvalue, can be chosen to be mutually orthogonal. Throughout the remainder of this book, we treat all the eigenfunctions of a hermitian operator as an orthogonal set. Extended orthogonality theorem The orthogonality theorem can also be extended to cover a somewhat more general form of the eigenvalue equation. For the sake of convenience, we present in detail the case of a single variable, although the treatment can be generalized to any number of variables. Suppose that instead of the eigenvalue equation (3.5), we have for a hermitian operator A^ of one variable A^\u142i(x) Æiw(x)\u142i(x) (3:18) where the function w(x) is real, positive, and the same for all values of i. Therefore, equation (3.18) can also be written as A^\ufffd\u142\ufffdj (x) Æ\ufffdj w(x)\u142\ufffdj (x) (3:19) Multiplication of equation (3.18) by \u142\ufffdj (x) and integration over x give \u142\ufffdj (x)A^\u142i(x) dx Æi \u142\ufffdj (x)\u142i(x)w(x) dx (3:20) Now, the operator A^ is hermitian with respect to the functions \u142i with a weighting function equaling unity, so that the integral on the left-hand side of equation (3.20) becomes \u142\ufffdj (x)A^\u142i(x) dx \u142i(x)A^ \ufffd\u142\ufffdj (x) dx Æ\ufffdj \u142\ufffdj (x)\u142i(x)w(x) dx where equation (3.19) has been used as well. Accordingly, equation (3.20) becomes 3.3 Hermitian operators 73 (Æi ÿ Æ\ufffdj ) \u142\ufffdj (x)\u142i(x)w(x) dx 0 (3:21) When j i, the integral in equation (3.21) cannot vanish because the product