Quantum dynamics: The Schroedinger equation

Sutee Boonchui, Wichit Sritrakool, Virulh Sa-Yakanit

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.

Daniel Bartolomeo, Ariel Caticha

In the Entropic Dynamics (ED) derivation of the Schroedinger equation the physical input is introduced through constraints that are implemented using Lagrange multipliers. There is one constraint involving a "drift" potential that correlates the motions of different particles and is ultimately responsible for entanglement. The purpose of this work is to deepen our understanding of the corresponding multiplier. Its main effect is to control the strength of the drift relative to the fluctuations. We show that ED exhibits a symmetry: models with different values of the multiplier can lead to the same Schroedinger equation; different "microscopic" or sub-quantum models lead to the same "macroscopic" or quantum behavior. As the multiplier tends to infinity the drift prevails over the fluctuations and the particles tend to move along the smooth probability flow lines. Thus ED includes the causal or Bohmian form of quantum mechanics as a special limiting case.

Steven Kenneth Kauffmann

The historical Klein-Gordon transformation of complex-valued first-order in time Schroedinger equations iterates these in a naively straightforward way which changes them into complex-valued second-order in time equations that have a plethora of extraneous solutions -- the transformation is an operator-calculus analogue of the squaring of both sides of an algebraic equation. The real and imaginary parts of a Schroedinger equation, however, are well known to be precisely the dynamical equation pair of the real-valued classical Hamiltonian functional which is numerically equal to the expectation value of that Schroedinger equation's Hermitian Hamiltonian operator. The purely real-valued second-order in time Euler-Lagrange equation of the corresponding classical Lagrangian functional is also isomorphic to that Schroedinger equation, and for symmetric Hamiltonians has exactly the same formal appearance as the corresponding naive complex-valued Klein-Gordon equation, but none of the latter's extraneous solutions. These quantum Schroedinger-equation isomorphisms to classical Euler-Lagrange equations are the technical manifestation of a key theoretical aspect of the principle of complementarity, one which is elegantly illustrated by the isomorphic free-photon wave-function complement to the vector potential of source-free classical electrodynamics.

Leon Bess