The Born rule, named after the physicist Max Born, is a fundamental principle in quantum mechanics that relates to the measurement of quantum systems. It provides a way to determine the probability of obtaining a particular outcome when measuring a physical property of a quantum particle.
According to the Born rule, the probability density of finding a quantum particle in a particular state is proportional to the square of the absolute value of its wavefunction. Mathematically, for a given state described by a wavefunction Ψ, the probability density P(x) of finding the particle at position x is given by:
P(x) = |Ψ(x)|^2
In the context of a particle in a box, we consider a one-dimensional quantum system where a particle is confined within a finite region. The box has infinite potential energy outside its boundaries, which results in the particle being confined to the region within the box.
The wavefunction Ψ(x) for a particle in a box can be obtained by solving the Schrödinger equation with appropriate boundary conditions. The square of the absolute value of this wavefunction, |Ψ(x)|^2, gives us the probability density of finding the particle at a particular position within the box.
The Born rule allows us to calculate the probability of finding the particle in a specific location. For example, if we want to determine the probability of finding the particle between positions x1 and x2, we integrate the probability density over that range:
P(x1 ≤ x ≤ x2) = ∫[x1,x2] |Ψ(x)|^2 dx
By applying the Born rule, we can analyze the behavior of particles in quantum systems and make probabilistic predictions about their measurements and outcomes.