Music and the Making of Modern Science

278 Chapter 18

so we must relinquish “ the ideas of ‘ place of the electron ’ and ‘ path of the electron. ’ If

these are not given up, contradictions remain. This contradiction has been so strongly felt

that it has even been doubted whether what goes on in the atom could ever be described

within the scheme of space and time ” — though Schr ö dinger continued to hope that his

comparison between geometrical and physical optics could provide some kind of intelli-

gibility.^13 But what could not be seen might be heard, so to speak; though Schr ö dinger

could not give a visual picture of his waves, their overtones were physically manifest in

atomic transitions and spectral lines.

In the years that followed, the increasingly sophisticated mathematical formalism of

quantum mechanics subsumed such reminiscences of acoustics or optics into abstract

vectors in a many-dimensional Hilbert space, a mathematical manifold on which the

machinery of the theory operated and which could be mined for observable predictions.

Schr ö dinger ’ s sense of the failure of intuition was increasingly borne out; as quantum

theory became more and more powerful in its predictive ability, it became less and less

visualizable, until physicists like P. A. M. Dirac and Richard Feynman abandoned any

pretense of trying to “ understand, ” in the sense that we believe we can understand ordinary

phenomena.^14 All that mattered was that quantum theory “ works ” : calculations gave exper-

imentally verifiable predictions, however statistical in nature.

Yet at the same time, Dirac asserted that “ it is more important to have beauty in one ’ s

equations than to have them fit experiment, ” a criterion he drew from the mathematician

Hermann Weyl and which led him to formulate the relativistic quantum equation now

named after him. This aesthetic criterion arguably is a veiled form of the search for

harmony that, as we have seen, was the heir of Pythagorean longings for a harmonious

and coherent mathematical theory that would connect with observable phenomena. Among

Figure 18.3

Schr ö dinger ’ s diagram of curving families of the action function W , from “ Quantization as a Problem of Proper

Values II ” (1926), showing the non-Euclidean effect of atomic potentials.