particlewavefunction

The particle wavefunction is a mathematical description of the quantum state of a particle in nonrelativistic quantum mechanics. In the position representation it is written as ψ(x,t), where x is the spatial coordinate and t is time. The squared magnitude |ψ(x,t)|^2 gives the probability density of finding the particle at position x at time t, with the state normalized so that ∫ |ψ(x,t)|^2 dx = 1. The wavefunction evolves deterministically according to the Schrödinger equation: iħ ∂ψ/∂t = H ψ, where H is the Hamiltonian operator. For a free particle, plane-wave solutions exist, and in general the wavefunction can be expressed as a superposition of energy eigenstates, each carrying a time-dependent phase factor e^{-iEt/ħ}.

The wavefunction is central to predicting measurement outcomes via the Born rule. Observables correspond to operators, and the probability of a given outcome is determined by projecting ψ onto the relevant eigenstate. Although the wavefunction contains all information about the quantum state, it is not itself directly observable; only probabilities and expectation values derived from it can be measured. The global phase of a wavefunction has no physical significance.

In multi-particle systems, the wavefunction becomes a function on a higher-dimensional configuration space, ψ(x1,x2,…,t), and may include spinor or other internal degrees of freedom. Entanglement occurs when the total wavefunction cannot be factored into separate single-particle components. The term “particle wavefunction” is often used to describe the state in a particular representation, especially the position or momentum representation, within the broader framework of quantum states in Hilbert space.