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Michael Weiss: Anschaulichkeit, Abscheulichkeit

My title comes from a famous pair of quotes, about the love lost between Schroedinger and Heisenberg. The first quote is in a footnote to Schroedinger's "equivalence" paper, where he proved the equivalence of his wave mechanics to Heisenberg's matrix mechanics.
My theory was inspired by L. de Broglie and by brief but infinitely far-seeing remarks of A. Einstein (Berl. Ber. 1925, p. 9ff.) I was absolutely unaware of any genetic relationship with Heisenberg. I naturally knew about his theory, but because of the to me very difficult-appearing methods of transcendental algebra and the lack of Anschaulichkeit [visualizability], I felt deterred, by it, if not to say repelled. [1]
Heisenberg responded in a letter to Pauli:
The more I think about the physical portion of the Schroedinger theory, the more repulsive [abscheulich] I find it....What Schroedinger writes about the visualizability of his theory 'is probably not quite right', in other words it's crap. [2]
Most of the philosophical debates swirling around quantum mechanics have to do with causality. We all know what a wet blanket Einstein was on Las Vegas night. (Probably still is. "Put those dice DOWN! I'm talking to you, God!")

But in the childhood of quantum theory, the matter of visualizability loomed just as large. In his second paper on wave mechanics, Schroedinger wrote: has even been doubted whether what goes on in an atom can be described within a scheme of space and time. From a philosophical standpoint, I should consider a conclusive decision in this sense as equivalent to a complete surrender. For we cannot really avoid our thinking in terms of space and time, and what we cannot comprehend within it, we cannot comprehend at all. There are such things but I do not believe that atomic structure is one of them. [3]
Schroedinger wrote to Willy Wien:
Bohr's standpoint, that a space-time description is impossible, I reject a limine... If [atomic research] cannot be fitted into space and time, then it fails in its whole aim and one does not know what purpose it really serves. [4]
Bohr and Heisenberg of course held a different opinion. The founding papers on matrix mechanics expressed the operational philosophy: "You got your equations, you got your observations, and they match. What more do you want? Shut up and calculate!" Of course, they had to say it more politely, at least in print. For example, here's the abstract, in full, of Heisenberg's famous paper:
The present paper seeks to establish a basis for theoretical quantum mechanics founded exclusively upon relationships between quantities which in principle are observable. [5]
And in the introduction to the "Dreimaennerarbeit", the paper that laid out the whole structure for the first time:
Admittedly, such a system of quantum-theoretical relations between observable quantities...would labor under the disadvantage of not being directly amenable to a geometrically visualizable interpretation, since the motion of electrons cannot be described in terms of the familiar concepts of space and time. [6]
but then they immediately point out what really counts: the equations of motion have the same form as in classical physics. Yet in the same paragraph, they concede:
In the further development of the theory, an important task will lie in the closer investigation of the nature of this correspondence and in the description of the manner in which symbolic quantum geometry goes over into visualizable classical geometry.
Echoes of this argument reverberate faintly today. Textbooks complain about the desire for pictures (see Feynman vol.III pp. 11-4, 11-5 for an example). Pop-science articles glory in "quantum weirdness".

Philosophically, I can't say Schroedinger's position makes sense to me. Our visual systems sport some pretty complicated circuity, a lot of it quite non-intuitive. (Read Hubel's Eye, Brain, and Vision for the nifty details.) It's first rate for watching movies and playing video games, and similar Cro-Magnon activities. It didn't evolve to help us understand quantum field theory!

Historically, common sense intuitions have a pretty poor track record. Who invented statistical mechanics anyway, a masochist during a New England winter? "Brrr, it's cold outside! Hit me harder, molecules, oh, that feels gooood..."

On the other hand, I'm a visualizer myself. I started the "photons, schmotons" thread with one question in mind: just how far can you visualize the QFT description of light? I had in mind something like the balloon analogy in GR: it's wrong because of (a), (b), and (c), but it's still useful because it does capture (e) and (f). Little did I know...

Perhaps by the time the infamous, never-ending, hydra-headed "Photons, Schmotons" thread runs its course, my question will be answered. For the rest of this post though I want to talk about the discarded pictures (discarded by QFT, at any rate). Presumably deadly experimental results could be marshalled to drive stakes through the hearts of all these alternatives, but I'll leave that for someone else to discuss. Pretend we've turned up an old family album in the attic. Each quaint sepia-toned photograph draws our interest and affection. Someone else can recount how prosperous-looking Uncle Max went bankrupt in 1926.

OK, let's say we're determined to visualize wave-particle duality, experiments be hanged! What are our options? I can think of four.

Photons with rhythm

Our photons are little ball-bearings, but with waves etched on the surface like a designer logo. I am reminded of Newton's light corpuscles, with their "Fits of easy Reflection" and "Fits of easy Transmission".

The Ten-Minute History of Science says, "Newton, light particles---BAD! Huyghens, light waves---GOOD!" It comes as a bit of surprise to learn that Newton's Opticks is filled with observations of interference and diffraction phenomena. Newton concluded that his corpuscles had to undergo a periodic change of state, swinging back and forth between Fits of easy Reflection and Fits of easy Transmission.

Newton's theory of light had three characteristics:

(I love the way Newton described polarization: the corpuscles have Sides.)

How does Huyghens stack up?

Yes, no periodicity in Huyghens! [7] Surprised me too. Score two out of three for Newton! [8]

Pilot waves

Of course we think of de Broglie, and this line of thought lead eventually to Bohm's interpretation of QM. But Newton again takes priority. The famous non-framer of hypotheses couldn't refrain from speculating:
...when a Ray of Light falls upon the Surface any pellucid Body, and is there refracted or reflected, may not Waves of Vibrations, or Tremors, be thereby excited... and are not these Vibrations propagated from the point of Incidence to great distances? And do they not overtake the Rays of Light, and by overtaking them successively, do they not put them into the Fits of easy Reflexion and easy Transmission described above? [9]

Continuous waves, discontinuous emission and/or absorption

I call this the microwave popcorn theory: microwaves continuously bath the popcorn, which discontinuously (and unpredictably) pops.

According to the pop-history of science, Planck's theory fell into this category. For example:

Imagine a sponge in a bathtub... According to Maxwell, when a sponge is squeezed it sends out its water in the the usual way and causes waves in the bathtub. Planck's sponge is of a rarer sort. Indeed it is more like a bunch of grapes than a sponge, consisting of myriads of tiny balloons of various sizes, each full of water. When this sponge is squeezed, the balloons burst one after the other, each shooting out its contents in a single quick explosion--- a bundle of water--- and setting up waves... Einstein, however, took the sponge right out of the bathtub... When he squeezed his sponge gently, water fell from it like shimmering drops of rain. [10]
A charming story, but historically all wrong! Kuhn [11] argues persuasively that Planck believed in a completely continuous theory--- continuous waves, continuous emission and absorption--- until 1908, after Einstein put forward his light quantum hypothesis.

However, in 1912 Planck did come up with his so-called "second theory", in which emission is discontinuous, while propagation and absorption remain continuous.

Kuhn's book has full details. Though Planck's second theory never made it to the big time, it did come up with two hits: zero-point energy made its first appearance here, and Bohr got some inspiration for his model of the atom.

Wave packets

Early on, Schroedinger suggested that his wavefunction (for an electron) meant that the charge really was spread out--- space was pervaded with a kind of electron goop. In his equivalence paper (the one with the "Anschaulichkeit" footnote), he notes:
There are today not a few physicists who, exactly in the sense of Mach and Kirchhoff, see the task of physical theory to be merely the most economial description of empirical connections between observable quantities... In this view, mathematical equivalence means almost the same as physical equivalence. [12]
So is matrix mechanics just as good as wave mechanics, or maybe even better, because it doesn't clutter up the story with fairy tales? No, say Schroedinger--- physicists need space-time (i.e., pictorial) descriptions to make progress. He then proposes an interpretation of the wavefunction psi: the real part of (psi d psi/dt) gives the spatial density of electric charge.

Schroedinger also constructed a wave-packet: a well-localized psi function that stays together in time. He did this for a harmonic oscillator potential (just our coherent states, I'll bet!), but he hoped originally to do the same in general.

All waves, no particles anywhere! Can it really be that simple? Schroedinger hoped so.

Schroedinger sent his papers to "grey eminence of theoretical physics", Hendrik Lorentz. (Lorentz incidentally was the first fellow to convince Planck that the black-body formula could not be derived without some sort of discontinuity assumption.)

Lorentz raised several objections [13]. First, he noted that psi is function of (x,y,z) only in the single-particle case. With two particles, psi becomes a function of six variables, the coordinates of both particles:

If I had to choose between wave mechanics and matrix mechanics, I would give preference to the former because of its greater Anschaulichkeit, so long as one is concerned only with the coordinates x,y,z. With a greater number of degrees of freedom, however, I cannot interpret physically the waves and vibrations in q-space and I must decide for matrix mechanics.
Lorentz also pointed out that the harmonic oscillator potential was quite special, and that in the field of a hydrogen atom, the wave packet would spread out rapidly.

I won't go through the rest of Lorentz's criticisms. Schroedinger's biographer notes:

Lorentz belonged to an older generation of physicists, and Schroedinger might have drawn from their discussions the conclusion that his new discoveries cannot be fitted into a classical framework at all.
So all these are wrong! But what's right? Stay tuned...

[1] Ann. Phys., v.79, 734-56; quoted and translated in Schroedinger: Life and Thought, by Walter Moore, CUP, 1989, p.211.

[2] Heisenberg to Pauli, 8 June 1926; quoted and translated in Uncertainty: The Life and Science of Werner Heisenberg, by David Cassidy, W.H. Freeman and Co., 1992, p.215.

[3] Moore, p.208.

[4] Moore, p.226.

[5] Heisenberg, "Quantum-Theoretical Re-interpretation of Kinematic and Mechanical Relations", in Sources of Quantum Mechanics, ed. B.L. van der Waerden, Dover, 1968.

[6] Born, Heisenberg, and Jordan, "On Quantum Mechanics II", in Sources of Quantum Mechanics.

[7] Or so says I. Bernard Cohen in the preface to the Dover edition of the Opticks (see page xlvii).

[8] Why not three out of three, if we believe in quantum mechanics? I side with I.Bernard Cohen: "...we must choose between (1) the historical or (2) the antiquarian approach to the development of science... the antiquarian's sifting of the disjecta membra of the Opticks (often out of context) for an occasional 'precursorship' of one or another 20th-century physical concept." (op. cit.)

[9] Newton, Opticks, III.1 Query 17. And yes, I'm being a bit antiquarian here.

[10] Banesh Hoffmann, The Strange Story of the Quantum, 2nd ed., Dover, 1959, p. 26.

[11] Kuhn, Black Body Theory and the Quantum Discontinuity, 1894-1912, OUP 1978.

[12] Moore, p.212.

[13] See the discussion in Moore, pp. 214-217. This is the source for the two quotes below.

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