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The wave packet shape of a quantum system: How are the properties and behavior of a quantum system depends fundamentally on uncertainty in its position?

Mohamed Mostafa

Published Nov 9, 2023• 0 citations
Publisher

Abstract

Abstract In this work we determine the parameter that characterized different wave packets from each other for different quantum systems. We find a specific definition for this parameter by relate the statistical distribution of the quantum system to the quantum fluctuations of the vacuum, and thus we find the mechanism by which the wave packet changes at every measurement-interaction with it. We also explain an important practical aspect of this mechanism.

DOI
10.21203/rs.3.rs-3575500/v1

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Abstract In this work we determine the parameter that characterized different wave packets from each other for different quantum systems. We find a specific definition for this parameter by relate the statistical distribution of the quantum system to the quantum fluctuations of the vacuum, and thus we find the mechanism by which the wave packet changes at every measurement-interaction with it. We also explain an important practical aspect of this mechanism.
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The quantum mechanics is most easily started from the experimentally observed discrete energy states of bound particles. The free particle may take on any energy whatever, which makes the quantum description of a free particle more awkward conceptually and mathematically than the quantum description of a bound particle. The construction of moving quantum spatial probabilities will require a superposition of stationary state wave functions of different energies. Any coherent superposition of energy eigen states for a particle represents a possible quantum state of that particle. The most important property of the spatial probability function for superposed energy states is that it changes with time. A moving probability that is limited spatially is called a packet and the corresponding quantum-mechanical description is called a packet state. The free-particle packet states examples satisfy the position-momentum uncertainty relation. Probably the most important application of the energy time uncertainty relation concerns the lifetimes of the excited states of atoms and nuclei.
The spread of the wave-function, or quantum uncertainty, is a key notion in quantum mechanics. At leading order, it is characterized by the quadratic moments of the position and momentum operators. These evolve and fluctuate independently from the position and momentum expectation values. They are extra degrees of quantum mechanics compared to classical mechanics, and encode the shape of wave-packets. Following the logic that quantum mechanics must be lifted to quantum field theory, we discuss the quantization of the quantum uncertainty as an operator acting on wave-functions over field space and derive its discrete spectrum, inherited from the sl2\textrm{sl}_{2}sl2​ Lie algebra formed by the operators x^2\hat{x}^{2}x^2, p^2\hat{p}^{2}p^​2 and xp^\widehat{xp}xp​. We further show how this spectrum appears in the value of the coupling of the effective conformal potential driving the evolution of extended Gaussian wave-packets according to Schrödinger equation, with the quantum uncertainty playing the same role as an effective intrinsic angular momentum. We conclude with an open question: is it possible to see experimental signatures of the quantization of the quantum uncertainty in non-relativistic physics, which would signal the departure from quantum mechanics to a QFT regime?
Abstract In this work, we prove that the behavior of quantum systems during various measurement processes is fully understood through the statistical structure in quantum mechanics represented by the uncertainty principle. We also prove that any measurement process is never different from any interaction between quantum particles, and this is evidenced only by the existence of a minimum uncertainty in position characteristic of each measurement system imposed by the statistical-mathematical structure of quantum mechanics.