Schrödinger's E = ℎν heresy

In 1926, after failing to convince Bohr and Heisenberg that wave
mechanics can get rid of quantum jumps, Schrödinger exclaimed:

𝐼𝑓 𝑤𝑒 ℎ𝑎𝑣𝑒 𝑡𝑜 𝑔𝑜 𝑜𝑛 𝑤𝑖𝑡ℎ 𝑡ℎ𝑒𝑠𝑒 𝑑𝑎𝑚𝑛𝑒𝑑
𝑞𝑢𝑎𝑛𝑡𝑢𝑚 𝑗𝑢𝑚𝑝𝑠, 𝑡ℎ𝑒𝑛 𝐼'𝑚 𝑠𝑜𝑟𝑟𝑦 𝑡ℎ𝑎𝑡 𝐼 𝑒𝑣𝑒𝑟
𝑔𝑜𝑡 𝑖𝑛𝑣𝑜𝑙𝑣𝑒𝑑. --𝐸. 𝑆𝑐ℎ𝑟𝑜𝑑𝑖𝑛𝑔𝑒𝑟

26 years later, in 1952, Schrödinger returned to this question in the
paper: '𝑨𝒓𝒆 𝒕𝒉𝒆𝒓𝒆 𝑸𝒖𝒂𝒏𝒕𝒖𝒎 𝑱𝒖𝒎𝒑𝒔?'

https://web.archive.org/web/20210914022807/www.ub.edu/hcub/hfq/sites/default/files/Quantum_Jumps_I.pdf

[Schrödinger reiterates that one of his main motivations for his intense
research into a more detailed atomic model was that the atomic jumps of
the Bohr model didn't sit well with him]:

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Bohr’s theory held the ground for about a dozen of years, scoring a
grand series of so marvelous and genuine successes, that we may well
claim excuses for having shut our eyes to its one great deficiency:
while describing minutely the so-called “stationary” states which the
atom had normally, i.e. in the comparatively uninteresting periods when
nothing happens, the theory was silent about the periods of transition
or “quantum jumps” (as one then began to call them). Since intermediary
states had to remain disallowed, one could not but regard the transition
as instantaneous; but on the other hand, the radiating of a coherent
wave train of 3 or 4 feet length, as it can be observed in an
interferometer, would use up just about the average interval between two
transitions, leaving the atom no time to “be” in those stationary
states, the only ones of which the theory gave a description.

This difficulty was overcome by quantum mechanics, more especially by
Wave Mechanics, which furnished a new description of the states;

... [Wave mechanics] is most easily grasped by the simile of a vibrating
string or drumhead or metal plate, or of a bell that is tolling. If such
a body is struck, it is set vibrating, that is to say it is slightly
deformed and then runs in rapid succession through a continuous series
of slight deformations again and again. There is, of course, an infinite
variety of ways of striking a given body, say a bell, by a hard or soft,
sharp or blunt instrument, at different points or at several points at a
time. This produces an infinite variety of initial deformations and
accordingly a truly infinite variety of shapes of the ensuing vibration:
the rapid “succession of cinema pictures,” so we might call it, which
describes the vibration following on a particular initial deformation is
infinitely manifold. But in every case, however complicated the actual
motion is, it can be mathematically analysed as being the superposition
of a discrete series of comparatively simple “proper vibrations,” each
of which goes on with a quite definite frequency. This discrete series
of frequencies depends on the shape and on the material of the body, its
density and elastic properties. It can be computed from the theory of
elasticity, from which the existence and the discreteness of proper
modes and proper frequencies, and the fact that any possible vibration
of that body can be analysed into a superposition of them, are very
easily deduced quite generally, i.e. for an elastic body of any shape
whatsoever.

The achievement of wave mechanics was, that it found a general model
picture in which the “stationary” states of Bohr’s theory take the role
of proper vibrations, and their discrete “energy levels” the role of the
proper frequencies of these proper vibrations; and all this follows from
the new theory, once it is accepted, as simply and neatly as in the
theory of elastic bodies, which we mentioned as a simile. Moreover, the
radiated frequencies, observed in the line spectra, are in the new
model, equal to the differences of the proper frequencies; and this is
easily understood, when two of them are acting simultaneously, on simple
assumptions about the nature of the vibrating “something.”

But to me the following point has always seemed the most relevant, and
it is the one I wish to stress here, because it has been almost
obliterated—if words mean something, and if certain words now in general
use are taken to mean what they say. The principle of superposition not
only bridges the gaps between the ‘stationary’ states, and allows, nay
compels us, to admit intermediate states without removing the
discreteness of the “energy levels” (because they have become proper
frequencies); but it completely does away with the prerogative of the
stationary states. The epithet stationary has become obsolete. Nobody
who would get acquainted with Wave Mechanics without knowing its
predecessor (the Planck-Einstein-Bohr-theory) would be inclined to think
that a wave-mechanical system has a predilection for being affected by
only one of its proper modes at a time. Yet this is implied by the
continued use of the words “energy levels,” “transitions,” “transition
probabilities.”

The perseverance in this way of thinking is understandable, because the
great and genuine successes of the idea of energy parcels has made it an
ingrained habit to regard the product of Planck’s constant ℎ and a
frequency as a bundle of energy, lost by one system and gained by
another. How else should one understand the exact dove-tailing in the
great “double-entry” book-keeping in nature?

𝐼 𝑚𝑎𝑖𝑛𝑡𝑎𝑖𝑛 𝑡ℎ𝑎𝑡 𝑖𝑡 𝑐𝑎𝑛 𝑖𝑛 𝑎𝑙𝑙 𝑐𝑎𝑠𝑒𝑠 𝑏𝑒
𝑢𝑛𝑑𝑒𝑟𝑠𝑡𝑜𝑜𝑑 𝑎𝑠 𝑎 𝑟𝑒𝑠𝑜𝑛𝑎𝑛𝑐𝑒 𝑝ℎ𝑒𝑛𝑜𝑚𝑒𝑛𝑜𝑛.

One ought at least to try, and look upon atomic frequencies just as
frequencies and drop the idea of energy-parcels. I submit that the word
'energy' is at present used with two entirely different meanings,
macroscopic and microscopic. Macroscopic energy is a ' quantity-concept
'. Microscopic energy, meaning ℎν, is a 'quality-concept' or
'intensity-concept'; it is quite proper to speak of high-grade and
low-grade energy according to the value of the frequency v. True, the
macroscopic energy is, strangely enough, obtained by a certain weighted
summation over the frequencies, and in this relation the constant ℎ is
operative. But this does not necessarily entail that in every single
case of microscopic interaction a whole portion ℎν of macroscopic energy
is exchanged. I is believe one allowed to regard microscopic interaction
as a continuous phenomenon without losing either the precious results of
Planck and Einstein on the equilibrium of (macroscopic) energy between
radiation and matter, or any other understanding of phenomena that the
parcel-theory affords.

The one thing which one has to accept and which is the inalienable
consequence of the wave-equation as it is used in every problem, under
the most various forms, is this : that the interaction between two
microscopic physical systems is controlled by a peculiar law of
resonance. This law requires that the difference of two proper
frequencies of the one system be equal to the difference of two proper
frequencies of the other:

(Eqn. I)  ν_1 - ν_1' = ν_2' - ν_2

The interaction is appropriately described as a gradual change of the
amplitudes of the four proper vibrations in question. People have kept
to the habit of multiplying this equation by ℎ and saying it means, that
the first system (index 1) has dropped from the energy level, ℎν_1 to
the level ℎν_1', the balance being transferred to the second system,
enabling it to rise from  ℎν_2 to ℎν_2'. This interpretation is
obsolete. There is nothing to recommend it, and it bars the
understanding of what is actually going on. It obstinately refuses to
take stock of the principle of superposition, which enables us to
envisage simultaneous gradual changes of any and all amplitudes without
surrendering the essential discontinuity, if any, namely that of the
frequencies.
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Obviously, my interest in Schrödinger's picture, is that it is
compatible with an aether, while discrete quantum jumps make no sense
from an aether point of view.