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Formula Infinite 1d square well Wave function    Quantum number    Length

Formula: Infinite 1d square well

Wave function

\(n\)-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

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

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


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