E. Schrödinger, “Die gegenwärtige Situation in der Quantenmechanik”, Naturwissenschaften 23 (1935) 807–12; 823–28; 844–49, excerpts by Gino W.
This informal review paper by one of the founders of quantum mechanics was prompted, like a “confession”, by the seminal paper by Einstein, Podolsky and Rosen (1935) on the question whether quantum mechanics is a complete description of physical reality. The infamous Gedankenexperiment of the cat (“Schrödinger’s cat”) appears here for the first time — although there are rumours that the idea has been prompted by Einstein in personal correspondence.
Model vs. Reality
In the first paragraph of his paper Schrödinger points out the relation between reality and our model of it. The model is an abstract construction that focuses on the important aspects and, in doing so, ignores particularities of a system. This approximation is arbitrarily set by humans, as he states: “Without arbitrariness there is no model.” But this simplification is done to achieve an exact way to calculate and herein lies a difficulty of quantum mechanics. This model gives less information for the (final) state of a system at given initial conditions than e.g. classic mechanics.
Schrödinger’s cat
In a later (5th) section, the famous cat experiment is described. A low-dose radioactive material is used as a quantum system which produces on average one radiation particle per hour. The cat is placed in a sealed box and killed via a mechanism if the radiative decay happens and is detected. This simple setup transferres the uncertainty of the quantum world to the macroscopic level as the cat clearly is. Applying strictly the linearity of quantum mechanics, the wave function of the cat becomes a superposition:
|Ψ⟩ = α|dead⟩ + β|alive⟩.
The “paradox” is that we only observe one of the two alternatives and have no means to demonstrate the interference between the two.
Statistics of a Model’s Variables in Quantum Mechanics
In the beginning of the 2nd paragraph of his paper, Schrödinger points out a belief, that he feels, will certainly undergo some variations, but still be preserved in its essence: The characteristics of nature are not described by a set of variables that determine each others values uniquely. As an example he writes about the Rutherford’s experiment with a system, consisting of an alpha particle and a gold atom’s nucleus, fully described by three spatial coordinates and three velocity components for each object (all in all 12 coordinates). It is not possible to assign a sharp value to each coordinate, as the certainty of one value causes an uncertainty of another. To determine the relationship between uncertainties of two different coordinates, they can be organized in canonically conjugated pairs. The most prominent example is probably given by position and momentum. The lower boundary of the product of their uncertainties (each preceded by a Δ) is given by Heisenberg’s inequality:
Δx Δp ≥ ℏ.
In general all values contain an uncertainty different from zero.
Another consequence is that a sharp outcome of a measurement can only be achieved if only one coordinate out of the pair is measured. He also states that the value of a variable at some time t, derived by some information gained in the past, contains uncertainty. After all, the focus on functions that describe the exact values for the characteristics of some system (at time t) is moved to functions describing the probability of measuring a certain value that one of the characteristics can obtain (at time t). Achieving an estimate of such a probability function needs a lot of identical repetitions of an experiment dedicated to the measurement of a certain quantitiy.
His 2nd paragraph concludes with the statement that quantum theory reveals the inability of the classical model to present the exact relationship between the quantities describing a system, but at least it delivers an idea of which quantities are actually measurable for a given system.
Transition from Classical to Quantum Description
In his 3rd paragraph he emphasizes the focus on variables known from classical models and the novel approach being the introduction of the already mentioned probability functions. On a system one measures the values for some chosen properties and derives probability functions in order to determine the expected values for other properties either at the same or at another time. The existence of system properties like the total energy E of an oscillator is mentioned, which are defined on a discrete set of values (multiples of Planck’s constant ℏ). This characteristic is independent of the kind of measurement and portrays the sharpness of these values (sharper than any possible measurement). Being able to only inherit an idea of what properties are measurable or describable and facing the struggle of not being able to determine the relationship between all properties, he indicates the difficulty of describing the system’s essential information in a probabilistic manner. This new concept leads him to the question of wether the new quantum mechanical and fundamental properties only owe their terms from the classical model. Finally, he states, that the interesting properties of a system are not vivid.
Reality of the Quantum Mechanical Description
In the 4th paragraph he discusses the reality of measurable quantities. As one can at most measure one half of a set of variables which describe a system completely, he asks wether the undetermined, “blurred” variables are part of reality. He also talks about the idea, that all quantities belong to reality and that it is only impossible to know each value simultaneously. Comparisons are made to the statistical theory of thermodynamics, where physicists are used to the circumstance of having limited knowledge about a systems characteristics and therefore deal with a set of possible states (Gibbs ensemble), which all fulfil those characteristics. It is shown that in quantum mechanics one cannot find such a set of states from which a specific one is measured in each particular case. Lastly, he argues that if you where to suppose a state (even if you do not know it in particular) for a system for every point of time, you could not make any statements without violating the rules of quantum mechanics.
Description and Consequences of Uncertainty
In the 5th paragraph he focuses on the idea of the dependence of the reality of a variable on its degree of quantum uncertainty. As a tool to describe the degree and kind of uncertainty the wave function is described. So instead of having sharp values for each variable, one determines sharp values for the uncertainty of a variable’s value. Even its equation of motion, if it is left to itself, is as exact as the corresponding classical equations. If the uncertainty ranges were to lie in atomic dimensions one could even use it instead of the classical counterpart. The only problem that could arise is the induction of uncertainties for macroscopic quantities by the uncertainty of a microscopic quantity. To illustrate this idea he introduces the Gedankenexperiment that became known as “Schrödinger’s Cat”: an uncertainty of the occurrence of an atomic decay leads to the uncertainty about the “life state” of a cat. This problem is one of the reasons, why caution is advised while dealing with uncertain or “blurred” models to depict reality.