The clocked Schroedinger equation was proposed by Sokolovski using the Feynman path integrals with constraint (Phys. Rev. A 52, R2, 1995). Sokolovski pointed out that the clocked Schroedinger equation (clocked SE) cannot be derived directly from the Schroedinger equation (SE). In this thesis, we show that the clocked Schroedinger equation can be derived by starting from the ordinary Schroedinger equation. We derive the clocked SE by two methods: (i) substitution of the wave function into the ordinary SE and (ii) reduction of a composite system, composed of the observed system and the apparatus system or the pointer, as defined by von Neumann. For applications, we provide the ket state in the Schroedinger picture from the entangled state and the differential equation for solving the probability amplitude of a quantum system as a function of time from the timeindependent SE. We use the Jaynes-Cummings model for an example.The clocked Schroedinger equation was proposed by Sokolovski using the Feynman path integrals with constraint (Phys. Rev. A 52, R2, 1995). Sokolovski pointed out that the clocked Schroedinger equation (clocked SE) cannot be derived directly from the Schroedinger equation (SE). In this thesis, we show that the clocked Schroedinger equation can be derived by starting from the ordinary Schroedinger equation. We derive the clocked SE by two methods: (i) substitution of the wave function into the ordinary SE and (ii) reduction of a composite system, composed of the observed system and the apparatus system or the pointer, as defined by von Neumann. For applications, we provide the ket state in the Schroedinger picture from the entangled state and the differential equation for solving the probability amplitude of a quantum system as a function of time from the timeindependent SE. We use the Jaynes-Cummings model for an example.

Jan 1, 20050 citations

The Schrödinger Equation

Rolf E. Hummel

Springer eBooksJan 1, 19930 citations

Derivation of Gross-Pitaevskii Version of Nonlinear Schroedinger Equation from Scale Invariance

Olga M. Kosheleva, Владик Крейнович

It is known that in the usual 3-D space, the Schroedinger equation can be derived from scale-invariance. In view of the fact that, according to modern physics, the actual dimension of proper space may be different from 3, it is desirable to analyze what happens in other spatial dimensions $D$ . It turns out that while for $D\ge 3$ we still get only the Schroedinger's equation, for $D=2$ , we also get the Gross-Pitaevskii version of a nonlinear Schroedinger equation that describes a quantum system of identical bosons, and for $D=1$ , we also get a new nonlinear version of the Schroedinger equation.It is known that in the usual 3-D space, the Schroedinger equation can be derived from scale-invariance. In view of the fact that, according to modern physics, the actual dimension of proper space may be different from 3, it is desirable to analyze what happens in other spatial dimensions $D$. It turns out that while for $D\ge 3$ we still get only the Schroedinger's equation, for $D=2$, we also get the Gross-Pitaevskii version of a nonlinear Schroedinger equation that describes a quantum system of identical bosons, and for $D=1$, we also get a new nonlinear version of the Schroedinger equation.

Matematičeskie struktury i modelirovanieJan 1, 20170 citations

The Schrödinger Equation

Springer eBooksJan 1, 20060 citations

D

. It turns out that while for

D\ge 3

we still get only the Schroedinger's equation, for

D=2

, we also get the Gross-Pitaevskii version of a nonlinear Schroedinger equation that describes a quantum system of identical bosons, and for

D=1

, we also get a new nonlinear version of the Schroedinger equation.