# Schrodinger Equation

## $\LARGE i\hbar\frac{\partial\psi}{{\partial}t}=-\frac{\hbar^2}{2m}\nabla^2\psi+V\psi$

### The Terms

$i$ - Imaginary unit, $i=\sqrt{-1}$

$\hbar$ - Planck's constant divided by $2\pi$  [Value: 1.051 × 10-34 m2 kg s-1]

$\psi$ - Wavefunction of the system - the probability amplitude for different configurations of the system at different times. Also known as the quantum state, this is the most complete description that can be given to a physical system.

$m$ - Mass of the particle [Units: kg]

$\nabla^2$ - Laplacian operator, $\nabla^2\mathbf{u}=\partial_{xx}^2\mathbf{u}+\partial_{yy}^2\mathbf{u}+\partial_{zz}^2\mathbf{u}$

$V$ - Potential energy [Units: J]

### What Does It Mean?

This is the Schrödinger equation for a single particle in a potential. It describes the evolution of the quantum state (wavefunction) of a physical system and solutions can describe both microscopic and macroscopic systems, from subatomic matter to perhaps even the whole universe.

### Did You Know?

The Schrödinger equation is synonymous with quantum theory (that's why we included it), but it is simply a non-relativistic wave equation and need not be related to quantum theory, and on the other hand you can have quantum theory without the Schrödinger equation (see for example spin systems which are finite dimensional or the Dirac equationfor relativisitc particles). The peculiarities of Quantum theory come form the meaning of $\psi$ and how it is used to make predictions - those are both very odd and poorly understood.

### Further information at Warwick

The Schrödinger equation isintroduced and used in the first year module &quot;PX101 Quantum Phenomena" and features in many later modules which involve quantum physics.