# Class: Quantum Mechanics

Quantum mechanics is one of the most classic examples of field theories in physics. The Schrödinger equation is a partial differential equation that describes how the quantum state of a physical system changes over time.

In this class, we simulate quantum mechanics through the evolution of the Schrödinger equation.

## The Schrödinger equation

Evolving according to the Schrödinger equation with electron mass

Dividing by the Hartree energy \(E_h = \frac{\hbar^2}{m_e a_0^2}\)

When expressing time in units of \(\frac{\hbar}{E_h}\), potential energy in units of \(E_h\) and length squared in units of

we get the Schrödinger equation in its dimensionless form

So

Atomic unit of | Value |
---|---|

Length | 0.529 Å (Angstrom) |

Energy | 27.2 eV (electron volts) |

Time | 24.2 aS (atto seconds) |

## The Born rule

The Born rule states that the probability \(p\) of measuring a particle in the interval \([a,b]\) is given by

## A wave packet

A Gaussian wave function is often called a wave packet and can visualize the position and motion of a particle in a quantum mechanical system. An initial wave function with a widt of \(\sigma\) is given by

so that \(|\psi|^2\) is a Gaussian distribution.
An initial velocity \(\mathbf v_0\) can be given to the wave packet by multiplying with a complex phase \(e^{\mathfrak i \mathbf v_0 \cdot \mathbf r}\).
Such an initial condition can be configured by the function `qm.conf_initial_condition_Gaussian`

.