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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space which transforms h\u142ij
into höij ˆ hA^\u142ij. This operator A^y is called the adjoint of A^. In bra space the
transformation is expressed as
hA^\u142ij ˆ h\u142ijA^y
Thus, for bras the operator acts on the vector to its left, whereas for kets the
operator acts on the vector to its right.
To find the relationship between A^ and its adjoint A^y, we take the scalar
product of hA^\u142 jj and j\u142ii
hA^\u142 jj\u142ii ˆ h\u142 jjA^yj\u142ii (3:33a)
or in integral notation …
(A^\u142 j)
\ufffd\u142i dx ˆ
…
\u142\ufffdj A^y\u142i dx (3:33b)
A comparison with equation (3.8) shows that if A^ is hermitian, then we have
A^y ˆ A^ and A^ is said to be self-adjoint. The two terms, hermitian and self-
adjoint, are synonymous. To find the adjoint of a non-hermitian operator, we
apply equations (3.33). For example, we see from equation (3.10) that the
adjoint of the operator d=dx is ÿd=dx.
Since the scalar product h\u142jöi is equal to höj\u142i\ufffd, we see that
hA^\u142 jj\u142ii ˆ h\u142ijA^j\u142 ji\ufffd (3:34)
Combining equations (3.33a) and (3.34) gives
h\u142 jjA^yj\u142ii ˆ h\u142ijA^j\u142 ji\ufffd (3:35)
If we replace A^ in equation (3.35) by the operator A^y, we obtain
h\u142 jj(A^y)yj\u142ii ˆ h\u142ijA^yj\u142 ji\ufffd (3:36)
where (A^y)y is the adjoint of the operator A^y. Equation (3.35) may be rewritten
as
h\u142ijA^yj\u142 ji\ufffd ˆ h\u142 jjA^j\u142ii
and when compared with (3.36), we see that
h\u142 jj(A^y)yj\u142ii ˆ h\u142 jjA^j\u142ii
We conclude that
(A^y)y ˆ A^ (3:37)
From equation (3.35) we can also show that
82 General principles of quantum theory
(cA^)y ˆ c\ufffdA^y (3:38)
where c is any complex constant, and that
(A^‡ B^)y ˆ A^y ‡ B^y (3:39)
To obtain the adjoint of the product A^B^ of two operators, we apply equation
(3.33a), first to A^B^, then to A^, and finally to B^
h\u142 jj(A^B^)yj\u142ii ˆ hA^B^\u142 jj\u142ii ˆ hB^\u142 jjA^yj\u142ii ˆ h\u142 jjB^yA^yj\u142ii
Thus, we have the relation
(A^B^)y ˆ B^yA^y (3:40)
If A^ and B^ are hermitian (self-adjoint), then we have (A^B^)y ˆ B^A^ and further, if
A^ and B^ commute, then the product A^B^ is hermitian or self-adjoint.
The outer product of a bra höj and a ket j\u142i is j\u142ihöj and behaves as an
operator. If we let this outer product operate on another ket j÷i, we obtain the
expression j\u142ihöj÷i, which can be regarded in two ways. The scalar product
höj÷i is a complex number multiplying the ket j\u142i, so that the complete
expression is a ket parallel to j\u142i. Alternatively, the operator j\u142ihöj acts on the
ket j÷i and transfroms j÷i into a ket proportional to j\u142i.
To find the adjoint of the outer product j÷ihöj of the ket j÷i and the bra höj,
we let A^ in equation (3.35) be equal to j÷ihöj and obtain
h\u142 jj(j÷ihöj)yj\u142ii ˆ h\u142ij(j÷ihöj)j\u142 ji\ufffd ˆ h\u142ij÷i\ufffdhöj\u142 ji\ufffd
ˆ h÷j\u142iih\u142 jjöi ˆ h\u142 jjöih÷j\u142ii ˆ h\u142 jj(jöih÷j)j\u142ii
Setting equal the operators in the left-most and right-most integrals, we find
that
(j÷ihöj)y ˆ jöih÷j (3:41)
Projection operator
We define the operator P^i as the outer product of j\u142ii and its corresponding bra
P^i \ufffd j\u142iih\u142ij \ufffd jiihij (3:42)
and apply P^i to an arbitrary ket jöi
P^ijöi ˆ jiihijöi
Thus, the result of P^i acting on jöi is a ket proportional to jii, the proportion-
ality constant being the scalar product h\u142ijöi. The operator P^i, then, projects
jöi onto j\u142ii and for that reason is known as the projection operator. The
operator P^2i is given by
P^2i ˆ P^i P^i ˆ jiihijiihij ˆ jiihij ˆ P^i
where we have noted that the kets jii are normalized. Likewise, the operator P^ni
3.6 Hilbert space and Dirac notation 83
for n . 2 also equals P^i. This property is consistent with the interpretation of
P^i as a projection operator since the result of projecting jöi onto jii should be
the same whether the projection is carried out once, twice, or multiple times.
The operator P^i is hermitian, so that the projection of jöi on j\u142ii is equal to
the projection of j\u142ii on jöi. To show that P^i is hermitian, we let j÷i ˆ
jöi ˆ jii in equation (3.41) and obtain P^yi ˆ P^i.
The expansion of a function f (x) in terms of the orthonormal set \u142i(x), as
shown in equation (3.27), may be expressed in terms of kets as
j f i ˆ
X
i
aij\u142ii ˆ
X
i
aijii
where j f i is regarded as a vector in ket space. The constants ai are the
projections of j f i on the \u2018unit ket vectors\u2019 jii and are given by equation (3.28)
ai ˆ hij f i
Combining these two equations gives equation (3.29), which when expressed
in Dirac notation is
j f i ˆ
X
i
jiihij f i
Since f (x) is an arbitrary function of x, the operator
P
ijiihij must equal the
identity operator, so that X
i
jiihij ˆ 1 (3:43)
From the definition of P^i in equation (3.42), we see thatX
i
P^i ˆ 1
Since the operator
P
ijiihij equals unity, it may be inserted at any point in an
equation. Accordingly, we insert it between the bra and the ket in the scalar
product of j f i with itself
h f j f i ˆ f
\ufffd\ufffd\ufffd\ufffd X
i
jiihij
 !\ufffd\ufffd\ufffd\ufffd f* + ˆ 1
where we have assumed j f i is normalized. This expression may be written as
h f j f i ˆ
X
i
h f jiihij f i ˆ
X
i
jhij f ij2 ˆ
X
i
jaij2 ˆ 1
Thus, the expression (3.43) is related to the completeness criterion (3.30) and
is called, therefore, the completeness relation.
84 General principles of quantum theory
3.7 Postulates of quantum mechanics
In this section we state the postulates of quantum mechanics in terms of the
properties of linear operators. By way of an introduction to quantum theory, the
basic principles have already been presented in Chapters 1 and 2. The purpose
of that introduction is to provide a rationale for the quantum concepts by
showing how the particle\u2013wave duality leads to the postulate of a wave
function based on the properties of a wave packet. Although this approach,
based in part on historical development, helps to explain why certain quantum
concepts were proposed, the basic principles of quantum mechanics cannot be
obtained by any process of deduction. They must be stated as postulates to be
accepted because the conclusions drawn from them agree with experiment
without exception.
We first state the postulates succinctly and then elaborate on each of them
with particular regard to the mathematical properties of linear operators. The
postulates are as follows.
1. The state of a physical system is defined by a normalized function Ø of the spatial
coordinates and the time. This function contains all the information that exists on
the state of the system.
2. Every physical observable A is represented by a linear hermitian operator A^.
3. Every individual measurement of a physical observable A yields an eigenvalue of
the corresponding operator A^. The average value or expectation value hAi from a
series of measurements of A for systems, each of which is in the exact same state
Ø, is given by hAi ˆ hØjAjØi.
4. If a measurement of a physical observable A for a system in state Ø gives the
eigenvalue ºn of A^, then the state of the system immediately after the measurement
is the eigenfunction (if ºn is non-degenerate) or a linear combination of eigenfunc-
tions (if ºn is degenerate) corresponding to ºn.
5. The time dependence of the state function Ø is determined by the time-dependent
Schro¨dinger differential equation
i"
@Ø
@ t
ˆ H^Ø
where H^ is the Hamiltonian operator for the system.
This list of postulates is not complete in that two quantum concepts are not
covered, spin and identical particles. In Section 1.7 we mentioned in passing
that an electron has an intrinsic angular momentum called spin. Other particles
also possess spin. The quantum-mechanical treatment of spin is postponed until
Chapter 7. Moreover, the state function for a system of two or more identical
and therefore indistinguishable particles requires special consideration and is
discussed in Chapter 8.
3.7 Postulates of quantum mechanics 85
State function
According to the first postulate, the state of a physical system