Matrix Representation of Linear Operators
Matrix Element
It is instructive to note that the principal advantage of linear operators is that their action on any vector is defined purely by their action on a set of basis vectors. Let us consider the linear operator F transforming the vector a via its components ak:
Similarly, in quantum mechanics the action of the linear operator F on the vector of state |д) is symbolically denoted as F|
This represents ^A,|F|fl^ via the probability amplitude (Aj.|a) where js the matrix element of the operator F in the ^.-representation and can be defined as the projection of F|A*) onto |Af). So, if all the vectors F|At) are known, consequently all the matrix elements Fik of the operator F are known in the particular basis. When F№ is known, then the transformation property of any arbitrary vector can be easily established. Relation 2.1 can be examined as the matrix multiplication of an n x n matrix against a single-column n-row matrix, where n is the dimensionality of the vector space. From 2.73, it is observed that once an orthonormal basis is specified, any operator can be written as a bilinear expression in the basis kets and bras, with the coefficients simply being all the matrix elements in that basis. In quantum mechanics it is required that all operators be self-adjoint and linear. This requirement is such that the superposition principle holds. In order to have meaningful and measurable observables with operators, it is necessary that the expectation (mean) values of these operators be real. We find if there is some sense in the complex conjugate of relation 2.2: For an arbitrary operator F we may select its adjoint F defined:
Comparing 2.3 with 2.4 while substituting in place of |a)the state vector (2f| then we have
The operators satisfying condition 2.6 are Hermitian. So, the operators for which the physical quantities are real are Hermitian. We examine the case for which the physical quantities are complex. Let that quantity be F, then we may take its complex conjugate F*. The operator corresponding to the quantity F* is denoted by F .It is the conjugate operator of F and it is, however, different from the complex conjugate of F . So, finally one simply transposes and complex conjugates a matrix representation of the operator F to get the • T corresponding matrix representation of F :
» t If we want to get the m, n element of F we go to the n, m element of F (the indices are reversed) and we take its complex conjugate. So, a Hermitian operator is thus represented by a Hermitian matrix. This implies one in which any two elements that are symmetric with respect to the principal diagonal are complex conjugates of each other. If we consider our condition, then
-t ~ * From here it is obvious that F however coincides with F . For real physical quantities, i.e., the operator coincides with its conjugate (the Hermitian operator is also called self-adjoint). From here, considering 2.3 then
Condition 2.8 is the reason for which the operator of a physical quantity should be self-adjoint and it also shows that the mean value of a physical quantity should be real. So, the necessary condition for the self-adjoint of operators of physical quantities stems from the condition that the mean values of these quantities should be real and so the diagonal elements of a Hermitian matrix are thus always real numbers. Hence the operator that is linear and self-adjoint is Hermitian. It is instructive to note that in quantum mechanics observables are associated with Hermitian operators. If the operator has to be transformed as
where U is a unitary operator then if F is Hermitian so F' is also Hermitian with F and F' having the same eigenvalues. It is instructive to note that complex numbers remain unchanged under this unitary transformation. Typically, we use a unitary transformation to diagonalize an operator: The new operator F' is diagonal in the original basis and has the same eigenvalues, in the same order, as F •
Linear Self-Adjoint (Hermitian Conjugate) Operators
from where
