Generalized Momentum Operator

In this research, we propose a generalized momentum operator upon imposing the spatial expansion provided that the EUP relation is confirmed. According to the EUP algebra, we use A(x) to be an auxiliary function ( or 1+ µ (x)) in the construction of our new momentum. Thus, we provide a formalism containing the auxiliary function in the real or complex domain upon which one has the freedom to define a hermitian or non-hermitian momentum operator. We start with introducing a generalized Lagrangian density affected by the proposed formalism. Our investigation is continued on the motion of a quantum particle under the variation of such Lagrangian density. Besides, we demonstrate the PT -symmetry field theory including the Ψ(x,t), Ψ∗ (x,t) or ΨPT (x,t) in the structure of the extended Lagrangian density. Having applied the principle of least action, the Euler-Lagrange equations lead to the corresponding Schrödinger equations. Upon finding the generalized Lagrangian density, we obtain the Hamiltonian density, momentum density and energy flux which are known to be the elements of the stress-energy tensor. The expectation value of the generalized Hamiltonian density for the hermitian and PT -symmetric fields are determined where it leads to the energy of the system. Thereupon, we extend the probability and particle current densities which significantly satisfy the continuity equation. Next, we develop further the concept of generalized momentum operator and elucidate the significance of our proposal considering once the real definition and once the PT -symmetric structure. With solving the eigen-value problems for the two combinations, we represent the eigen-values and eigen-functions of some examples of the generalized momentum operator. In accordance with the generalized Schrödinger equation, the kinetic energy operator is rebuilt and consequently the Hamiltonian operator is identified based on the imposed potential energy and the new kinetic energy. The corresponding differential equations are declared and the exact solutions are computed using the variable transformation method. Accordingly, we transform the extended Schrödinger equation from x-space into the target space, here z-space, with the manner whose energy spectrum remains identical. Later, we demonstrate an illustrative examples based on the idea of a step momentum operator which is flexible to have a hermitian or the PT -symmetric Hamiltonian operator. We employ the formalism such that expresses a sudden change in the momentum of a system at a specific point. To show that, we use a step auxiliary function and develop the Schrödinger equation considering a quantum particle inside a square well. The outcomes yield infinite bound states with real energy spectrum for the particles with hermitian step momentum, it is finite for a particle with PT -symmetric momentum. Afterwards, we study the PT -symmetric Hamiltonian in two dimensions. Having considered standard kinetic energy, a two dimensional complex harmonic oscillator potential is introduced which is invariant under the parity and time reversal operator. The Schrödinger equation yields real eigen-values with complex eigen-functions. We also construct the coherent state of the system by using a superposition of 12 eigen-functions. Utilizing the complex correspondence principle for the probability density, we investigate the possible modifications in the probability densities due to the non-hermitian aspect of the Hamiltonian.