Schrödinger'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 wave-function $\psi(x,t)$. This wave-function evolves in time according to Schrödinger's equation:

$${\rm i}\,\hbar\,\frac{\partial\psi}{\partial t} = -\frac{\hbar^2}{2\,m}\frac{\partial^2\psi}{\partial x^2} + V(x)\,\psi.$$ (119)

The wave-function is interpreted as follows: $\vert\psi(x,t)\vert^{\,2}$ 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

$$P_{x\,\in\, a:b}(t) = \int_{a}^{b}\vert\psi(x,t)\vert^{\,2}\,dx.$$ (120)

Note that this quantity is real and positive definite.