# Formula Infinite 1d square well Wave function    Quantum number    Length

## Wave function

Unit
$$n$$-th wave function $$\psi_n$$ is the solution of the Schrödinger equation for a bound particle (for example an electron) in a one-dimensional, infinite potential well. The squared magnitude $$|\psi_n|^2$$ of the wave function is used to calculate the probability of determining a particle at a particular location $$x$$ in the potential well.

Outside the potential well, the wave function vanishes. Thus, the probability of finding the particle outside the potential well is zero.

## Quantum number

Unit
The quantum number $$n$$ takes discrete values: $$n ~=~ 1,2,3...$$.

For $$n ~=~ 1$$ the wave function $$\psi_1(x)$$ of a bound particle in the ground state is:$\psi_1(x) ~=~ \sqrt{\frac{2}{L}}\sin\left(\frac{\pi}{L}\,x\right)$

## Space coordinate

Unit
Space coordinate of the one-dimensional potential box. This determines the limits $$x = 0$$ and $$x = L$$ of the potential well.

## Length

Unit
Length of the one-dimensional potential well where the potential is zero.