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

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need to determine the coefficients c1, c2 such that B^\u142i âi\u142i, that is c1B^\u142i1 c2B^\u142i2 âi(c1\u142i1 c2\u142i2) If we take the scalar product of this equation first with \u142i1 and then with \u142i2, we obtain c1(B11 ÿ âi) c2 B12 0 c1 B21 c2(B22 ÿ âi) 0 where we have introduced the simplified notation Bjk \ufffd h\u142ij j B^\u142iki These simultaneous linear homogeneous equations determine c1 and c2 and have a non-trivial solution if the determinant of the coefficients of c1, c2 vanishes B11 ÿ âi B12 B21 B22 ÿ âi \ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd 0 or â2i ÿ (B11 B22)âi B11 B22 ÿ B12 B21 0 This quadratic equation has two roots â(1)i and â (2) i , which lead to two corresponding sets of constants c (1) 1 , c (1) 2 and c (2) 1 , c (2) 2 . Thus, there are two distinct functions \u142(1)i and \u142 (2) i 78 General principles of quantum theory \u142(1)i c(1)1 \u142i1 c(1)2 \u142i2 \u142(2)i c(2)1 \u142i1 c(2)2 \u142i2 which satisfy the relations B^\u142(1)i â(1)i \u142(1)i B^\u142(2)i â(2)i \u142(2)i and are, therefore, simultaneous eigenfunctions of the commuting operators A^ and B^. This analysis can be extended to three or more operators. If three operators A^, B^, and C^ have a complete set of simultaneous eigenfunctions, then the argument above shows that A^ and B^ commute, B^ and C^ commute, and A^ and C^ commute. Furthermore, the converse is also true. If A^ commutes with both B^ and C^, and B^ commutes with C^, then the three operators possess simultaneous eigenfunctions. To show this, suppose that the three operators commute with one another. We know that since A^ and B^ commute, they possess simultaneous eigenfunctions \u142i such that A^\u142i Æi\u142i B^\u142i âi\u142i We next operate on each of these expressions with C^, giving C^A^\u142i A^(C^\u142i) C^(Æi\u142i) Æi(C^\u142i) C^B^\u142i B^(C^\u142i) C^(âi\u142i) âi(C^\u142i) Thus, the function C^\u142i is an eigenfunction of both A^ and B^ with eigenvalues Æi and âi, respectively. If Æi and âi are non-degenerate, then there is only one eigenfunction \u142i corresponding to them and the function C^\u142i is proportional to \u142i C^\u142i ªi\u142i and, consequently, A^, B^, and C^ possess simultaneous eigenfunctions. For degenerate eigenvalues Æi and/or âi, simultaneous eigenfunctions may be constructed using a procedure parallel to the one described above for the doubly degenerate two-operator case. We note here that if A^ commutes with B^ and B^ commutes with C^, but A^ does not commute with C^, then A^ and B^ possess simultaneous eigenfunctions, B^ and C^ possess simultaneous eigenfunctions, but A^ and C^ do not. The set of simultaneous eigenfunctions of A^ and B^ will differ from the set for B^ and C^. An example of this situation is discussed in Chapter 5. 3.5 Simultaneous eigenfunctions 79 In some of the derivations presented in this section, operators need not be hermitian. However, we are only interested in the properties of hermitian operators because quantum mechanics requires them. Therefore, we have implicitly assumed that all the operators are hermitian and we have not bothered to comment on the parts where hermiticity is not required. 3.6 Hilbert space and Dirac notation This section introduces the basic mathematics of linear vector spaces as an alternative conceptual scheme for quantum-mechanical wave functions. The concept of vector spaces was developed before quantum mechanics, but Dirac applied it to wave functions and introduced a particularly useful and widely accepted notation. Much of the literature on quantum mechanics uses Dirac\u2019s ideas and notation. A set of complete orthonormal functions \u142i(x) of a single variable x may be regarded as the basis vectors of a linear vector space of either finite or infinite dimensions, depending on whether the complete set contains a finite or infinite number of members. The situation is analogous to three-dimensional cartesian space formed by three orthogonal unit vectors. In quantum mechanics we usually (see Section 7.2 for an exception) encounter complete sets with an infinite number of members and, therefore, are usually concerned with linear vector spaces of infinite dimensionality. Such a linear vector space is called a Hilbert space. The functions \u142i(x) used as the basis vectors may constitute a discrete set or a continuous set. While a vector space composed of a discrete set of basis vectors is easier to visualize (even if the space is of infinite dimensionality) than one composed of a continuous set, there is no mathema- tical reason to exclude continuous basis vectors from the concept of Hilbert space. In Dirac notation, the basis vectors in Hilbert space are called ket vectors or just kets and are represented by the symbol j\u142ii or sometimes simply by jii. These ket vectors determine a ket space. When a ket j\u142ii is multiplied by a constant c, the result c j\u142ii jc\u142ii is a ket in the same direction as j\u142ii; only the magnitude of the ket vector is changed. However, when an operator A^ acts on a ket j\u142ii, the result is another ket jöii jöii A^j\u142ii jA^\u142ii In general, the ket jöii is not in the same direction as j\u142ii nor in the same direction as any other ket j\u142 ji, but rather has projections along several or all basis kets. If an operator A^ acts on all kets j\u142ii of the basis set, and the resulting set of kets jöii jA^\u142ii are orthonormal, then the net result of the 80 General principles of quantum theory operation is a rotation of the basis set j\u142ii about the origin to a new basis set jöii. In the situation where A^ acting on j\u142ii gives a constant times j\u142ii (cf. equation (3.5)) A^j\u142ii jA^\u142ii Æij\u142ii the ket jA^\u142ii is along the direction of j\u142ii and the kets j\u142ii are said to be eigenkets of the operator A^. Although the expressions A^j\u142ii and jA^\u142ii are completely equivalent, there is a subtle distinction between them. The first, A^j\u142ii, indicates the operator A^ being applied to the ket j\u142ii. The quantity jA^\u142ii is the ket which results from that application. Bra vectors The functions \u142i(x) are, in general, complex functions. As a consequence, ket space is a complex vector space, making it mathematically necessary to introduce a corresponding set of vectors which are the adjoints of the ket vectors. The adjoint (sometimes also called the complex conjugate transpose) of a complex vector is the generalization of the complex conjugate of a complex number. In Dirac notation these adjoint vectors are called bra vectors or bras and are denoted by h\u142ij or hij. Thus, the bra h\u142ij is the adjoint j\u142iiy of the ket j\u142ii and, conversely, the ket j\u142ii is the adjoint h\u142ijy of the bra h\u142ij j\u142iiy h\u142ij h\u142ijy j\u142ii These bra vectors determine a bra space, just as the kets determine ket space. The scalar product or inner product of a bra höj and a ket j\u142i is written in Dirac notation as höj\u142i and is defined as höj\u142i ö\ufffd(x)\u142(x) dx The bracket (bra-c-ket) in höj\u142i provides the names for the component vectors. This notation was introduced in Section 3.2 as a shorthand for the scalar product integral. The scalar product of a ket j\u142i with its corresponding bra h\u142j gives a real, positive number and is the analog of multiplying a complex number by its complex conjugate. The scalar product of a bra h\u142 jj and the ket jA^\u142ii is expressed in Dirac notation as h\u142 jjA^j\u142ii or as h jjA^jii. These scalar products are also known as the matrix elements of A^ and are sometimes denoted by Aij. To every ket in ket space, there corresponds a bra in bra space. For the ket 3.6 Hilbert space and Dirac notation 81 cj\u142ii, the corresponding bra is c\ufffdh\u142ij. We can also write cj\u142ii as jc\u142ii, in which case the corresponding bra is hc\u142ij, so that hc\u142ij c\ufffdh\u142ij For every linear operator A^ that transforms j\u142ii in ket space into jöii jA^\u142ii, there is a corresponding linear operator A^y in bra