A Rigorous Real Time Feynman Path Integral and Propagator
نویسنده
چکیده
Abstract. We will derive a rigorous real time propagator for the Non-relativistic Quantum Mechanic L transition probability amplitude and for the Non-relativistic wave function. The propagator will be explicitly given in terms of the time evolution operator. The derivation will be for all self-adjoint nonvector potential Hamiltonians. For systems with potential that carries at most a finite number of singularity and discontinuities, we will show that our propagator can be written in the form of a rigorous real time, time sliced Feynman path integral via improper Riemann integrals. We will also derive the Feynman path integral in Nonstandard Analysis Formulation. Finally, we will compute the propagator for the harmonic oscillator using the Nonstandard Analysis Feynman path integral formuluation; we will compute the propagator without using any knowledge of classical properties of the harmonic oscillator.
منابع مشابه
A Rigorous Real Time Feynman Path Integral
where φ, ψ ∈ L, H = −~ 2m ∆+V (~x) is essentially self-adjoint, H̄ is the closure ofH , and φ, ψ, V each carries at most a finite number of singularities and discontinuities. In flavor of physics literature, we will formulate the Feynman path integral with improper Riemann integrals. In hope that with further research we can formulate a rigorous polygonal path integral, we will also provide a No...
متن کاملQuadratic Actions, Semi-classical Approximation, and Delta Sequences in Gaussian Analysis
A mathematically rigorous realization of Feynman integrals is given. The construction works for quadratic actions (there is only a restriction to certain time intervals). These techniques enable the calculation of the semi-classical approximation for a given Feynman propagator. Finally, delta sequences in Gaussian analysis are presented and their connection to semi-classical approximation is di...
متن کامل0 Rigorous Real - Time Feynman Path Integral for Vector Potentials
Abstract. In this paper, we will show the existence and uniqueness of a real-time, time-sliced Feynman path integral for quantum systems with vector potential. Our formulation of the path integral will be derived on the L transition probability amplitude via improper Riemann integrals. Our formulation will hold for vector potential Hamiltonian for which its potential and vector potential each c...
متن کاملPath Integral and the Induction Law
We show how the induction law is correctly used in the path integral computation of the free particle propagator. The way this primary path integral example is treated in most textbooks is a little bit missleading. ⋆ e-mail: [email protected] † e-mail: [email protected] 1 The path integral quantization method was developed in detail by Feynman [1] in 1948. Feynman developed some earlier ideas...
متن کاملPath probabilities of continuous time random walks
Employing the path integral formulation of a broad class of anomalous diffusion processes, we derive the exact relations for the path probability densities of these processes. In particular, we obtain a closed analytical solution for the path probability distribution of a Continuous Time Random Walk (CTRW) process. This solution is given in terms of its waiting time distribution and short time ...
متن کامل