A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system in position or momentum space. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements performed on the system can be derived from it. The most common symbols used to denote the wave function, are the Greek letters \(\psi \) or \(\Psi \) (lowercase and uppercase psi, respectively).

In quantum mechanics, the expected value is the probabilistic expected value of the result (measurement) of an experiment. It can be considered as an average of all possible measurement results, weighted by their probability.
Consider an operator \( \hat{A} \). The expectation value is then \( \langle A \rangle = \langle \psi |\hat{A}| \psi \rangle \) in Dirac notation with \( | \psi \rangle \) a normalized state vector.