Schrodinger equation

See: Time independent Schrodinger equation Typically when we refer to the Schrodinger equation we are talking about the time independent one The Schrödinger equation is a fundamental equation in quantum mechanics that describes how the wave function of a physical system evolves over time. It is named after the Austrian physicist Erwin Schrödinger, who formulated it in 1925.

The time-dependent Schrödinger equation for a non-relativistic particle (i.e., a particle not moving close to the speed of light) is given by:

$i\hslash \frac{\partial }{\partial t}\Psi (\mathbf{r},t)=\hat{H}\Psi (\mathbf{r},t)$

Here’s what the terms in the equation represent:

• $i$ is the imaginary unit $i^2=-1$.
• $\hslash$ is the reduced Planck’s constant, which is a fundamental constant in quantum mechanics.
• $\frac{\partial }{\partial t}$denotes the partial derivative with respect to time (t).
• $\Psi (\mathbf{r},t)$ is the wave function, which is a mathematical function that contains all the information about a quantum system. It depends on the spatial coordinates $r$ and time $t$.
• $\hat{H}$ is the Hamiltonian operator, which represents the total energy operator of the system.

The time-independent Schrödinger equation, which is a special case for systems with constant energy (i.e., systems not subject to time-dependent external forces), is given by:

$\hat{H}\psi (\mathbf{r})=E\psi (\mathbf{r})$

Here, $\psi (r)$ is the spatial part of the wave function, which depends only on the spatial coordinates $\mathbf{r}$, and $E$ represents the energy of the system.

Solving the Schrödinger equation allows us to determine the allowed energy levels and corresponding wave functions for a given quantum system. The wave function provides a complete description of the system’s quantum state, and from it, we can calculate various observables, such as position, momentum, and energy.

It’s important to note that the Schrödinger equation is a cornerstone of non-relativistic quantum mechanics, applicable to systems that do not involve speeds approaching the speed of light. For relativistic particles, a different equation, the Dirac equation, is used.