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Formula Infinite 1d square well Wave function    Quantum number    Length

Formula
Formula: Infinite 1d square well

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.