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

## Wave function

`\( \psi_n \)`Unit

`\( \frac{1}{\sqrt{\text m}} \)`

\(n\)-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

`\( n \)`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

`\( x \)`Unit

`\( \text{m} \)`

Space coordinate of the one-dimensional potential box. This determines the limits \( x = 0 \) and \( x = L \) of the potential well.

## Length

`\( L \)`Unit

`\( \text{m} \)`

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