Erwin Schrödinger

Discussion of Probability Relations between Separated Systems

The probability relations which can occur between two separated physical systems are discussed, on the assumption that their state is known by a representative in common. The two families of observables, relating to the first and to the second system respectively, are linked by at least one match between two definite members, one of either family. The word match is short for stating that the values of the two observables in question determine each other uniquely and therefore (since the actual labelling is irrelevant) can be taken to be equal. In general there is but one match, but there can be more. If, in addition to the first match, there is a second one between canonical conjugates of the first mates, then there are infinitely many matches, every function of the first canonical pair matching with the same function of the second canonical pair. Thus there is a complete one-to-one correspondence between those two branches (of the two families of observables) which relate to the two degrees of freedom in question. If there are no others, the one-to-one correspondence persists as time advances, but the observables of the first system (say) change their mates in the way that the latter, i.e. the observables of the second system, undergo a certain continuous contact-transformation.

Probability relations between separated systems

The paper first scrutinizes thoroughly the variety of compositions which lead to the same quantum-mechanical mixture (as opposed to state or pure state ). With respect to a given mixture every state has a definite probability (or mixing fraction) between 0 and 1 (including the limits), which is calculated from the mixtures Statistical Operator and the wave function of the state in question. A well-known example of mixtures occurs when a system consists of two separated parts. If the wave function of the whole system is known, either part is in the situation of a mixture, which is decomposed into definite constituents by a definite measuring programme to be carried out on the other part. All the conceivable decompositions (into linearly independent constituents) of the first system are just realized by all the possible measuring programmes that can be carried out on the second one. In general every state of the first system can be given a finite chance by a suitable choice of the programme. It is suggested that these conclusions, unavoidable within the present theory but repugnant to some physicists including the author, are caused by applying non-relativistic quantum mechanics beyond its legitimate range. An alternative possibility is indicated.

ARE THERE QUANTUM JUMPS?PART I*

PHYSICAL science, which aims not only at devising fascinating new experiments, but at obtaining a rational understanding of the results of observations, incurs at present, so I believe, the grave danger ot getting severed from its historical background. The innovations of thought in the last 50 years, great and momentous and unavoidable as they were, are usually overrated compared with those of the preceding century ; and the disproportionate foreshortening by timeperspective, of previous achievements on which all our enlightenment m modern times depends, reaches a disconcerting degree according as earlier and earlier centuries are considered. Along with this disregard for histoncal linkage there is a tendency to forget that all science is bound up with human culture in general, and that scientific findings, even those which at the moment appear the most advanced and esoteric and difficult to grasp, are meaningless outside their cultural context. A theoretical science, unaware that those of its constructs considered relevant and momentous are destined eventually to be

Contributions to Born’s new theory of the electromagnetic field

Born’s theory starts from describing the field by two vectors (or a “six-vector”), B, E, the magnetic induction and electric field-strength respectively. A second pair of vectors (or a second six-vector) H, D, is introduced, merely an abbreviation, if you please, for the partial derivatives of the Lagrange function with respect to the components of B and E respectively (though with the negative sign for E). H is called magnetic field and D dielectric displacement. It was pointed out by Born that it is possible to choose the independent vectors in different ways. Four different and, to a certain extent, equivalent and symmetrical representations of the theory can be given by combining each of the two “magnetic” vectors with each of the two “electric” vectors to form the set of six independent variables. Every one of these four representations can be derived from a variation principle, using, of course, entirely different Lagrange functions—physically different, that is, though their analytic expressions by the respective variables are either identical or very similar to each other. In studying Born’s theory I came across a further representation, which is so entirely different from all the aforementioned, and presents such curious analytical aspects, that I desired to have it communicated. The idea is to use two complex combinations of B, E, H, D as independent variables, but in such a way that their “conjugates,” i. e. , the partial derivatives of L , equal their complex conjugates.

Must the Photon Mass be Zero

The old query concerning longitudinal waves, which already beset the elastic theory of light, has in our day revived in the form expressed in the title. Maxwell’s field laws are a singular limiting case in that they admit but transversal waves. If this held only in however close an approximation, some fundamental laws of radiation would seem to be affected by a factor 3/2, on account of the ‘third degree of freedom’. If so, this would render even Maxwell’s theory suspect, for we are loath to accept as an adequate description of nature a limiting case whose predictions differ grossly and discontinuously from those reached by a sufficiently close approach to the limit. We show here in the simple, if fictitious, example of an ideal conductor, that by extending Proca’s field equations in a plausible fashion to the interior of matter the discontinuity is avoided and the correct factors (not 3/2 thereof) are already reached with a rest-mass at the upper limit, imposed anyhow by other well-known considerations.

The philosophy of experiment

SummaryThe accepted outlook in quantum mechanics (q.m.) is based entirely on its theory of measurement. Quantitative results of observations are regarded as the only accessible reality, our only aim is to predict them as well as possible from other observations already made on the same physical system. This pattern is patently taken over from the positional astronomer, after whose grand analytical tool (analytical mechanics) q.m. itself has been modelled. But the laboratory experiment hardly ever follows the astronomical pattern. The astronomer can do nothing but observe his objects, while the physicist can interfere with his in many ways, and does so elaborately. In astronomy the time-order ofstates is not only of paramount practical interest (e.g. for navigation), but it was and is the only method of discovering thelaw, known by now in its general features (Newton). The physicist is nearly always still out for discovering thelaw (technically speaking; a Hamiltonian); this he rarely, if ever, attempts by following a single system in the time-succession of its states, which in themselves are of no interest. The accepted foundation of q.m. claims to be intimately linked with experimental science. But actually it is based on a scheme of measurement which, because it is entirely antiquated, is hardly fit to describe any relevant experiment that is actually carried out, but a host of such as are for ever confined to the imagination of their inventors.RiassuntoL’interpretazione più comune della meccanica quantistica (m.q.) è interamente fondata sulla sua teoria della misura. Come sola realtà accessibile si considerano i risultati quantitativi delle osservazioni, il nostro unico fine essendo quello di predirli per quanto possibile a partire da altre osservazioni già fatte sullo stesso sistema fisico. Questo schema è interamente suggerito dalla astronomia di posizione, sul cui grande strumento analitico (la meccanica analitica) è stata modellata la stessa m.q.. Ma le esperienze di laboratorio ben di rado seguono lo schema dell’astronomia. L’astronomo non può che osservare i suoi oggetti, mentre il fisico può influenzare i propri in molte maniere, e anzi lo fa in modo elaborato. In astronomia la sequenza temporale deglistati è non solo di enorme interesse pratico (per esempio per la navigazione), ma è stata ed è il solo modo di scoprire lalegge, che si è finito per conoscere nei suoi aspetti generali (Newton). Il fisico ancora oggi si propone di scoprire la legge (in terminologia tecnica: una Hamiltoniana); ma raramente, o mai, egli cerca di raggiungere lo scopo seguendo la successione temporale degli stati di un singolo sistema, che non sono di per sè di interesse fisico. L’interpretazione più comune della m.q. si vanta di essere intimamente legata alla scienza sperimentale. Ma in realtà è basata su uno schema di misura che, essendo interamente antiquato, è ben poco adatto a descrivere qualunque esperienza che venga realmente eseguita, ma piuttosto una schiera di esperienze per sempre limitate alla immaginazione dei loro inventori.

Might perhaps energy be a merely statistical concept

SummaryArguments are given in favour of the opinion that the quantum-mechanical frequency, multiplied by Planck’s constant, has for microscopic systemsnot the meaning of energy. It is suggested, that the concept of energy and its conservation, just like that of entropy and its increase, has merely a statistical meaning, the energy of a macroscopic system being the product of Planck’s constant and a weighted average of the frequencies in question. The wide-spread attitude that the claim for an objective description of physical reality must be given up, is rejected on the ground that the so-called external world is built up exclusively of elements of the single minds, and is characterized as what is common to all, recognized by every healthy and sane person. Hence the demand for a non-subjective description is inevitable, of course without prejudice whether it be deterministic or otherwise.RiassuntoSi danno argomenti in favore dell’opinione che in meccanica quantistica la frequenza moltiplicata per la costante di Plancknon ha per i sistemi microscopici il significato di energia. Si esprime l’opinione che i concetti di energia e della sua conservazione, al pari di quelli di entropia e del suo aumento, hanno solo un significato statistico, l’energia di un sistema macroscopico essendo il prodotto della costante di Planck per una media ponderata delle frequenze in questione. L’opinione diffusa che il proposito di dare una descrizione obiettiva della realtà fisica debba essere abbandonato è respinta basandosi sul fatto che il cosiddetto mondo esterno è costituito soltanto di elementi delle singole menti ed è caratterizzato come ciò che è comune a tutti, e riconosciuto da ogni persona sana e ragionevole. Donde è inevitabile la richiesta di una descrizione non soggettiva, naturalmente senza pregiudizio del fatto che essa sia deterministica o di altra natura.

The fundamental idea of wave mechanics

On passing through an optical instrument, such as a telescope or a camera lens, a ray of light is subjected to a change in direction at each refracting or reflecting surface. The path of the rays can be constructed if we know the two simple laws which govern the changes in direction: the law of refraction which was discovered by Snellius a few hundred years ago, and the law of reflection with which Archimedes was familiar more than 2,000 years ago. As a simple example, Fig. 1 shows a ray A-B which is subjected to refraction at each of the four boundary surfaces of two lenses in accordance with the law of Snellius.

Bohrs neue Strahlungshypothese und der Energiesatz

vom Gesichtspunkt der Schliegung oder Aufsprengung von-Ringsystemen erneut zu erforschen sein. Selbstverst~ndlich muB auch der Bildung tier Proteine im Pflanzenreich erneute Aufmerksamkeit zugewandt werden. Es ist wohl m6glich, dab zun~chst nicht Aminos~uren, sondern direkt Anhydride gebildet werden, und zwar k6nnten vielleicht Beziehungen zu den Elementark6rpern der Polysaccharide und insbesondere der StXrkeanteile vorhanden sein. Es set noch ganz besonders auf die Beziehungen yon Diketopiperazinen zu Bausteinen bzw. Strukturen yon AIkatoiden hingewiesen. Auch in dieser Richtung ergeben sick yon der entwicketten Auffassung der Struktur der Proteine aus ganz neue Ausblicke. Ich bin mir wohl bewul3t, dab noch eine weite Strecke mfihevoller Forscherarbeit zurfickzulegen SCNR6DINGER: Bohrs neue Strahlungshypothese und der Energiesatz. [ Die Natur[wissenschafte~

The Wave Equation for Spin 1 in Hamiltonian Form

If from the differential equations that hold in a Proca field you select the ten that express the time derivatives of the ten components involved, i. e. of the ‘electromagnetic’ field and its potential vector, you obtain right away for the ten-componental entity an equation that may be said to be at the same time of the Schrödinger, the Dirac and the Kemmer type. The four 10 x 10-matrices that occur as coefficients are Hermitian and satisfy Kemmer’s commutation rules. The fifth is easily constructed. Those of the Proca equations that were not included are merely injunctions on the initial value. They are expressed by one matrix equation, that makes it evident that, once posited, they are preserved. The three spin matrices are indicated. The spin number is 1 or 0, but the aforesaid injunctions exclude 0.