This Cheat Sheet provides a quick reference to some of the main equations used in quantum physics.

## Quantum physics and the Hamiltonian

One of the central problems of quantum mechanics is to calculate the energy levels of a system. The energy operator, called the *Hamiltonian**,** *abbreviated *H, *gives you the total energy. Finding the energy levels of a system breaks down to finding the eigenvalues of the problem

The eigenvalues can be found by solving the equation:

## The Heisenberg uncertainty principle

In quantum physics, you encounter the Heisenberg uncertainty principle, which says that the better you know the position of a particle, the less you know the momentum, and vice versa. In the *x* direction, for example, that looks like this:

where D*x* is the measurement uncertainty in the particle’s *x* position, D*p** _{x}* is its measurement uncertainty in its momentum in the

*x*direction, and

This relation holds for all three dimensions:

## The Schrödinger equation

When a quantum mechanical state can be described by a wave function,

then this is a solution of the Schrödinger equation, which is written in terms of the potential

and energy

like so:

The Schrödinger equation work in three dimensions as well:

## Spin operators and commutation in quantum physics

Don’t think quantum physics is devoid of anything but dry science. The fact is that it’s full of relationships, they’re just commutation relationships — which are pretty dry science after all. In any case, among the angular momentum operators L_{x}*,* L_{y}*, *and L* _{z}*, are these commutation relations:

All the orbital angular momentum operators, such as L_{x}*,* L_{y}*,* and L_{z}*,* have analogous spin operators: S_{x}*,* S_{y}*,* and S_{z}. And the commutation relations work the same way for spin:

## The Compton effect

In quantum physics, you may deal with the Compton effect of X-ray and gamma ray qualities in matter. To calculate these effects, use the following formula, which assumes that the light is represented by a photon with energy E = *h*u and that its momentum is *p* = E/*c*: