Hermiticity momentum operator

The hermitian character of an operator depends not only on the operator itself, but also on the functions on which it acts and on the range of integration. An operator may be hermitian with respect to one set of functions, but not with respect to another set. It may be hermitian with respect to a set of functions defined over one range of variables, but not with respect to the same set over a different range. For example, the hermiticity of the momentum operator p is dependent on the vanishing of the functions ipi at infinity. 

On account of the hermiticity of the operators, only even values of L are allowed. Since the D operators are tensor operators of rank 1, the only allowed values are L = 0,2. The L = 0 contribution has been treated in type (1). Hence, only the L = 2 contribution must be considered here. The matrix elements of the operators (4.122) with L = 2 are difficult to evaluate. Nonetheless, by making use of the angular momentum algebra, they can be evaluated in explicit... 

Equation of quantum state. The Dirac bra-ket formalism of quantum mechanics. Representation of the wave-momentum and coordinates. The adjunct operators. Hermiticity. Normal and adjunct operators. Scalar multiplication. Hilbert space. Dirac function. Orthogonality and orthonormality. Commutators. The completely set of commuting operators. 

As a useful illustration let s check the coordinate, momentum and Hamiltonian hermiticity. The position operator is hermitic... 

THEOREM Quantum hermitic operators of coordinate and momentum fulfill the Newtonian laws of motion... 

Recognizing the fact that representation of observable quantum operators, in various bases, is made by (hermitic) matrices, Heisenberg had generalized the commutation rules to operators and thus to matrix level, while this way constructing the so-called quantum matrix mechanics. It is basically founded by the commutation rules among the coordinate [2] momentum [- ] matrices. 

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