This yields to a Hamiltonian whose eigenvalues are the square of the imaginary part of the Riemann zeros, and also the functional determinant of this Hamiltonian operator is just the Riemann Xi function.
22.
In RS theory one considers an unperturbed Hamiltonian operator \ hat { H } _ { 0 }, to which a small ( often external ) perturbation \ hat { V } is added:
23.
To apply the Schr�dinger equation, the Hamiltonian operator is set up for the system, accounting for the kinetic and potential energy of the particles constituting the system, then inserted into the Schr�dinger equation.
24.
This means that the state at a slightly later time differs from the state at the current time by the result of acting with the Hamiltonian operator ( multiplied by the negative imaginary unit, ).
25.
As an application, we consider the Schr�dinger equation, or equivalently, the Hamiltonian operator " H " models the total energy observable of a quantum mechanical system "'S " '.
26.
"Fractional quantum oscillator " introduced by Nick Laskin ( see, Ref . [ 2 ] ) is the fractional quantum mechanical model with the Hamiltonian operator H _ { \ alpha, \ beta } defined as
27.
The tensor product factorization is only possible if the orbital and spin angular momenta of the particle are separable in the Hamiltonian operator underlying the system's dynamics ( in other words, the Hamiltonian can be split into the sum of orbital and spin terms ).
28.
The Lagrangian approach with field interpretation of is the subject of QFT rather than RQM : Feynman's path integral formulation uses invariant Lagrangians rather than Hamiltonian operators, since the latter can become extremely complicated, see ( for example ) S . Weinberg ( 1995 ).
29.
Solving the latter in any particular case requires specifying the Hamiltonian operator, which includes a generic part related to kinetic energy and specific part related to how the particle ( s ) potential energy can be expected to change as a function of position and / or time.
30.
A very important aspect of the Hamiltonian operator is that it only acts at vertices ( a consequence of this is that Thiemann's Hamiltonian operator, like Ashtekar's operator, annihilates non-intersecting loops except now it is not just formal and has rigorous mathematical meaning ).
