In classical mechanics, if a particle of mass m subject to a force F is to move along a specified x-axis, the position of the particle at any time is given by x(t). From its position, we can determine its velocity v=dx/dt, its momentum p=mv, its kinetic energy $E_{k} = \frac{1}{2} mv^{2}$ and so on.
However, in the microscopic world, the state of an electron (information about its position x, energy, momentum and so on) is described by a wave function or probability amplitude function ψ (x,t). Playing the role of Newton’s equation of motion in classical mechanics is a new equation called the Schrodinger equation, put forward by Austrian pysicist Erwin Schrodinger in 1926.
The wave function ψ(x,t) for a given system can be obtained by solving the Schrodinger equation.
The wave function ψ can, in general, take on positive and negative values, however, it is the square of the amplitude of the wave function |ψ (x,t)|2 that gives the probability of finding the particle at a position x at time t.
|ψ (x,t)|2 dx = Probability of finding a particle between position (x + dx) at time t.
