Normalization Of The Wave Function

The wave function ψ is not an observable quantity. It manifests itself only on the statistical distribution of particle detection.

In one-dimension, the quantity $| \psi |^{2} \, dx$ represents the probability of finding the particle associated with the wave function ψ(x) in the interval dx at some position x.

i.e. $\text{Probability} = | \psi |^{2} \, dx$

Thus, ψ2 is interpretated as the probability per unit length (also called the probability density) that we will find a particle.

Note: In three-dimensional space, $| \psi |^{2}$ is regarded as the probability per unit volume.

But the particle must be somewhere. Therefore, if we sum the probabilities of finding the particle in all such intervals dx along the x axis, the result must be 1. There must be a 100% chance that we will find the particle somewhere. Since the probability density may vary with position, that sum becomes an integral, and we have

$$\int\limits_{- \infty}^{+ \infty} | \psi |^{2} \, dx = 1$$

The above equation is called the normalization condition.

Once we have a solution ψ(x) to the Schrodinger equation, this condition can be used to set the overall amplitude of the wave function ψ.

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