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.

Back To Quantum Mechanics

Mini Physics

As the Administrator of Mini Physics, I possess a BSc. (Hons) in Physics. I am committed to ensuring the accuracy and quality of the content on this site. If you encounter any inaccuracies or have suggestions for enhancements, I encourage you to contact us. Your support and feedback are invaluable to us. If you appreciate the resources available on this site, kindly consider recommending Mini Physics to your friends. Together, we can foster a community passionate about Physics and continuous learning.

Leave a Comment

This site uses Akismet to reduce spam. Learn how your comment data is processed.