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3.1: Schrodinger's Equation

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  • Consider a dynamical system consisting of a single non-relativistic particle of mass \(m\) moving along the \(x\)-axis in some real potential \(V(x)\). In quantum mechanics, the instantaneous state of the system is represented by a complex wavefunction \(\psi(x,t)\). This wavefunction evolves in time according to Schrödinger’s equation: \[\label{e3.1} {\rm i}\,\hbar\,\frac{\partial\psi}{\partial t} = -\frac{\hbar^{\,2}}{2\,m}\frac{\partial^{\,2} \psi}{\partial x^{\,2}} + V(x)\,\psi.\] The wavefunction is interpreted as follows: \(|\psi(x,t)|^{\,2}\) is the probability density of a measurement of the particle’s displacement yielding the value \(x\). Thus, the probability of a measurement of the displacement giving a result between \(a\) and \(b\) (where \(a<b\)) is \[\label{e3.2} P_{x\,\in\, a:b}(t) = \int_{a}^{b}|\psi(x,t)|^{\,2}\,dx.\] Note that this quantity is real and positive definite.