and this kind of expression is referred to by physicists as a “completeness rela-
tion”, since it requires that the set of|j〉be a basis with no missing elements.
The operator
Pj=|j〉〈j|
is the projection operator onto thej’th basis vector.
Digression. In this book, all of our indices will be lower indices. One way
to keep straight the difference between vectors and dual vectors is to use upper
indices for components of vectors, lower indices for components of dual vectors.
This is quite useful in Riemannian geometry and general relativity, where the
inner product is given by a metric that can vary from point to point, causing
the isomorphism between vectors and dual vectors to also vary. For quantum
mechanical state spaces, we will be using a single, standard, fixed inner product,
so there will be a single isomorphism between vectors and dual vectors. In this
case the bra-ket notation can be used to provide a notational distinction between
vectors and dual vectors.
4.5 Adjoint operators
WhenV is a vector space with inner product, the adjoint ofLcan be defined
by:
Definition(Adjoint operator). The adjoint of a linear operatorL:V→Vis
the operatorL†satisfying
〈Lv,w〉=〈v,L†w〉
for allv,w∈V.
Note that mathematicians tend to favorL∗as notation for the adjoint ofL, as
opposed to the physicist’s notationL†that we are using.
In terms of explicit matrices, sincelLvis the conjugate-transpose ofLv, the
matrix forL†will be given by the conjugate-transposeLTof the matrix forL:
L†jk=LkjIn the real case, the matrix for the adjoint is just the transpose matrix. We
will say that a linear transformation is self-adjoint ifL†=L, skew-adjoint if
L†=−L.
4.6 Orthogonal and unitary transformations
A special class of linear transformations will be invertible transformations that
preserve the inner product, i.e., satisfying
〈Lv,Lw〉=〈v,w〉