The Schrödinger equation

The fathers of matrix quantum mechanics believed that the quantum particles are unvisualizable and pop into existence only when measured. Challenging this view, in 1926 Erwin Schrödinger developed his wave equation that describes the quantum particles as packets of quantum probability amplitudes evolving in space and time. Thus, Schrödinger visualized the unvisualizable and lifted the veil that has been obscuring the wonders of the quantum world.

The Schrödinger equation governs the time evolution of closed quantum physical systems \begin{equation}\imath\hbar\frac{\partial}{\partial t}\,|\Psi(x,y,z,t)\rangle=\hat{H}\,|\Psi(x,y,z,t)\rangle\label{eq:1}\end{equation} where $\imath$ is the imaginary unit, $\hbar$ is the reduced Planck constant, $\frac{\partial}{\partial t}$ indicates a partial derivative with respect to time, $\Psi$ is the wave function of the quantum system, $x$, $y$, $z$, $t$ are the three position coordinates and time respectively, and $\hat{H}$ is the Hamiltonian operator corresponding to the total energy of the system. The solution of the Schrödinger equation is the quantum wave function $\Psi$ of the system. At each point $(x,y,z)$ in space at a time $t$, the value of the quantum wave function is a complex number $\Psi(x,y,z,t)$, referred to as a quantum probability amplitude. Since the quantum wave function provides a complete description of the quantum state of a physical system, it follows that the fabric of the quantum world is made of quantum probability amplitudes $\Psi(x,y,z,t)$. The squared modulus $\left|\Psi(x,y,z,t)\right|^{2}$ of each quantum probability amplitude gives a corresponding quantum probability for a physical event to occur at the given point in space and time. The quantum probabilities do not arise due to our ignorance of what the state of the quantum system is, but rather represent inherent propensities of the quantum systems to produce certain outcomes under experimental measurement.

Despite that the Schrödinger equation can be easily written in its general mathematical form, solving it for an arbitrary Hamiltonian $\hat{H}$ is extremely difficult and usually done only approximately with a finite numerical precision. Understanding the general properties of the Schrödinger equation and its solutions is essential for modern developments in theoretical physics, quantum information theory, quantum chemistry, biology, neuroscience, cognitive science and artificial intelligence.

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