Chapter 13
Inner Product Space
Thus far we have treated the theory of linear vector spaces. The vector spaces,
however, were somewhat “structureless,” and so it will be desirable to introduce a
concept of metric or measure into the linear vector spaces. We call a linear vector
space where the inner product is defined an inner product space. In virtue of a
concept of the inner product, the linear vector space is given a variety of structures.
For instance, introduction of the inner product to the linear vector space immediately
leads to the definition of adjoint operators and Gram matrices.
Above all, the concept of inner product can readily be extended to a functional
space and facilitate understanding of, e.g., orthogonalization of functions, as was
exemplified in Parts I and II. Moreover, definition of the inner product allows us to
relate matrix operators and differential operators. In particular, it is a key issue to
understand logical structure of quantum mechanics. This can easily be understood
from the fact that Paul Dirac, who was known as one of prominent founders of
quantum mechanics, invented bra and ket vectors to represent an inner product.
13.1 Inner Product and Metric
Inner product relates two vectors to a complex number. To do this, we introduce the
notation jai and hbj to represent the vectors. This notation is due to Dirac and widely
used in physics and mathematical physics. Usually jai and hbj are called a “ket”
vector and a “bra” vector, respectively, again due to Dirac. Alternatively, we may
call haj an adjoint vector of jai. Or we denote haj jai
{ . The symbol “{” (dagger)
means that for a matrix its transposed matrix should be taken with complex conjugate matrix elements. That is, a ij
À Á { ¼ a
Ã
ji
. If we represent a full matrix, we have
© Springer Nature Singapore Pte Ltd. 2020
S. Hotta, Mathematical Physical Chemistry,
https://doi.org/10.1007/978-981-15-2225-3_13
523
Inner Product Space
Thus far we have treated the theory of linear vector spaces. The vector spaces,
however, were somewhat “structureless,” and so it will be desirable to introduce a
concept of metric or measure into the linear vector spaces. We call a linear vector
space where the inner product is defined an inner product space. In virtue of a
concept of the inner product, the linear vector space is given a variety of structures.
For instance, introduction of the inner product to the linear vector space immediately
leads to the definition of adjoint operators and Gram matrices.
Above all, the concept of inner product can readily be extended to a functional
space and facilitate understanding of, e.g., orthogonalization of functions, as was
exemplified in Parts I and II. Moreover, definition of the inner product allows us to
relate matrix operators and differential operators. In particular, it is a key issue to
understand logical structure of quantum mechanics. This can easily be understood
from the fact that Paul Dirac, who was known as one of prominent founders of
quantum mechanics, invented bra and ket vectors to represent an inner product.
13.1 Inner Product and Metric
Inner product relates two vectors to a complex number. To do this, we introduce the
notation jai and hbj to represent the vectors. This notation is due to Dirac and widely
used in physics and mathematical physics. Usually jai and hbj are called a “ket”
vector and a “bra” vector, respectively, again due to Dirac. Alternatively, we may
call haj an adjoint vector of jai. Or we denote haj jai
{ . The symbol “{” (dagger)
means that for a matrix its transposed matrix should be taken with complex conjugate matrix elements. That is, a ij
À Á { ¼ a
Ã
ji
. If we represent a full matrix, we have
© Springer Nature Singapore Pte Ltd. 2020
S. Hotta, Mathematical Physical Chemistry,
https://doi.org/10.1007/978-981-15-2225-3_13
523