The Amazing Story of Quantum Mechanics by James Kakalios

forces do not change with time, as in the electrical attraction between the negatively charged electron and the positively charged protons in the atomic nucleus. In this case we need only to know how the electrical force varies with the separation between the two charges, to determine the potential V at all points in space.
    I’ve belabored the fact that V can take on different values depending on which point in space x we are, and at what time t we consider, because what is true of V is also true of the wave function Ψ. Mathematically, an expression that does not have one single value, but can take on many possible values depending on where you measure it or when you look, is called a “function.” Anyone who has read a topographic map is familiar with the notion of functions, where different regions of the map, sometimes denoted by different colors, represent different heights above sea level. This is why Ψ is called a “wave function. ” It is the mathematical expression that tells me the value of the matter-wave depending on where the electron is (its location in space x) and when I measure it (at a time t). Though as we’ll see in the next chapter with Heisenberg, where and when get a little fuzzy with quantum objects.
    For an electron near a proton, such as a hydrogen atom, the only force on the electron is the electrostatic attraction. Since the nature of the electrical attraction does not change with time, the potential V will depend only on how far apart in space the two charges are from each other. One can then solve the Schrödinger equation to see what wave function Ψ. will be consistent with this particular V. The wave function Ψ will also be a mathematical function that will take on different values depending on the point in space. Now, here’s the weird thing (among a large list of “weird things” in quantum physics). When Schrödinger first solved this equation for the hydrogen atom, he incorrectly interpreted what Ψ represented—in his own equation!
    Schrödinger knew that Ψ itself could not be any physical quantity related to the electron inside the atom. This was because the mathematical function he obtained for Ψ involved the imaginary number i. Any measurement of a real, physical quantity must involve real numbers, and not the square root of -1. But there are mathematical procedures that enable one to get rid of the imaginary numbers in a mathematical function. Once we know the wave function Ψ, then if we square it—that is, multiply it by itself soΨ×Ψ* = Ψ 2 (pronounced “sigh-squared”)—we obtain a new mathematical function, termed, imaginatively enough, Ψ 2 . 26 Why would we want to do that? What physical interpretation should we give to the mathematical function Ψ 2 ?
    Schrödinger noticed that Ψ 2 for the full three-dimensional version of the matter-wave equation had the physical units of a number divided by a volume. The wave function Ψ itself has units of 1/(square root (volume)). This is what motivated the consideration of Ψ 2 rather than Ψ—for while there are physically meaningful quantities that have the units of 1/volume, there is nothing that can be measured that has units of 1/(square root (volume)). He argued that if Ψ 2 were multiplied by the charge of the electron, then the result would indicate the charge per volume, also known as the charge density of the electron. Reasonable—but wrong. Ψ 2 does indeed have the form of a number density, but Schrödinger himself incorrectly identified the physical interpretation of solutions to his own equation.
    Within the year of Schrödinger publishing his development of a matter-wave equation, Max Born argued that in fact Ψ 2 represented the “probability density” for the electron in the atom. That is, the function Ψ 2 tells us the probability per volume of finding the electron at any given point within the atom. Schrödinger thought this was nuts, but in fact Born’s interpretation is accepted by all