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Quantum mechanics describes the microscopic properties of nature in a regime where classical mechanics no longer applies. It explains phenomena such as the wave-particle duality, quantization of energy, and the uncertainty principle and is generally used in single-body systems. Use the quantum-field-theory tag for the theory of many-body quantum-mechanical systems.

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Who knows experiments on wave packets spread?

A few years late but here's a realization of Gaussian wave packet spreading in continuous space that uses few Lithium atoms and an optical lattice: https://arxiv.org/abs/2404.05699 A brief summary of …
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Time evolution of a state on adding a constant to the Hamiltonian of the Schrodinger equation

Adding a constant to a potential energy function does not change the dynamics (time evolution) of an object in Newtonian physics. I expect the same to be true in quantum mechanics but for some reason, …
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Local charge conservation in quantum mechanics

The description of charged particles in electrodynamics obeys the continuity equation, $$ \nabla \cdot \textbf{J} = -\frac{\partial \rho}{\partial t} $$ With the common physical understanding of this …