The probability interpretation and Born rule

What does the wavefunction mean?

The wavefunction $psi(x,t)$ is not directly a classical physical wave like a water wave. It is a probability amplitude. The measurable probability density for finding a particle near position $x$ at time $t$ is given by the Born rule:

$P(x,t)=|psi(x,t)|^2$

More precisely, the probability of finding the particle between $x$ and $x+dx$ is

$dP=|psi(x,t)|^2dx$

The wavefunction itself can be complex, but $|psi|^2=psi^*psi$ is real and nonnegative.

Probability over an interval

The probability of finding the particle in an interval $[a,b]$ is

$P(aleq xleq b)=int_a^b |psi(x,t)|^2dx$

For a particle that must be somewhere on the x-axis, total probability is

$int_{-infty}^{infty}|psi(x,t)|^2dx=1$

This is the normalization condition.

Amplitudes and interference

Quantum mechanics adds amplitudes, not probabilities. If a particle can reach a detector by two indistinguishable alternatives with amplitudes $psi_1$ and $psi_2$, the total amplitude is

$psi=psi_1+psi_2$

and the probability is

$|psi|^2=|psi_1+psi_2|^2$

This includes cross terms that produce interference. If alternatives are distinguishable, interference disappears and probabilities add more classically.

Measurement outcomes

Before measurement, a quantum state can be a superposition of possible outcomes. Measurement yields one outcome, with probabilities determined by the Born rule. For position measurement, $|psi(x,t)|^2$ gives the distribution of possible positions.

The Born rule does not generally predict the exact result of a single measurement. It predicts probabilities for many identically prepared systems.

Expectation value

The average value expected from many position measurements is

angle=int_{-infty}^{infty}x|psi(x,t)|^2dx$$ This expectation value is not necessarily the most likely value. It is a statistical mean over repeated measurements. For an observable represented by operator $hat{A}$, the expectation value is $$langle A angle=int psi^*hat{A}psi,dx$$ assuming the wavefunction is normalized. ## Probability current Probability is conserved locally through a continuity equation: $$rac{partial |psi|^2}{partial t}+rac{partial j}{partial x}=0$$ where the probability current for a one-dimensional particle is $$j=rac{hbar}{2mi}left(psi^*rac{partialpsi}{partial x}-psirac{partialpsi^*}{partial x} ight)$$ This resembles conservation laws in fluids and electromagnetism. ## Philosophical impact The Born rule introduced fundamental probability into physics. In classical mechanics, probability often reflects ignorance of exact initial conditions. In quantum mechanics, even a maximally specified state generally predicts probabilities, not definite values for all observables. Different interpretations of quantum mechanics disagree about what this means, but they agree on the Born rule's experimental predictions. ## The big idea The Born rule connects the mathematical wavefunction to measurement. The squared magnitude $|psi|^2$ is probability density, amplitudes interfere, and expectation values predict statistical averages. Quantum mechanics is fundamentally a theory of probability amplitudes rather than deterministic trajectories.