The Schrödinger Wave Equation: An Elementary Idea

Since an electron in an atom has an associated wave (de Broglie, Section 2.4) and its exact path cannot even in principle be known (Heisenberg, Section 2.5), the correct way to describe an electron's behaviour in an atom is not to solve for a classical orbit but to solve a wave equation for it. Erwin Schrödinger (1926) wrote down exactly such an equation, now called the Schrödinger wave equation, in its time-independent form:

Here $\hat{H}$ (read "H-hat") is a mathematical operator called the Hamiltonian, built from the kinetic and potential energy of the electron; $E$ is the total energy of the electron; and $\psi$ (psi) is the wave function -- a mathematical function of the electron's coordinates that contains all the information that can, in principle, be known about the state of that electron. At this elementary stage, the exact mathematical form of $\hat{H}$ and the method of solving the equation are not needed; what matters is what the solutions mean.

What the wave function represents. The wave function $\psi$ itself has no direct physical meaning and can even take negative or complex values. Its square, $\psi^2$ (or, more precisely, $|\psi|^2$), does have a direct physical meaning: at any given point in space, $\psi^2$ is proportional to the probability of finding the electron at that point. This is the probability-density interpretation, and it is the precise mathematical replacement for Bohr's idea of a definite orbit -- instead of the electron following one fixed path, quantum mechanics describes a probability distribution in space, with the electron more likely to be found where $\psi^2$ is large and less likely to be found where it is small. …