In quantum mechanics, the wave function is a mathematical function that describes the state of a quantum system. It is denoted by the symbol Ψ (psi) and depends on the position and time variables of the system.
The wave function contains information about the probabilities of different outcomes when measuring physical quantities, such as position or momentum, of a quantum particle. The square of the wave function, |Ψ|^2 (psi squared), represents the probability density function. It gives the probability of finding the particle at a specific position in space.
Mathematically, the probability density function is calculated by taking the absolute value squared of the wave function:
|Ψ|^2 = Ψ * Ψ^*,
where Ψ^* is the complex conjugate of the wave function Ψ. The probability density function is always a positive real number and integrates to unity over all space, ensuring that the total probability of finding the particle somewhere is equal to 1.
It's important to note that the interpretation of the wave function and its square is a subject of ongoing debate and various interpretations exist within the realm of quantum mechanics. However, the probabilistic interpretation described here is the most widely accepted and utilized interpretation in the field.