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  "body": "\n## Definition\n\nLet $n \\ge 1$, let $\\hbar \u003e 0$ and $m \u003e 0$ be constants, and let $V \\colon \\R^n \\to \\R$ be a measurable function, the *potential*.\nLet $\\psi(x, t)$, for $x \\in \\R^n$ and $t \\in \\R$, be an unknown complex-valued function, and let $\\Delta$ be the [[Laplacian|0E0B6RA8K0]] in $x$.\nThe *Schrödinger equation* with potential $V$ is the [[partial differential equation|GC5BDZ6QBC]]\n$$i\\hbar \\, \\frac{\\partial \\psi}{\\partial t}(x, t) = -\\frac{\\hbar^2}{2m} \\Delta \\psi(x, t) + V(x) \\psi(x, t).$$\nWith $H = -\\frac{\\hbar^2}{2m}\\Delta + V$, the [[Schrödinger operator|89AA6XJY5K]], it reads $i\\hbar\\, \\partial_t \\psi = H\\psi$.\nThe *time-independent* Schrödinger equation is the eigenvalue equation $H\\varphi = E\\varphi$; if it holds, then $\\psi(x, t) = e^{-iEt/\\hbar}\\varphi(x)$ solves the time-dependent equation.\n\n## Remarks\n\nIn mathematics one often sets $\\hbar = 1$ and $2m = 1$, giving $i\\partial_t u = -\\Delta u + Vu$.\n\nWhen $H$ is self-adjoint on $L^2(\\R^n)$, Stone's theorem gives the solution $\\psi(t) = e^{-itH/\\hbar}\\psi(0)$ for every initial value in $L^2(\\R^n)$, and $e^{-itH/\\hbar}$ is unitary, so $\\norm{\\psi(t)}_{L^2} = \\norm{\\psi(0)}_{L^2}$.\n\nExample: with $V = 0$, $\\hbar = 1$, $2m = 1$ and $k \\in \\R^n$, the plane wave $u(x, t) = e^{i(k \\cdot x - \\abs{k}^2 t)}$ is a solution.\nIndeed $i\\partial_t u = i \\cdot (-i\\abs{k}^2) u = \\abs{k}^2 u$ and $-\\Delta u = \\abs{k}^2 u$.\n\nNon-example: the heat equation $\\partial_t u = \\Delta u$ differs only by the factor $i$, but it is not a Schrödinger equation.\nIts solutions decay in $L^2$ instead of keeping their norm: $u = e^{i k\\cdot x - \\abs{k}^2 t}$ solves it, and its modulus $e^{-\\abs{k}^2 t}$ decreases.\n\n## Explanations\n\n### L1 human\n\nThe Schrödinger equation says that the time derivative of the wave function, times $i\\hbar$, equals the energy operator applied to the wave function.\nThe factor $i$ makes solutions oscillate and spread rather than decay, and it keeps the total probability equal to one.\n",
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