The wave function is a concept in quantum mechanics which is used to describe the features of the wave nature of an electron. We describe the wave of an electron using a mathematical function represented by the Greek letter psi, Ψ.
De Broglie was the first to suggest the wave like properties of an electron. He also presented a formula to calculate this wavelength, λ = h / mv, where h is Planck constant, m is mass of the electron and v is velocity of the electron. Then in 1926, Erwin Schrodinger, an Austrian-Irish physicist formulated an equation which now bares his name, the Schrodinger Equation. It is given as-
ignore the equation above, there is no need to understand that complex equation, we just need to know what this equation shows us. The equation is used to solve psi (Ψ) on a graph as a function of x. for example, look at this graph below, the graph of psi. psi is taken on the y-axis and the distance from the center of an atom or “r” is taken on x-axis.
Using this wave function, we can calculate the probability of finding an electron in a certain region of space around around the nucleus of atom. Interestingly, it was not Erwin Schrodinger who came up with this interpretation of the wave function but it was a physicist named Max Born. Schrodinger assumed that the graph of a wave function represents charge density of an electron or the way in which an electron distributes its charge in space. But then Max Born came up and said, “No, the wave function does not represent charge density, but a probability density. If you square the wave function, you get the probability of locating an electron in a point in space”. So it is Ψ which gives us the shape of a wave and Ψ2 which gives us the probability density of an electron. In the graph, the points where the maxima or minima are the greatest, depict the highest probability of finding a certain electron in that point in space. Whereas, there is a lesser probability of finding an electron at points which have lower maxima and minima. The points where the wave function touches the x-axis have the zero probability of finding an electron.
The Schrodinger equation is one of the most important formula in all of quantum mechanics. The equations helps us in locating the most probable region of space where an electron can be found. But a question still remains till date, what is the wave function or the wave of an electron made of? Is it just a mathematical thing or does it have any physical or mechanical properties. This is a highly debated topic among physicists and I believe one of you readers might solve this mystery.