In nonrelativistic quantum mechanics, the state of a particle is represented by a wavefunction $\psi(x,t)$ or more generally by a vector in Hilbert space. The wavefunction contains all probabilistic information needed to predict measurement outcomes.

A physically valid wavefunction must be normalized so that $$\int_{-\infty}^{\infty} |\psi(x,t)|^2\,dx = 1.$$ This ensures the particle is found somewhere with total probability $1$.

If $\psi_1$ and $\psi_2$ are allowed states, then $c_1\psi_1 + c_2\psi_2$ is also an allowed state. Superposition is one of the most distinctive features of quantum theory and leads directly to interference.

For a closed system, time evolution is governed by the Schrödinger equation $$i\hbar \frac{\partial \psi}{\partial t} = \hat H\psi,$$ where $\hat H$ is the Hamiltonian operator.

The wavefunction is a calculational object with direct operational meaning through measurement probabilities. Different interpretations disagree on what it 'really is,' but they agree on the predictions of standard quantum mechanics.