3.1: Schrodinger's Equation

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.