Equivalence Principle: Tunnelling, Quantized Spectra and Trajectories from the Quantum HJ Equation
نویسندگان
چکیده
A basic aspect of the recently proposed approach to quantum mechanics is that no use of any axiomatic interpretation of the wave function is made. In particular, the quantum potential turns out to be an intrinsic potential energy of the particle, which, similarly to the relativistic rest energy, is never vanishing. This is related to the tunnel effect, a consequence of the fact that the conjugate momentum field is real even in the classically forbidden regions. The quantum stationary Hamilton–Jacobi equation is defined only if the ratio ψ/ψ of two real linearly independent solutions of the Schrödinger equation, and therefore of the trivializing map, is a local homeomorphism of the extended real line into itself, a consequence of the Möbius symmetry of the Schwarzian derivative. In this respect we prove a basic theorem relating the request of continuity at spatial infinity of ψ/ψ, a consequence of the q ←→ q duality of the Schwarzian derivative, to the existence of L(R) solutions of the corresponding Schrödinger equation. As a result, while in the conventional approach one needs the Schrödinger equation with the L(R) condition, consequence of the axiomatic interpretation of the wave function, the equivalence principle by itself implies a dynamical equation that does not need any assumption and reproduces both the tunnel effect and energy quantization.
منابع مشابه
Energy quantisation and time parameterisation
We show that if space is compact, then trajectories cannot be defined in the framework of the quantum Hamilton–Jacobi (HJ) equation. The starting point is the simple observation that when the energy is quantised it is not possible to make variations with respect to the energy, and the time parameterisation t − t0 = ∂ES0, implied by Jacobi’s theorem, which leads to the group velocity, is ill def...
متن کاملQuantum Trajectories, State Diffusion and Time Asymmetric Eventum Mechanics
We show that the quantum stochastic Langevin model for continuous in time measurements provides an exact formulation of the Heisenberg uncertainty error-disturbance principle. Moreover, as it was shown in the 80’s, this Markov model induces all stochastic linear and non-linear equations of the phenomenological ”quantum trajectories” such as quantum state diffusion and spontaneous localization b...
متن کاملThe Equivalence Postulate of Quantum Mechanics
Implementation of the equivalence principle, stating that all physical systems can be connected by a coordinate transformation (trivializing map) to the free system with vanishing energy, uniquely leads to the quantum analogue of the Hamilton–Jacobi (HJ) equation. A property of the formalism is manifest p–q duality, a consequence of the involutive nature of the Legendre transformation and of th...
متن کاملEquivalence Principle, Planck Length and Quantum Hamilton–Jacobi Equation
The Quantum Stationary HJ Equation (QSHJE) that we derived from the equivalence principle, gives rise to initial conditions which cannot be seen in the Schrödinger equation. Existence of the classical limit leads to a dependence of the integration constant l = l1 + il2 on the Planck length. Solutions of the QSHJE provide a trajectory representation of quantum mechanics, which, unlike Bohm’s the...
متن کاملModulation Response and Relative Intensity Noise Spectra in Quantum Cascade Lasers
Static properties, relatively intensity noise and intensity modulation response in quantum cascade lasers (QCLs) studied theoretically in this paper. The present rate equations model consists of three equations for the electrons density in the conduction band and one equation for photons density in cavity length. Two equations were derived to calculate the noise and modulation response. Calcula...
متن کامل