Three Scientific Revolutions: How They Transformed Our Conceptions of Reality (26 page)

How different his present dissatisfaction with matrices is from his youthful intention of replacing physical objects with mathematical symbols.

But as might be expected, not all the physicists at the time were disdainful of visual model building, the Viennese-born Erwin Schrödinger (1887–1961) being an outstanding example. Attracted to de Broglie's discovery that particles also have wave properties that are physical and observable, he was the physicist Einstein consulted when asked by the committee examining de Broglie's doctoral dissertation of its credibility, but also being undecided Einstein did not give his approval until Schrödinger assured him that the thesis had merit.

Schrödinger, influenced by Einstein's belief that fields should replace material particles as the fundamental reality along with de Broglie's discovery that particles have wave properties, decided to see if a complete wave theory could be formulated to replace Bohr's electronic solar model. De Broglie had shown that although mass and momentum were considered properties of particles, they also could be depicted as functions of waves. Such equivalences had previously been discovered: Einstein showing the equivalence of mass and energy (
E
=
m
c
²
) and Planck that energy could be equated with frequency (e =
hv
). Since waves have energy and energy has mass, it was possible that material particles could be depicted as waves as well as particles.

Starting with the classical wave equation that describes the spatial properties of electromagnetic waves, Schrödinger began investigating whether a wave equation could be found to describe the wave properties of subatomic particles, one that would supplement the equations of Newtonian mechanics. In four papers he presented his new theory of “wave mechanics” that appeared in
Annalen der Physik
(Annals of Physics) from January to April 1926 with the title “Quantization as an Eigenvalue Problem.”

Among his publications he proposed that waves should be considered the basic reality and introduced what is now known as the “celebrated” Schrödinger's wave equation
that contains the famous scalar wave function Ψ(psi), along with the Hamiltonian H, “which is simply the observable corresponding to the energy of the system under consideration.”
⁹⁶
As explained by Pais, in replacing particles with waves he

suggested that waves are the basic reality, particles are only derivative things. In support of this monistic view he considered a wave packet made up out of linear harmonic oscillator wavefunctions . . . a superposition of eigenfunctions so chosen that at a given time the packet looks like a blob localized in a more or less small region. . . . He examined what happened to his packet in the course of time and found: “Our wave packet holds
permanently together
, does
not
expand over an ever greater domain in the course of time.” This result led him to anticipate that a particle is nothing more nor less than a very confined packet of waves, and that, therefore, wave mechanics would turn out to be a branch of classical physics, a new branch, to be sure, yet as classical as the theory of vibrating strings or drums or balls.
⁹⁷

According to the equation, between measurements the state vector known as the “wave function” moves in an undisturbed, regular way as described in classical physics until it is measured, which then causes it to collapse into an eigenvalue or single value that produces the observation. Thus its state is uncertain until measured. Owing to its being an extension of classical wave theory, formulated in a mathematics more familiar than matrix mechanics, and providing a visualizable explanation, it was acclaimed by most physicists. As described by Crease and Mann,

the mathematics Schrödinger used was much easier for physicists to understand. . . . If it was hard to imagine how a solid object like an atom could really be made out of waves—what was making the waves?—many physicists had confidence that Schrödinger, a clever fellow, would figure out the answer.
⁹⁸

Moreover, as also explained by Crease and Mann, Schrödinger even proposed an explanation as to how particles could be considered as waves.

A particle was in reality nothing but “a group of waves of relatively small dimensions in every direction,” that is, a sort of tiny clump of waves, its behavior governed by wave interactions. Ordinarily, the bundle of waves was small enough that one could think of it as a dot, a point, a particle in the old sense. But in the microworld, Schrödinger argued, this approximation broke down. There it became useless to talk about particles. At very small distances, “we
must
proceed strictly according to the wave theory, that is, we must proceed from the
wave
equation
, and not from the fundamental equation of mechanics, in order to include all possible processes.” (p. 56)

Unfortunately, these expectations turned out to be illusory. Instead of determinate or distinct portrayals of the electron states of the atom, the solutions to Schrödinger's wave equation produced small cloudlike images reminding one of Rorschach ink blots. Yet even Born, who contributed to the article creating matrix mechanics, after reading Schrödinger's first paper wrote that he was drawn to the traditional aspects of Schrödinger's wave mechanics, a view that angered Heisenberg.

But then, surprisingly, on April 12, 1926, after a very careful perusal of both matrix mechanics and Schrödinger's wave mechanics,

Pauli sent a lengthy letter to Jordan in which he proved that the two approaches were identical [or more accurately stated mathematically equivalent]. Schrödinger himself proved the same thing, a little less completely, a month later. . . . In the equivalence paper, Schrödinger mentions
pro forma
, that it was really impossible to decide between the two theories—and then went on to argue fiercely the merits of wave mechanics. (p. 57; brackets added)

Yet despite his attraction to the more traditional approach of Schrödinger's wave mechanics compared to matrix mechanics, during his investigations Max Born made a discovery described in two papers titled (in translation), “Quantum Mechanics of Collision Phenomena,” published in the
Zeitschrift für Physik
in June and July of 1926 that challenged Schrödinger's claim that wave mechanics, based on measurements of actual waves, was closer to classical physics than matrix mechanics, which dealt only with abstract numerical matrices. The June paper discovered an indeterminacy or uncertainty in Schrödinger's method of determining the position of alleged particles by measuring the density of wave packets.

Calling it the “measurement problem,” Born found that the impact of the measurement would actually produce a “scattering” of the waves in the “wave packet” causing an indeterminacy in the measurement. Producing an unavoidable uncertainty or probability in the measurements in wave mechanics contrary to the strict causality and determinism in classical mechanics, he concluded that this showed it was not closer to traditional physics as Schrödinger claimed. According to Pais: “
It is the first paper to contain the quantum mechanical probability concept.
”
⁹⁹
In his June paper Born described the scattering by a wave function Ψ
_mn
' where the label
n
symbolizes the initial beam direction, while
m
denotes some particular direction of observation of the scattered particles. At that point Born introduced quantum mechanical probability: “Ψ
_mn
determines the probability for the scattering of the electron . . . into the direction [
m
].” (p. 286)

In the second paper, published in July, he interpreted Schrödinger's wave function |Ψ|
²
as the probability for locating the “particle” at the point of greatest density in the wave packet, adding to the measuring probability the probability of quantum states. Although Born had originally believed that Schrödinger's wave mechanics led back to a more traditional interpretation of subatomic physics, his probabilistic interpretations convinced him otherwise. As stated in his autobiography:

Schrödinger believed . . . that he had accomplished a return to classical thinking; he regarded the electron not as a particle but as a density distribution given by the square of his wave function |Ψ|
^2.
. He argued that the idea of particles and of quantum jumps be given up altogether; he never faltered in this conviction. . . . I, however, was witnessing the fertility of the particle concept every day in . . . brilliant experiments on atomic and molecular collisions and was convinced that particles could not simply be abolished. A way had to be found for reconciling particles and waves.
¹⁰⁰

Just as Newton's conception of absolute space and time that were based on measurements made by rods and clocks that were unaffected by the relatively slight velocities of the earth had to be revised when Einstein discovered that when approaching the velocity of light measuring rods contract, clocks slow down, and mass increases (to account for the invariant velocity of light), so measurements of the subatomic or quantum world, assuming that they would follow the same Newtonian calculation method, when actually measured, would have to be radically revised.

As usual, physicists were confounded when they encountered the wave-particle duality, the statistical nature of quantum mechanics, and the uncertainty principle due to the interacting measurements at the subatomic level of inquiry that refuted the Newtonian assumption of the universality of the laws of nature at all levels or scales of inquiry. Here again we encounter a further aspect of the third radical revision in our conceptions of reality at different dimensions or levels of inquiry. The bewilderment decreased somewhat when it was discovered that the formalisms of Dirac's theory, Schrödinger's wave mechanics, and Heisenberg's matrix mechanics were equivalent: according to Emilio Segrè, “[f]or all three the essential relation that produces the quantification is
pq – qp = h/2πi
. . . [while] for Heisenberg
p
and
q
are matrices; for Schrödinger
q
is a number and
p
the differential operator
p
=
h
/2π
i
∂/∂
q
. . . [and] for Dirac
p
and
q
are special numbers obeying a noncommutative algebra. . . .”
¹⁰¹
But the dispute continued with Bohr inviting Schrödinger to his Institute in Copenhagen on October 27, 1926, to discuss their theoretical differences with such intensity that Schrödinger became ill from the tension during the exchange, even though Bohr had the reputation of being a “very considerate and friendly person by nature.” Yet no resolution was reached.

After Schrödinger left Copenhagen Bohr carried on his dispute with the same intensity with Heisenberg, who was an associate at his Institute at the time. Trying to resolve their differences with Bohr defending the view that the solution depended on forging the correct
conceptual framework
and Heisenberg insisting, as usual, that the resolution would depend upon devising the correct
mathematical formalism
, they, too, arrived at an impasse. Frustrated and exhausted by these intense discussions, Bohr decided to take a skiing trip to Norway to relax leaving Heisenberg at the Institute to pursue his investigation.

Concentrating on his measurement problem, as a result of his discussion with Bohr, Heisenberg decided to investigate the difficulty involved in measuring the position and momentum of a particle under a gamma ray microscope. The latter is used because its short wavelength provides great accuracy in determining the position, but according to Planck's formula ε =
hv
, a short wavelength also has a high frequency with high energy such that the interaction between the wave and the particle adversely affects the precision of the momentum measurement. To reduce the inaccuracy of the latter a longer wavelength is required, but that produces less certainty in the position measurement.

Rather than trying to remove the discrepancy, by the end of February 1927, Heisenberg had decided to accept it as unavoidable and devise a formula to state what initially came to be known as the famous “uncertainty or indeterminacy relations.” An appreciation of the radical change involved is seen if contrasted with what was taken for granted in classical mechanics as stated by Pierre-Simon Laplace in 1886.

An intellect which at a given instant knew all the forces acting in nature, and the position of all things of which the world consists . . . would embrace in the same formula the motions of the greatest bodies in the universe and those of the slightest atoms; nothing would be uncertain for it, and the future, like the past would be present to its eyes.
¹⁰²

It was this assurance that nature is governed by exact laws that would disclose a final knowledge of the universe
at all dimensions
that Heisenberg was rebutting. Having accepted the conjugate indeterminacy, Heisenberg sought a mathematical formula that would describe the resultant uncertainty. Although the conditions necessary for measuring the
conjoined
values of the
conjugate magnitude's
position and momentum, along with energy and time, could not be precisely measured, either of the dimensions alone could be exactly determined, but the more precise the measurement of one the less precise the measurement of the other. As Heisenberg expressed this mathematically: if the
uncertainty
in accuracy of the measurement of each of the interdependent
conjugate
attributes is represented by the delta symbol (Δ), then the product of the
conjoined
magnitudes momentum
p
and position
q
cannot be reduced to less than Planck's constant barred, Δ
p
× Δ
q
must be equal to or greater than
ħ
. The second uncertainty states that in the time interval Δ
t
the energy can only be measured with an accuracy equal to or greater than
ħ
.