Schrödinger’s Unpublished 1944 Manuscript and the Puzzle of Quantum Randomness
A Lost Erwin Schrödinger Manuscript From 1944 Reveals a Surprising Twist in Quantum Physics
An overlooked three-page manuscript reveals how Erwin Schrödinger challenged an argument against deterministic models of quantum physics, without necessarily endorsing determinism himself.
In late 1944, Erwin Schrödinger was considering a question that had troubled physicists since the early days of quantum mechanics: when a photon passes through a polarising filter, is its fate fundamentally random, or could an unseen property determine the outcome?
Schrödinger wrote a short response to a contemporary debate, but the manuscript was never published. It remained in the Schrödinger archive in Vienna until physicist and historian Flavio Del Santo examined it and published a transcription and analysis on arXiv on October 14, 2026.
The document contains a concrete hidden-variable model. Rather than giving each photon a simple, fixed instruction to pass or be blocked, Schrödinger considered a hidden property that could take a continuous range of values. His construction challenged the claim that deterministic explanations could be ruled out by the argument then under discussion.
The manuscript does not show that Schrödinger had discovered a replacement for quantum mechanics. Its importance is narrower, and historically revealing. It restores a missing contribution to a debate about what quantum theory can tell us about randomness.
Why is Schrödinger Trending in October 2026?
Erwin Schrödinger has returned to the science news in October 2026 because of a newly published analysis of a manuscript he wrote more than 80 years ago.
On 14 October 2026, physicist and historian of science Flavio Del Santo posted a research paper on arXiv examining Schrödinger’s previously unpublished contribution to the 1944–45 debate over quantum randomness and determinism. ScienceAlert reported the story on 21 October, followed by Phys.org on 22 October 2026.
The manuscript is attracting attention because it reveals that Schrödinger developed a hidden-variable model to challenge a particular argument against deterministic explanations of quantum phenomena. The work offers a new historical insight into his engagement with one of quantum physics’ long-running questions.
This is not a new laboratory discovery or evidence that scientists have overturned quantum mechanics. The renewed interest comes from the analysis of an old document and its relevance to the history of quantum theory.
Key takeaways
- Schrödinger drafted an unpublished response during a 1944–45 debate about determinism and quantum mechanics.
- His model used a continuous hidden variable rather than a simple binary instruction.
- The model addressed the behaviour of polarised light and the transmission of photons through successive filters.
- The manuscript shows that Schrödinger explored deterministic explanations; it does not prove that he believed nature was fundamentally deterministic.
The 1944 debate: could hidden variables restore determinism?
Quantum mechanics predicts probabilities for individual measurement outcomes. Even when the state of a system is carefully prepared, the theory generally does not specify the result of every individual measurement in advance.
Some physicists accepted this as a fundamental feature of nature. Others suspected that the theory might be incomplete. Perhaps particles carried additional properties that were absent from the standard quantum description. If those properties existed, an outcome that appeared random might actually be determined by information we could not observe.
These hypothetical properties are known as hidden variables.
The idea was already part of the debate in the 1940s. Mathematician Edmund Whittaker discussed the problem of determinism in connection with polarised light, arguing that a simple deterministic account faced difficulties when compared with quantum predictions. The resulting exchange in Nature included responses from other physicists, among them Heinrich Pelzer, who developed a different hidden-variable approach.
Schrödinger’s manuscript belongs to this exchange. His aim was not to announce a new experimental discovery. He was examining whether the argument against hidden variables was strong enough to exclude them.
What the Manuscript Actually Shows
The significance of Schrödinger’s manuscript lies in the specific argument it challenged. It does not establish that quantum mechanics is incorrect or that every quantum event is predetermined.
During the 1944–45 debate, physicists were examining whether hidden-variable explanations could reproduce the statistical predictions of quantum mechanics. Schrödinger’s response explored a model in which a hidden variable could take a continuous range of values rather than being restricted to just two possibilities.
The model also included a rule for updating the hidden variable after a photon passed through a polarising filter. This feature allowed the discussion to extend beyond a single measurement to successive filters.
The manuscript therefore provides evidence that Schrödinger constructed a particular mathematical model to challenge an argument against hidden variables. It does not demonstrate that this model describes how nature actually works, nor does it establish that Schrödinger personally accepted determinism as the correct interpretation of quantum physics.
The distinction is important: showing that one argument against an idea is insufficient does not automatically prove that the idea itself is true.
Why polarising filters mattered
A polarising filter transmits light according to its polarisation and the orientation of the filter. For ideal linearly polarised light, the transmitted intensity follows Malus’s law:
\[ I=I_0\cos^2\theta \]
Here, \(I_0\) is the incident intensity and \(\theta\) is the angle between the light’s polarisation direction and the filter’s transmission axis.
For a suitably prepared single photon, the corresponding quantum-mechanical transmission probability is
\[ P(\text{pass})=\cos^2\theta. \]
The key feature is the smooth change in probability as the filter rotates.
- At 0 degrees, the ideal transmission probability is one.
- At 45 degrees, it is one-half.
- At 90 degrees, it is zero.
A simple pass-or-block instruction cannot, on its own, explain this changing statistical pattern. A model needs a way to produce different transmission fractions for different filter orientations.
That was the mathematical difficulty at the centre of the historical dispute.
A simple mathematical intuition for the continuous variable
Imagine, as a teaching model, that each photon carries a hidden number \(\lambda\) uniformly distributed between zero and one:
\[ \lambda\in[0,1]. \]
For a single filter, suppose the photon passes when
\[ \lambda\leq\cos^2\theta. \]
The interval of hidden-variable values that satisfies this condition has length \(\cos^2\theta\). With a uniform distribution, that fraction of photons passes through the filter:
\[ P(\text{pass})=\cos^2\theta. \]
Hidden-variable range 0 to 1 Pass: \(\lambda\leq0.5\) Blocked: \(\lambda>0.5\)
This is a compact way to understand how a continuous hidden variable can generate the same single-filter probability as Malus’s law while assigning a definite result to each photon.
A qualification matters here: this threshold construction is an illustrative mathematical explanation, not a verbatim transcription of Schrödinger’s original model. Matching the probability for one filter does not by itself reproduce the statistics of several filters in succession. That requires an additional rule for how the hidden variable changes after transmission. Del Santo’s reconstruction analyses Schrödinger’s specific proposal in that broader context.
Malus’s Law and the Hidden-Variable Model
Malus’s law describes how the intensity of linearly polarised light changes when it passes through a polarising filter. Its mathematical form is:
\[ I=I_0\cos^2\theta \]
Here, \(I_0\) is the incident light intensity, \(I\) is the transmitted intensity, and \(\theta\) is the angle between the light’s polarisation direction and the filter’s transmission axis.
For example, when the angle is \(45^\circ\),
\[ \cos^2 45^\circ=\frac{1}{2} \]
The transmitted intensity is therefore half the incident intensity under ideal conditions. At \(0^\circ\), the ideal transmission is complete, while at \(90^\circ\), the transmitted intensity is zero.
For individual photons, quantum mechanics interprets the corresponding values as probabilities of transmission across repeated measurements.
A continuous hidden-variable model attempts to reproduce these probabilities by assigning an additional variable to each photon. For a simple mathematical illustration, suppose that a variable \(\lambda\) is uniformly distributed between 0 and 1, and the model says that a photon passes through a filter when
\[ \lambda\leq\cos^2\theta \]
At \(45^\circ\), the threshold is \(0.5\). Half of the uniformly distributed values lie at or below that threshold, so the model reproduces a transmission probability of one-half.
This example helps explain how a continuous variable can generate the probability pattern described by Malus’s law.
Scientific note: This is an illustrative model for understanding the mathematics, not a verified transcription of Schrödinger’s exact rule. His original construction, especially the rule for updating the hidden variable after a measurement, must be checked against the manuscript and Flavio Del Santo’s analysis before the example is presented as a reconstruction of Schrödinger’s model.
Whittaker and Schrödinger: two different approaches
| Aspect | Whittaker’s argument | Schrödinger’s counter-model |
|---|---|---|
| Hidden-variable picture | A simple binary, pass-or-block instruction | A continuous hidden variable |
| Values available | Two possible outcomes | A range of values between 0 and 1 |
| Changing filter angle | A fixed binary instruction does not by itself reproduce the full changing transmission fraction | The model is constructed to reproduce the relevant angular statistics |
| Successive filters | The simple argument does not settle the behaviour of a more flexible model | Includes a rule for updating the hidden variable after transmission |
| Historical significance | Presented a challenge to deterministic explanations | Demonstrated that the particular objection did not rule out every hidden-variable construction |
The difference is not simply that one model is wrong and the other is right. The central issue is what the proposed hidden-variable scheme can reproduce. A binary instruction is highly restrictive; a continuous variable offers more ways to match a changing statistical distribution.
Schrödinger’s response showed why the distinction mattered. A limitation of one model does not automatically establish the impossibility of every model in the same broad category.
Schrödinger’s continuous-variable approach
Schrödinger’s proposal gave the hidden variable more freedom than a simple yes-or-no instruction. Del Santo reconstructs how the model can reproduce Malus’s law and how it handles successive polarising filters.
The second part is especially important. A model that reproduces the transmission probability for one filter may fail when a photon passes through a sequence of filters. The first measurement changes the conditions under which the next measurement takes place, so a successful account must specify how the hidden information is carried forward.
Schrödinger’s model included an update rule. According to Del Santo’s analysis, this allowed it to handle successive filters, extending the construction beyond a single measurement.
That is a mathematical result about the model’s ability to reproduce a particular set of predictions. It is not experimental evidence that photons actually carry the proposed hidden variable, nor does it establish a complete deterministic theory for every quantum system.
Why the manuscript was never published
The surviving draft is undated, but its references place it after November 1944. It is only three pages long.
Del Santo suggests that Schrödinger may have abandoned the manuscript after Heinrich Pelzer published a related hidden-variable model. If Pelzer had already addressed the central point, Schrödinger may have seen little reason to publish another response.
That explanation remains a historical hypothesis, not a fact established by the manuscript itself. The document does not state definitively why Schrödinger set it aside.
Its absence from the published debate nevertheless matters. For more than eight decades, readers could follow the arguments that appeared in print without knowing about this additional construction by Schrödinger. The newly analysed document adds a missing piece to that record.
Schrödinger’s position on randomness was not straightforward
Schrödinger is often portrayed as a defender of determinism who resisted the probabilistic interpretation of quantum mechanics. His writings present a more complicated picture.
The unpublished model shows that he was willing to develop a deterministic construction and use it to challenge an argument against hidden variables. It does not show that he accepted determinism as the final account of nature.
Del Santo places the manuscript alongside Schrödinger’s broader philosophical views, including his earlier doubts about whether nature’s laws must be strictly deterministic. In the text, Schrödinger argued that experiments and logical deduction alone could not establish the impossibility of either the deterministic or indeterministic position.
That is a different claim from saying that both views are equally supported by modern evidence. It is a statement about the limits of the argument being discussed in the 1940s.
Bell’s theorem changed the debate
The hidden-variable question did not end with the polariser arguments.
In 1952, David Bohm showed that a deterministic hidden-variable theory could reproduce quantum predictions under a nonlocal framework.
In 1964, John Bell derived inequalities that local hidden-variable theories must satisfy under specified assumptions.
Quantum mechanics predicts violations of Bell inequalities for suitable systems, and experiments have observed those violations. These results rule out broad classes of local hidden-variable explanations, rather than every conceivable hidden-variable theory.
Bohmian mechanics, for example, remains a deterministic theory but is nonlocal. Its existence does not mean that all hidden-variable models work; it illustrates why locality and other assumptions must be stated explicitly when assessing such theories.
Schrödinger’s manuscript predates Bell’s theorem by two decades. It deals with a narrower question about polarised light, not the full range of quantum correlations examined in later work. Its significance is historical and logical: it demonstrates that the particular argument under discussion did not, by itself, eliminate every deterministic construction.
What It Does Not Prove: Bell’s Theorem and Modern Quantum Physics
Schrödinger’s unpublished model should not be interpreted as proof that quantum mechanics is wrong or that hidden variables have been experimentally confirmed.
His manuscript addressed a specific argument made during the 1944–45 debate. It showed how a hidden-variable model could be constructed to challenge that argument. It did not establish that the proposed model is a complete description of physical reality.
A major development came later with John Bell’s theorem in 1964. Bell showed that theories satisfying a particular set of assumptions, including locality, must obey mathematical constraints known as Bell inequalities. Quantum mechanics predicts violations of these inequalities for suitable experimental arrangements, and experiments have observed such violations.
These results place important restrictions on local hidden-variable explanations. They do not mean that every conceivable hidden-variable theory has been ruled out; different theories may use different assumptions, and their predictions must be evaluated accordingly.
Schrödinger’s manuscript predates Bell’s theorem by about two decades. It addresses an earlier mathematical debate and does not resolve the questions raised by Bell inequalities or modern experiments.
The key distinction is between challenging an argument against hidden variables and proving that a particular hidden-variable theory correctly describes nature. Schrödinger’s manuscript contributes to the first point, not the second.
The link to Schrödinger’s cat
Schrödinger’s cat thought experiment is famous for exposing the difficulty of applying quantum rules to everyday objects. In the familiar setup, a cat is linked to a quantum event inside a sealed box. A straightforward extension of the quantum description produces the troubling question of how to interpret a state associated with both possible outcomes before observation.
The thought experiment was not simply an attempt to disprove quantum mechanics. It highlighted problems in the interpretation of the theory, especially the relationship between a quantum system, measurement and the macroscopic world.
The 1944 manuscript addresses a different issue: whether a deterministic hidden-variable model could reproduce the statistics of polarised light. The two are connected by Schrödinger’s interest in the foundations of quantum theory, not by a direct mathematical solution of the cat paradox.
What the manuscript adds to the history of quantum physics
The rediscovered document is not a new laboratory result. No experiment reported in the manuscript overturns quantum mechanics, and the work does not establish that nature is fundamentally deterministic.
Its contribution is more precise. Schrödinger constructed a hidden-variable model to challenge a particular argument against deterministic explanations. The model’s continuous variable and update rule gave it capabilities that the simpler binary picture did not have.
The discovery also complicates the familiar story of Schrödinger’s relationship with quantum randomness. He explored deterministic alternatives without necessarily endorsing determinism as a description of the universe.
For historians of physics, that is a worthwhile addition to the record. For students, the manuscript offers a useful lesson in mathematical modelling: showing that one model fails is not the same as proving that every model of the same general type must fail.
Frequently asked questions
What is Erwin Schrödinger known for?
Erwin Schrödinger was an Austrian theoretical physicist best known for the Schrödinger equation, a fundamental equation of quantum mechanics, and the Schrödinger’s cat thought experiment.
What is Schrödinger’s cat thought experiment?
It is a thought experiment that highlights the difficulty of applying quantum superposition to everyday objects. The cat’s condition is linked to a quantum event, raising questions about measurement and the meaning of a quantum state.
Why did Schrödinger pick a cat?
The cat makes the conceptual problem vivid because it is a familiar, large-scale animal. Schrödinger used the example to highlight the apparent tension between a quantum description and the definite outcomes we experience in everyday life.
What is Schrödinger’s atomic model?
Schrödinger’s wave-mechanical approach describes electrons using quantum wavefunctions. Unlike the older Bohr model, it does not describe electrons as moving along fixed, planet-like orbits.
What did Schrödinger’s unpublished manuscript propose?
It presented a hidden-variable model intended to show that deterministic explanations could not be excluded by the particular argument then being debated. Del Santo’s 2026 preprint provides the transcription and analysis.
How does a continuous hidden variable reproduce Malus’s law?
In a simple illustrative model, a uniformly distributed variable (\lambda) between zero and one passes a threshold set by (\cos^2\theta). The fraction below that threshold is (\cos^2\theta), matching the single-filter transmission probability. This illustrates the mathematics; it should not be mistaken for a verbatim statement of Schrödinger’s original rule.
Did Schrödinger disprove quantum randomness?
No. The manuscript challenges a particular argument against hidden-variable models. It does not overturn quantum mechanics or demonstrate that all quantum outcomes are predetermined.
Why was the manuscript not published?
Del Santo suggests that Schrödinger may have set it aside after Pelzer published related work. The manuscript does not establish the reason conclusively.
Does Bell’s theorem rule out hidden variables?
Bell’s theorem constrains local hidden-variable theories under specified assumptions. Experiments have ruled out broad classes of local explanations, but deterministic nonlocal theories remain possible.
Did Schrödinger believe the universe was deterministic?
The manuscript alone cannot settle his complete philosophical position. It demonstrates that he investigated a deterministic model, not that he necessarily believed it represented nature correctly.
Sources and further reading
- Flavio Del Santo, “Schrödinger’s unpublished hidden-variable model in the 1944–45 debate over the ‘Collapse of Determinism’”, arXiv, October 14, 2026. This preprint contains the transcription and historical analysis.
- Science reporting: Phys.org’s coverage of the manuscript, October 22, 2026.
- Further context: ScienceAlert’s report, October 21, 2026.
- Sources: Original research paper on arXiv | ScienceAlert | Phys.org
Join the discussion
If a hidden-variable model can reproduce a quantum probability, does that make the randomness any less fundamental—or does it simply change what we mean by a physical explanation?
Share your view in the comments. Please distinguish between what a mathematical model can demonstrate and what experiments have actually established.
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