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  "body": "\n## Summary\n\nThe Schrödinger equation $i\\hbar\\, \\partial_t \\psi = -\\frac{\\hbar^2}{2m} \\Delta \\psi + V \\psi$ describes how the wave function $\\psi$ of a quantum particle of mass $m$ in a potential $V$ evolves in time.\nIt is a linear dispersive partial differential equation with complex-valued solutions.\nIts right-hand side is a [[Schrödinger operator|89AA6XJY5K]] applied to $\\psi$.\nThe solutions are given by a unitary group of operators, so the total probability $\\int \\abs{\\psi}^2 \\, dx$ is conserved.\nIt is the basic equation of non-relativistic quantum mechanics and a model problem for dispersive equations.\n\n## Explanations\n\n### L1 human\n\nIn quantum mechanics a particle is described by a complex wave, and $\\abs{\\psi(x, t)}^2$ is the probability density of finding it at $x$ at time $t$.\nThe Schrödinger equation says how this wave moves: the kinetic energy term spreads it out, and the potential bends it.\nA free particle is a superposition of plane waves $e^{i(k \\cdot x - \\abs{k}^2 t)}$, in units where $\\hbar = 1$ and $2m = 1$.\nWaves with larger $k$ move faster, so a localised wave packet spreads out in time, which is the typical behaviour of dispersive equations.\n",
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