Wave function

mathematical description of the quantum state of a system; complex-valued probability amplitude, and the probabilities for the possible results of measurements made on the system can be derived from it

In quantum mechanics, the Wave function, usually represented by Ψ, or ψ, describes the probability of finding an electron somewhere in its matter wave. To be more precise, the square of the wave function gives the probability of finding the location of the electron in the given area, since the normal answer for the wave function is usually a complex number. The wave function concept was first introduced in the Schrödinger equation.

Mathematical interpretation

The formula for finding the wave function (i.e., the probability wave), is below:

${\displaystyle i\hbar {\frac {\partial }{\partial t}}\Psi (\mathbf {x} ,\,t)={\hat {H}}\Psi (\mathbf {x} ,\,t)}$

where i is the imaginary number, ψ (x,t) is the wave function, ħ is the reduced Planck constant, t is time, x is position in space, Ĥ is a mathematical object known as the Hamiltonian operator. The reader will note that the symbol ${\displaystyle {\frac {\partial }{\partial t}}}$  denotes that the partial derivative of the wave function is being taken.

Probabilistic nature

Note that the wave in question, commonly referred to as Schrodinger's Wave, "is not in physical space", and does not give the probability of where an object is -- the object has no physical existence. The wave function gives "the probability that if you look, you will observe (emphasis mine) the object at a particular place". When nothing disturbs it, an electron does not exist in any place. "Quantum mechanics brings probability to the heart of the evolution of things."[1][2]

References

1. Rovelli, Carlo (2017). Reality is not what it seems: the elementary structure of things. Translated by Carnell, Simon; Segre, Erica (1st American ed.). New York, New York: Riverhead Books. ISBN 978-0-7352-1392-0.
2. Rosenblum, Bruce; Kutner (2011). Quantum enigma: physics encounters consciousness (2nd ed.). Oxford: Oxford Univ. Press. ISBN 978-0-19-975381-9.