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An important subclass of projectors are the projectors P onto the basis
i
states themselves. These are given simply by
P = i i (5.40)
i
| ih |
satisfying the equation
P P =P δ . (5.41)
i j i ij
5.2.4 Adjoints
The adjoint A of a complex matrix A is the matrix obtained by transposing
†
and complex conjugating the matrix elements
(A ) =A . (5.42)
ij † ∗ji
In order to define the abstract notion of adjoint operators, the action of an
operatoronabravectorhastobedefined. Thisisapointwheresomeconfusion
might arise as to how employ the notational system.
As arguedin [Dirac], the inner product of a bra vector y with a ket vector
h |
Ax isacomplex numberthatdepends linearlyon x , thereforeitcanlikewise
| i | i
be considered as the inner product of x with some, as yet undefined, bra
| i
vector. This bra vector depends linearly on y , so it can be considered as the
h |
resultofapplying alinearoperatorto y . Since this linearoperatoris uniquely
h |
determinedbytheoriginallinearoperatorAitcanbeconsideredtobethesame
operator.
Then, choosing the convention of writing the action of the linear operator
A on y as y A, i.e with the operator to the right of the bra, we get the two
h | h |
ways of writing the inner product discussed above
y (Ax ) or ( y A)x .
h | | i h | | i
107
But from the linearity, this ’product’ is clearly associative, and we can it write
simply as
y Ax
h | | i
where the operator can be considered to act either to the right or to left. The
correctness of this is also born out by writing out the product concretely as
matrices and row and column vectors.
The adjoint of the operator A is defined as that operator A , which acting
†
on an arbitrary bra vector x, yields the same vector as the dual to the vector
h |
Ax , or in formulas
| i
xA† =(Ax )†. (5.43)
h | | i
Again, this definition can be justified using linearity.
From (5.43) follows the important property
y A x = xAy . (5.44)
† ∗
h | | i h | | i
This equation could in fact be used as an alternative definition of the ad-
joint. Let us derive it since the short calculation illustrates the workings of
the formalism. Start with the right hand side, taking the complex conjugate of
xAy
h | | i
xAy ∗ = x(Ay ) ∗ = Ay † x† = y A† x
h | | i h | | i | i h | h | | i
(cid:0) (cid:1) (cid:0) (cid:1)
where in the first step parenthesis are introduced to emphasizes which parts of
theexpressionsaregroupedtogether,nextequations(5.23)and(5.24)areused,
and finally the definition of the adjoint (5.43) is employed.
Note that in terms of matrices, the adjoint is the same as the conjugate-
transpose, or A =(A )T
† ∗
Hermitean and unitary operators
Of special interest in quantum mechanics are hermitean and unitary opera-
tors. They play the roles of representing observable quantities and generators
of transformations respectively.
An operator A is said to be hermitean or self-adjoint if
A =A. 5.45
†
An operator U is said to be unitary if
U =U 1. 5.46
† −
Hermitean operators corresponds to observable physical quantities. Unitary
operators corresponds to transformations of states.
108
5.2.5 Composition of operators
Sincealinearoperatoractingonastateisagainastate,compositionofoperators
is naturally defined as
(AB)x =A(B x )=AB x . 5.47
| i | i | i
Composition is associative
A(BC)=(AB)C =ABC 5.48
so there is no need to use parentheses.
The ’product’ of two operatorsA and B can be concretely realizedin terms
of ordinary matrix multiplication given matrix representationsof the operators
(in the same basis). As in matrix multiplication, the product is in general
not commutative. The non-commutativity of quantum mechanical operators
is an important property of quantum mechanics and leads to the celebrated
uncertaintyrelationsconnectingresultsofmeasurementsofnon-commutingob-
servables. But we will come to this is due time.