Principles of Quantum Mechanics   as Applied to Chemistry and Chemical Physics

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 ˆ âi\u142i, that is
c1B^\u142i1 ‡ c2B^\u142i2 ˆ 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 ˆ 0
c1 B21 ‡ c2(B22 ÿ âi) ˆ 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
B11 ÿ âi B12
B21 B22 ÿ âi
\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd ˆ 0
â2i ÿ (B11 ‡ B22)âi ‡ B11 B22 ÿ B12 B21 ˆ 0
This quadratic equation has two roots â(1)i and â
i , which lead to two
corresponding sets of constants c
1 , c
2 and c
1 , c
2 . Thus, there are two
distinct functions \u142(1)i and \u142
78 General principles of quantum theory
\u142(1)i ˆ c(1)1 \u142i1 ‡ c(1)2 \u142i2
\u142(2)i ˆ c(2)1 \u142i1 ‡ c(2)2 \u142i2
which satisfy the relations
B^\u142(1)i ˆ â(1)i \u142(1)i
B^\u142(2)i ˆ 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 ˆ Æi\u142i
B^\u142i ˆ âi\u142i
We next operate on each of these expressions with C^, giving
C^A^\u142i ˆ A^(C^\u142i) ˆ C^(Æi\u142i) ˆ Æi(C^\u142i)
C^B^\u142i ˆ B^(C^\u142i) ˆ C^(âi\u142i) ˆ 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 ˆ 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 ˆ 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 ˆ A^j\u142ii ˆ 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 ˆ 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 ˆ jA^\u142ii ˆ 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 ˆ h\u142ij
h\u142ijy ˆ 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 ˆ
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 ˆ c\ufffdh\u142ij
For every linear operator A^ that transforms j\u142ii in ket space into jöii ˆ jA^\u142ii,
there is a corresponding linear operator A^y in bra