4 The fuzziness relation

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 ≥ 12.

Remem­bering de Broglie’s rela­tion p = k, 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/​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.


A fuzzy momentum

Figure 2.4.1 A fuzzy posi­tion, rep­re­sented by a prob­a­bility den­sity func­tion, at two dif­ferent times. Above: if an object with this posi­tion has an exact momentum, it moves by an exact dis­tance; the fuzzi­ness of its posi­tion does not increase. Below: if the same object has a fuzzy momentum, it moves by a fuzzy dis­tance; as a result, its posi­tion grows fuzzier.


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).