In the same year that Schrödinger published the equation which now bears his name, the non-relativistic theory was completed by Max Born’s insight that the solutions of the Schrödinger equation are tools for calculating probabilities of measurement outcomes. Specifically, if ψ is associated with a particle, its magnitude squared, |ψ|2, is a (time-dependent) probability density, in the sense that the probability of detecting the particle in a region R, by a measurement made at the time t, is given by
p(R,t) = ∫R d3r |ψ(x,y,z,t)|2.
The symbol ∫R d3r indicates a summation over all points contained in the 3-dimensional region R: every point (x,y,z) contributes a complex number |ψ(x,y,z,t)|2. (For the purists: the summation actually extends over mutually disjoint subsets that together are coextensive with R, in the limit that the volumes of the individual subsets tend to zero.)
Like any probability density, |ψ(x,y,z,t)|2 defines, for each spatial dimension, a mean or “expected” value and a standard or “root-mean-square” deviation from the mean. With respect to the x-axis, we shall denote these by <x> and Δx, respectively.
Defining ψ(k,t) as the complex number a(k)[1:−ω(k)t] — a function of k and t — we can cast Eq. (2.3.5) into the form
(2.4.1) ψ(x,t) = (1/√(2π)) ∫dk ψ(k,t) [1:kx].
This tells us that ψ(k,t) is the Fourier transform of ψ(x,t) and thus can be written as
(2.4.2) ψ(k,t) = (1/√(2π)) ∫dx ψ(x,t) [1:−kx].
If two functions are related like (2.4.1) and (2.4.2), the standard deviations they define satisfy the inequality
Δx Δk ≥ 1/2.
Remembering de Broglie’s relation p = ℏk, we arrive at the “uncertainty relation” for the x-components of a particle’s position and momentum, first derived by Werner Heisenberg, also in 1926:
(2.4.3) Δx Δpx ≥ ℏ/2.
Bohr, as you will remember, postulated the quantization of angular momentum in an effort to explain the stability of atoms. An atom “occupies” hugely more space than its nucleus (which is tiny by comparison) or any one of its electrons (which do not appear to “occupy” any space at all). How then does an atom come to “occupy” as much space as it does, without collapsing? The answer is: because of the “uncertainties” in both the positions and the momenta of its electrons relative to its nucleus. It is these “uncertainties” that “fluff out” matter.
Except that “uncertainty” cannot then be the right word. What “fluffs out” matter cannot be our very own, subjective uncertainty about the values of the relative positions and momenta of its constituents. It has to be an objective fuzziness of these values.
Consider again the lowly hydrogen atom. Intuitively it seems clear enough that the fuzziness of the electron’s position relative to the proton can be at least partly responsible for the amount of space that the atom “occupies.” (For a hydrogen atom in its ground state, this is a space roughly one tenth of a nanometer across.) But being fuzzy is not enough. This position must also stay fuzzy, and that is where the fuzziness of the corresponding momentum comes in.
Let r stand for the radial component of the electron’s position relative to the nucleus. The standard deviation Δr is a measure of the fuzziness of this position. If the electrostatic attraction between the (negatively charged) electron and the (positively charged) proton were the only force at work, it would cause a decrease in Δr, and the atom would collapse as a result. The stability of the atom requires that the electrostatic attraction be counterbalanced by an effective repulsion. Since we already have a fuzzy relative position, the absolutely simplest way of obtaining such a repulsion — and a darn elegant way at that — is to let the corresponding relative momentum be fuzzy, too. As Fig. 2.4.1 illustrates, a fuzzy momentum causes a fuzzy position to grow more fuzzy. If the electrostatic attraction were absent, the fuzziness of the momentum would causes an increase in Δr. In its presence, equilibrium is possible.
But if a stable equilibrium is to be maintained, more is needed. If the mean distance between the electron and the nucleus decreases, their electrostatic attraction increases. A stable equilibrium is possible only if the effective repulsion increases at the same time. We therefore expect a decrease in Δr to be accompanied by an increase in Δpr, the fuzziness of the radial component of the corresponding momentum, and we expect an increase in Δr to be accompanied by a decrease in Δpr. We therefore expect the product of Δr and Δpr to have a positive lower limit. In other words, the stability of the atom implies a relation of the form (2.4.3).