Introduction: What Does ψ Mean?
Max Born realized that \(|\psi|^2\) gives the probability density for finding the particle. This Born rule is the link between the mathematical formalism and experimental observation.
Born's Rule
The probability of finding a particle between \(x\) and \(x + dx\) is:
\[P(x)dx = |\psi(x)|^2 dx\]More generally, for any observable \(\hat{A}\) with eigenstate \(|a\rangle\):
\[P(a) = |\langle a|\psi\rangle|^2\]Why Squared?
Complex amplitudes can interfere: \(|\psi_1 + \psi_2|^2 = |\psi_1|^2 + |\psi_2|^2 + 2\text{Re}(\psi_1^*\psi_2)\)
The cross term gives quantum interference—absent in classical probability.
Worked Examples
Example 1: Probability in a Region
Probability of finding particle between \(x = a\) and \(x = b\):
\[P(a \leq x \leq b) = \int_a^b |\psi(x)|^2 dx\]Example 2: Double-Slit Interference
Two paths with amplitudes \(\psi_1\) and \(\psi_2\):
\[P = |\psi_1 + \psi_2|^2 = |\psi_1|^2 + |\psi_2|^2 + 2|\psi_1||\psi_2|\cos\phi\]The interference term creates the famous pattern.
The Quantum Connection
Born's rule is sometimes called "Born's interpretation" because it interprets the abstract wavefunction physically. Why probability equals amplitude squared (not cubed, not something else) is a deep question still debated in foundations of physics. But it works—every quantum experiment confirms it.