Quantum mechanics

The wavefunction and its probabilistic interpretation

The wavefunction ψ encodes probability amplitudes; |ψ|² is the position probability density, normalized so total probability is one.

In quantum mechanics, a particle's pure state is described by a complex wavefunction $\psi(\mathbf r,t)$ . It is neither a trajectory nor a material density; it lets us calculate probabilities of measurement outcomes.

\rho(\mathbf r,t)=|\psi(\mathbf r,t)|^2,\qquad P(\Omega,t)=\int_{\Omega}|\psi(\mathbf r,t)|^2\,d^3r

Definition: Born probability density

$|\psi|^2$ is probability per unit volume. The probability of finding the particle in a region $\Omega$ is the density integrated over that region; in SI, $|\psi|^2$ has units m $^{-3}$ .

Observe

|\psi|^2

for a superposition of two box states. Change the energy splitting to see the density oscillate while normalization is preserved.

Normalization and measurement

\int_{\mathbb R^3}|\psi(\mathbf r,t)|^2\,d^3r=1

For a particle confined to one dimension, the probability of finding it between $a$ and $b$ is $P=\int_a^b|\psi(x,t)|^2dx$ . The density can change with time, but unitary evolution of a closed system preserves its integral over all space at one.

Example: Probability in half a box

A particle in the ground state of an infinite well $0<x<L$ has $\psi_1=\sqrt{2/L}\sin(\pi x/L)$ . What is the probability of finding it in the left half?

Solution

$P=\int_0^{L/2}(2/L)\sin^2(\pi x/L)dx=1/2$ . The symmetry of $|\psi_1|^2$ about $L/2$ gives the same result.

For a position measurement, the probability of finding a particle in $[a,b]$ is $P=\int_a^b |\psi(x,t)|^2\,dx$ . A state normalized on the full line integrates to one, and probabilities for disjoint regions add. Complex amplitudes can interfere before squaring: for two indistinguishable paths, $|\psi_1+\psi_2|^2$ contains a cross term that adding separate probabilities would miss. Thus measurement is not merely sampling a pre-existing classical distribution; the chosen observable determines the possible outcomes and their probabilities. In three dimensions replace $dx$ by the volume element $d^3r$ .

Quick check

Which quantity gives the probability density for finding a particle at $\mathbf r$ ?

What does normalization of a wavefunction over all space require?

References

David J. Griffiths, Darrell F. Schroeter (2018). Introduction to Quantum Mechanics