Normalization of Wave Function

Given a wave function $\psi = A \sin(2x)$ for $x \in [-\pi, \pi]$ and $\psi (x) = 0$ otherwise, find the value of $A$ such that the wave function is normalized.

Normalization is a fundamental concept in quantum mechanics, reflecting the probabilistic nature of quantum states. A wave function must be normalized so that the total probability of finding the particle within the entire space is one. This ensures that the wave function accurately represents a valid quantum state. The act of normalizing involves integrating the square of the absolute value of the wave function over the entire space it occupies and setting the result equal to one. For wave functions expressed in a piecewise manner — such as the given scenario where the wave function is non-zero over a specific range and zero elsewhere — normalization requires careful integration only over the non-zero region. The amplitude A in the wave function is adjusted to ensure this condition holds. By focusing on the region of interest and applying the integral condition, one can solve for the normalization constant.