12
2 Function Spaces and Matrices
For linearly independent functions, we have
∀ k = l ::f k |f l =0
(2.1)
The basis is orthonormal if all vectors are in addition normalized to +1:
∀ k ::f k |f k =1
(2.2)
This result can be summarized with the help of the Kronecker delta, δ ij , which is
zero unless the subscript indices are identical, in which case it is unity. Hence, for
an orthonormal basis set,
f k |f l =δ kl
(2.3)
In quantum mechanics, the bra-function of f k is simply the complex-conjugate function, ¯
f k , and the bracket or scalar product is defined as the integral of the product of
the functions over space:
f k |f l =
¯
f k f l dV
(2.4)
One thus also has
f k |f l ==f l |f k
(2.5)
2.2 Linear Operators and Transformation Matrices
A linear operator is an operator that commutes with multiplicative scalars and is
distributive with respect to summation: this means that when it acts on a sum of
functions, it will operate on each term of the sum:
ˆ
Rc|f k =c ˆ
R|f k
ˆ
R
|f k +|f l
= ˆ
R|f k + ˆ
R|f l
(2.6)
If the transformations of functions under an operator can be expressed as a mapping of these functions onto a linear combination of the basis vectors in the function
space, then the operator is said to leave the function space invariant. The corresponding coefficients can then be collected in a transformation matrix. For this purpose, we arrange the components in a row vector, (|f 1 , |f 2 ,...,|f n ), as agreed
upon in Chap. 1. This row precedes the transformation matrix. The usual symbols
are ˆ
R for the operator and D(R) for the corresponding matrix:
ˆ
R
|f 1 |f 2 ···|f n
=
|f 1 |f 2 ···|f n
⎛
⎝ D(R)
⎞
⎠
2 Function Spaces and Matrices
For linearly independent functions, we have
∀ k = l ::f k |f l =0
(2.1)
The basis is orthonormal if all vectors are in addition normalized to +1:
∀ k ::f k |f k =1
(2.2)
This result can be summarized with the help of the Kronecker delta, δ ij , which is
zero unless the subscript indices are identical, in which case it is unity. Hence, for
an orthonormal basis set,
f k |f l =δ kl
(2.3)
In quantum mechanics, the bra-function of f k is simply the complex-conjugate function, ¯
f k , and the bracket or scalar product is defined as the integral of the product of
the functions over space:
f k |f l =
¯
f k f l dV
(2.4)
One thus also has
f k |f l ==f l |f k
(2.5)
2.2 Linear Operators and Transformation Matrices
A linear operator is an operator that commutes with multiplicative scalars and is
distributive with respect to summation: this means that when it acts on a sum of
functions, it will operate on each term of the sum:
ˆ
Rc|f k =c ˆ
R|f k
ˆ
R
|f k +|f l
= ˆ
R|f k + ˆ
R|f l
(2.6)
If the transformations of functions under an operator can be expressed as a mapping of these functions onto a linear combination of the basis vectors in the function
space, then the operator is said to leave the function space invariant. The corresponding coefficients can then be collected in a transformation matrix. For this purpose, we arrange the components in a row vector, (|f 1 , |f 2 ,...,|f n ), as agreed
upon in Chap. 1. This row precedes the transformation matrix. The usual symbols
are ˆ
R for the operator and D(R) for the corresponding matrix:
ˆ
R
|f 1 |f 2 ···|f n
=
|f 1 |f 2 ···|f n
⎛
⎝ D(R)
⎞
⎠