A Compendium of Quantum Physics

Concepts, Experiments, History and Philosophy


Edited by Friedel Weinert, Klaus Hentschel and Dan Greenberger

I.                    The objective of this project is to publish a Compendium to the Quantum World, which covers both technical and interpretational aspects of quantum theory. This Compendium is committed to an educational aim: to aid students of physics and philosophy at both a graduate and undergraduate level and in many different educational settings to gain a lucid understanding of the terms involved. Each article will be structured around three themes: a clear explanation of each term with illustrations where possible; an indication of the first appearance of the term in the technical literature and a brief list of relevant literature. The Compendium will also contain a Glossary of English, French and German terminology. Such glossary reflects the fact that quantum mechanics was created in these three languages. It will help all those who need to read and write in different languages. (Some examples are given in Section VI.)

II.                 The rationale for this compendium is threefold. Firstly, there is no consistent use of the technical terminology in the physical and philosophical literature. No such Compendium for Quantum Mechanics exists at present. [John Gribbin’s Companion of the Cosmos (1996) does the job for astronomy. CUP has published Compendiums to Darwin, the Mind etc.]. Secondly, quantum mechanics continues to preoccupy physicists and philosophers so that there is a need for a comprehensive, consistent terminology. By providing authoritative entries in this Compendium, we aim to contribute to a more consistent use amongst physicists and philosophers. We also aim to provide a comprehensive terminology to complement existing textbooks. For instance, many textbooks on modern physics do not even mention the Bell inequalities. Thirdly, the field has undergone rapid development in recent years. Many once purely theoretical ideas, like entanglement and the double-slit experiment, have become experimental realities. The programme of Decoherence and new interpretations of quantum theory, in particular the Consistent Histories Approach, have added a new impetus and put the familiar approaches to quantum mechanics into a new light. Quantum physics stands on the threshold of unimagined new technological advances in quantum cryptography, information and teleportation.

III.               The market for such a Compendium is worldwide. It will be a timely publication because it will record established and new concepts in quantum mechanics at a time of major transition from foundational to applicable work. Already the changes in quantum mechanics are attracting the attention of a wider public. We expect there to be a strong demand for such a Compendium: firstly, because of its educational aim and secondly because of the international dimension we intend to give it. Whilst the four main editors will write some of the articles, we will solicit specialized articles from leading international figures in the field. The Compendium will provide an up-to-date and technically precise dictionary of quantum mechanical terms. The third reason is the continued fascination with quantum mechanics both in the areas of physics and the history and philosophy of science. Many universities across the world have established Schools or Departments with the specific purpose of teaching history and philosophy of science.  

IV.              There are no direct competitors for this project. As far as we are aware a Handbook of the Philosophy of Science is being prepared under the editorship of John Earman and Jeremy Butterfield for North Holland Publishing Company. Oxford University Press is preparing a Companion to the Philosophy of Physics. We should also mention the Stanford Encyclopædia of Philosophy, which contains many entries related to quantum mechanics. The aims of these works will be more general and dealing with philosophical issues only. The entries in these books will generally be longer (as for instance in the Stanford Encyclopædia) and mainly addressed to philosophers. In the Stanford Encyclopædia one looks in vain, for instance, for entries on which-way experiments, the Franck-Hertz experiment etc.  We aim at a broader audience: to publish a reference work for physicists, philosophers and historians alike. We also aim in a different direction: to provide reliable descriptions and illustrations of key terminology in comprehensible and comprehensive form; to derive the philosophical consequences of quantum mechanics out the scientific problem situation; to identify the historical roots of the terminology; and to aid readers with targeted bibliographical references.


Note: For reasons of length and economy it is best not to include entries on particular physicists, as this information is easily obtainable elsewhere.

IV. Entries and Contributors

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VI.               Glossary: English, German, French

·        Angular Momentum/Drehimpuls/moment angulaire

·        Annihilation operator/Vernichtungsoperator/opérateur d’annihilation

·        Bell inequalities/Bellsche Ungleichung/Inégalités de Bell

·        Blackbody Radiation/Hohlraumstrahlung/rayonnement du corps noir

·        Brownian motion/ Brownsche Bewegung/Brownsche Molekularbewegung/ Mouvement brownien

·        Collapse of wavefunction/Kollaps or Reduktion der Wellenfunktion/réduction du paquet d’ondes

·        Creation Operator/Erzeugungsoperator/ opérateur de création

·        Delayed Choice Experiment/Experiment mit verzögerter Wahl/experience à choix retardé

·        Detached observer/aussenstehender Beobachter/observateur détaché

·        Double slit or two-slit experiment/Doppelspalt – or Zwei-Löcher Experiment/Expérience des fentes d’Young or Expérience à doubles fentes

·        Entanglement/Verschränkung/Intrication

·        Gauge theories/Eichtheorien/Théorie de jauge

·        Mixed states :

§         Mixture of states/Gemenge/ vrai mélange

§         Improper mixture/Gemisch/mélange, impropre

·        Hidden parameters/verborgene Variable/Variables cachés

·        Large-angle scattering/Rückwärtsstreuung/ diffusion à grand angle

·        Many-worlds interpretation/Viele-Welten-Interpretation/Interprétation multimondes

·        Relative states interpretation/relative Zustände Interpretation/théorie de la relativité des états

·        Measurement Problem/Messproblem/problème de la mesure

·        Occam’s razor/Occams Rasiermesser/rasoir d’Occam

·        Observable, non-commuting/ Observable, nichtvertauschbare/observable non commutantes

·        Observable, physical quantity, measurable quantity/Observable, physikalische Grösse, Messgrösse, beobachtbare Grösse/observable propriété physique, propriété mesurable

·        Operator, self-adjunct/Operator, selbstadjungierter/opérateur autoadjoint

·        Pauli exclusion principle/Pauliprinzip/Principe de Pauli

·        Pilot wave/Führungswelle)/onde pilote

·        Plum pudding model/Rosinenkuchenmodell/modèle du gâteau aux raisins

·        Quantum Eraser/Quantenlöscher/gomme quantique

·        Schrödinger equation/Schrödinger-Gleichung/èquation de Schrödinger

·        Space quantization/Richtungsquantelung/

·        Spin/Spin/Spin

·        States/Zustände/états

·    Decaying states/zerfallende Zustände/états se désintégrant

·    Smeared-out states/verschmierte Zustände/ états étalés ou non-localisés

·    Excitation states/Anregungszustände/ états d’excitation

·    Excited states/erregte Zustände/états excités

·    Pure states/reiner Fall/état pur

·    Mixed state/Gemenge/

·    State reduction/Zustandsreduktion/réduction du paquet d’ondes

·        Superposition/Superposition or kohärente Überlagerung/Superposition

·        Superselection Rules/Superauswahlregeln/règle de supersélection

·        Trace/Spur/trace

·        Tunnel effect/Tunneleffekt/effet tunnel

·        Wave function/Wellenfunktion/Fonction d’onde

·        Wave packet/ Wellenpaket/paquet d’ondes

·        Wave-particle duality/Welle-Teilchen Dualismus/dualité onde-particule

·        Which way experiments/welcher-weg Experimente/


Note: Many thanks to Michel Le Bellac for his help with the French terms.

Worked Examples


·        The famous Compton experiment concentrates on the wave rather than the particle aspect of quantum phenomena. It had been observed that the wavelength of X-rays is increased when they are scattered off matter. Arthur Compton (1892-1962) showed that this behaviour could be explained by assuming that the X-rays were photons. When photons are scattered off electrons, part of their energy is transferred to the electrons. The loss of energy is translated into a reduction of frequency, which in turn leads to a lengthening of the wavelength of the scattered photons. This happens because the relation E = hn = hc/l holds. In these experiments, first carried out between 1919 and 1922, the scattering of X-rays is treated as a collision of photons with electrons (Figure I).


Figure I: Compton’s Model of the Scattering process


Incident Photon,Recoiling Electron,Scattered Photon,l









The wavelength of the scattered photon, l, can be related to its initial wavelength, lo, to the electron mass, me, and the scattering angle, q, by the relation. We should note that Compton was not content with stating the equation. He also sought an explanation. Compton’s description of his model conveys the flavour of a mechanistic explanation.



From the point of view of the quantum theory, we may suppose that any particular quantum of X-rays is not scattered by all the electrons in the radiator, but spends all of its energy upon some particular electron. This electron will in turn scatter the ray in some definite direction, at an angle with the incident beam. This bending of the path of the quantum of radiation results in a change in its momentum. As a consequence, the scattering electron will recoil with a momentum equal to the change in momentum of the X-ray. The energy in the scattered ray will be equal to that in the incident ray minus the kinetic energy of the recoil of the scattering electron; and since the scattered ray must be a complete quantum, the frequency will be reduced in the same ratio as is the energy. Thus on the quantum theory we should expect the wave-length of the scattered X-rays to be greater than that of the incident rays.

In terms of a causal account, the effect is the increase in wavelength of the scattered photon, caused by a collision with an electron. Note that Compton’s explanation dispenses with the above-stated Compton scattering formula, i.e. the precise numerical determination of the wavelenth, l, of the scattered photon. 




[1] A. H. Compton: A Quantum Theory of the Scattering of X-Rays by Light Elements. The Physical Review 21, 483-502 (1923a)

[2] A. H. Compton: The Spectrum of Scattered X-Rays. The Physical Review 22, 409-413 (1923b)

[3] W.Bothe, H. Geiger: Über das Wesen des Comptoneffekts. Zeitschrift für Physik 32, 639 (1925)



[4] M. H. Shamos: Great Experiments in Physics (Dover 1987/New York 1959, 348-58)

[5] R. Stuewer: The Compton Effect- Turning Point in Physics (Science History, New York 1975)

[6] B. Falkenburg: Teilchenmetaphysik (Spektrum, Heidelberg, Berlin, Oxford 1995, 102-5)



The Davisson-Germer experiment (1927) was the first measurement of the wavelengths of electrons. C. J. Davisson, who worked in the Bell Research Laboratories, received the Nobel Prize in Physics for the year 1937 together with George P.- Thomson from the University of Aberdeen in Scotland, who independently also found experimental indications of electron diffraction. According to the →Copenhagen Interpretation of Quantum Mechanics,  →wave-particle duality leads to particles also exhibiting wave-like properties like extension in space and interference.


Clinton J. Davisson (1881-1958) and Lester H. Germer (1896-1971) investigated the reflection of electron beams on the surface of nickel crystals. When the beam strikes the crystal, the nickel atoms in the crystal scatter the electrons in all directions. Their detector measured the intensity of the scattered electrons  with respect to the incident electron beam. Their normal polycrystalline samples exhibited a very smooth angular distribution of scattered electrons. In early 1925, one of their samples was inadvertently recrystallized in a laboratory accident that changed its structure into nearly monocrystalline form. As a result, the angular distribution manifested sharp peaks at certain angles. As Davisson and Germer soon found out, other monocrystalline samples also exhibited such anomalous patterns, which differ with  chemical constitution, angle of incidence and orientation of the sample. . Only in late 1926 did they understand what was going on, when Davisson attended the meeting of the British Association for the Advancement of Science in Oxford. There Born spoke about →de Broglie’s matter-waves and Schrödinger’s →wave mechanics. Their later measurements completely confirmed the quantum mechanical predictions for electron wavelength l as a function of momentum p: l=h/p. But their initial experiments (unlike G.P. Thomson’s) were conducted in the context of industrial materials research on filaments for vacuum tubes, not under any specific theoretical guidance.

The phenomenon of electron diffraction is quite general and can be explained by the wave nature of atomic particles. Planes of atoms in the crystal (Bragg planes) are regularly spaced and can produce a constructive interference pattern, if the so-called Bragg condition (nl = 2 d sinq = Dsinf, where d is is the spacing of atomic planes and D is the spacing of the atoms in the crystal) is satisfied. This condition basically states that the reflected beams from the planes of atoms in the crystal will give an intensity maximum, or interfere constructively, if the distance, which the wave travels between two successive planes (2 d sinq), amounts to a whole number of wavelengths (nl, n = 1,2,3...). This is illustrated in Figure I.



Figure I: Davisson-Germer Experiment

                Scattering of electrons by a crystal for 54eV electrons.



                                                                          Source                               Detector


Electron beam
Reflected beam




                        ·    ·    ·    ·    ·    ·    ·    · Atoms in crystal

·    ·    ·    ·    ·    ·    ·    ·

·    ·    ·    ·    ·    ·    ·    ·

D·    ·    ·    ·    ·    ·    ·    ·

Bragg planes





In their experiment, Davisson and Germer found that the intensity reached a maximum at f = 50º (for an initial kinetic energy of the electrons of 54 eV, normal incidence as indicated and f as the scattering angle.). From a philosophical point of view this experiment reveals a striking feature. It demonstrates the existence of de Broglie waves. Yet we can speak of causation, not in a deterministic but in a probabilistic sense. There is clearly, on the observational level, a conditional dependence of the intensity of the reflected beam on the set of antecedent conditions. These antecedent conditions are also conditionally prior to their respective effects. There is of course no local causal mechanism, as the causal situation covers a stream of particles. There is only a certain likelihood that one particular particle in these experiments will be scattered in a particular direction.


But sufficiently much is known about scattering of atomic particles to establish a causal dependence between the antecedent and consequent conditions. In the Davisson-Germer experiment the wavelength of the electron beam, scattered at 50º, is 0.165nm. This is the effect to which specific antecedent conditions correspond: the electron beam has initial kinetic energy of 54 eV; the lattice spacing of the nickel atoms is known, from which the spacing of the Bragg planes can be calculated; the condition for constructive interference is also known. There is quite a general dependence of the interference effects on the regular spacing of the atom planes in the crystal. It is used regularly in the study of atomic properties and is completely analogous to the use of X-raydiffraction by Max von Laue, Paul Knipping and Walter Friedrich in 1912. Under certain conditions, particles such as electrons thus exhibit wave-like characteristics like electromagnetic radiation.




[1] C. Davisson, L. H. Germer: Diffraction of Electrons By A Crystal of Nickel. Physical Review 30/6, 705-40 (1927)



[2] A. P. French, E. F. Taylor: An Introduction to Quantum Physics (Chapman & Hall 1979; Stanley Thornes Publishers, Cheltenham 1998, 64-72)

[3] K. Krane: Modern Physics (John Wiley & Sons, New York 1983, 89-93)

[4] A. Russo: Fundamental research at Bell laboratories: The discovery of electron diffraction. Historical Studies in the Physical Sciences 12, 117-160 (1981)

[5] P. A. Tipler: Modern Physics (Worth Publishers, New York 1978, 169-73)



·        Franck-Hertz experiment In 1913 Bohr took Rutherford’s nucleus model of the hydrogen atom as the basis for his quantized →atom model. Although it was not the  first, it was the first successful atom model.  A year later,– two Berlin experimenters, James Franck (1882-1964) and Gustav Hertz (1887-1975), unaware of Bohr's model and its implications, performed an experiment which later turned out to be one of its strongest corroborations. For the so-called Franck-Hertz experiment, they were awarded the Nobel Prize for Physics in 1925. In this experiment electrons are ejected from a cathode, C, into a tube filled with mercury gas (see Figure I). The energy of the electrons can be increased in a controllable manner by accelerating them towards the positively charged grid, G, through the potential difference Va. Electrons fly through the grid towards anode A. Between G and A, a small retarding voltage, Vr, decelerates the electrons. They will only reach the anode A, if their energies V exceed Vr, where they will be recorded by the ammeter A.










                                                                      +                      +         

Figure I: Franck-Hertz Experiment (1914)

Collisions between the atoms and the electrons will occur. Only electrons with sufficient energy will cause the mercury atoms to make transitions to higher states of energy. The electrons will lose their energy to the atoms. When Va = 4,9 V the curve drops very sharply.

The two experimenters initially thought they had measured mercury's ionization potential.

As Bohr pointed out in August 1915 but Franck and Hertz only realized in 1917, the Bohr atomic model provides a perfect explanation for this behaviour. The electrons near the grid lose all their energy to the mercury atoms and are unable to overcome the small retarding potential, Vr, to reach the anode. A drop in the current, Ia, is observed. When Va = 9.8 V, another drop in the curve occurs. The electrons either excite the atoms to higher energy levels or lose 4.9V more than once. The excited mercury atoms in turn will return to their ground energy state and emit photons with energies corresponding to the energy intake. The experiment displayed the loss of the electronic energy at discrete levels. Later, more precise experiments confirmed that the higher states of energy of the atoms corresponded to the discrete energy levels calculated from the Bohr model. The observable results are shown in Figure II.


As in quantum mechanics there is a traditional distinction between the wave and the particle picture, we should note that the Franck-Hertz experiment illustrates the particle picture of quantum mechanical processes. (For the wave-picture see Stern-Gerlach experiment and Davisson-Germer experiment) In this experiment the particle picture gives rise to a probabilistic notion of causality, since we are not in a position to predict which electron will collide with which mercury atom and how much energy it will transfer.




[1] J. Franck, G. Hertz: Über Zusammenstöße zwischen Elektronen und den Molekülen des Quecksilberdampfes und die Ionisierungsspannung desselben. Verhandlungen der Deutschen Physikalischen Gesellschaft 16, 457-67 (1914) [together with supplementary texts and commentary by Armin Hermann in: Die Elektronenstoß­versuche (Battenberg, München 1967 = Dokumente der Naturwissenschaft, 9)]

[2] J. Franck, P. Jordan: Anregung von Quantenspüngen durch Stöße (Springer, Berlin 1926)



[3] A. P. French, E. F. Taylor: An Introduction to Quantum Physics (Chapman & Hall 1979; Stanley Thornes Publishers, Cheltenham 1998, 30-1)

[4] K. Krane: Modern Physics (John Wiley & Sons, New York 1983, 169-70)

[5] P. A. Tipler.: Modern Physics (Worth Publishers, New York 1978, 155-7)

[6] H. Geiger, K.  Scheel eds.: Quanten (Springer, Berlin 1926) (= Handbuch der Physik,  23)

[7] J. Lemmerich: Max Born, James Franck: Physiker in ihrer Zeit (Reichert, Wiesbaden 1982)

[8] J. Lemmerich: Aufrecht im Sturm (Spektrum, Berlin 2005; biography of J. Franck)

[9] J. Kuczera: Gustav Hertz (Teubner, Leipzig 1985)




The Stern-Gerlach experiments (SG experiments) were prepared and carried out by Otto Stern (1888-1969) and his junior collaborator Walther Gerlach (1889-1979) between 1921 and 1925. [1]-[6] According to modern textbook interpretations the experiments established experimentally the so-called quantization of angular momentum and therefore the discreteness of the magnetic moment of atomic particles. This phenomenon is known as ‘space quantization’ (Richtungsquantelung) of angular momentum. As indicated below, the actual historical context, in which the experiments were carried out, is more complex. Quantization of angular momentum means that particles like electrons orbit the nucleus only in certain permitted planes.  The experiments demonstrated, for the first time, the idea, proposed by Arnold Sommerfeld (1868-1951) in 1916, of the quantization of the orbital planes of the electron in the atom. The orbital planes of electrons do not only possess discrete sizes and shapes. These orbital planes must also be inclined in certain ways. They must have discrete spatial orientations in relation to a co-ordinate system like an external magnetic field. The size, shape and orientation of the orbital planes are indicated by quantum numbers (n, l, ml). In addition it became clear in 1925 that a quantum number for intrinsic angular momentum, s, was needed. These quantum numbers specify the state of the atoms in an atom beam. When a beam of atoms is sent through a non-uniform magnetic field, this discrete spatial orientation will be revealed on a screen mounted behind the magnet. Stern and Gerlach therefore ran these experiments on beams of silver atoms in inhomogeneous magnetic fields. The purpose of the SG experiments is to maximize the effect of magnetic field gradients, , on the silver atoms. It is necessary for the magnetic field to be inhomogeneous so that the magnetic moments of the particles feel a net force acting on them. In fact, in a non-uniform magnetic field, with gradient, the magnetic dipole moments, μ, experience both a torque, which makes them align with the magnetic field, Bz, but also a net force, which leads to their displacement.    In a typical Stern-Gerlach experiment, the magnetic field will split the beam into two parts and send the silver atoms either into the upper or the lower beam. Two scenarios can be distinguished:


1)      The beam of silver atoms – silver atoms have 47 electrons - is sent through the magnet but the magnet is switched off. A screen mounted behind the magnet will record the impact of the atoms. When the magnet is switched off, one central trace will be recorded after the passage of the atom beam () because no deflection is experienced by the atoms in this state (Figure I).


2)      The magnet is now switched on when the beam of atoms is sent through. Depending on the precise state of the atom beam, specified by its quantum numbers, and assuming the simplest case, two traces will appear on the screen. The effect of the magnet will be an intensity shift. When the magnet was switched off the intensity maximum was in the centre of the screen. But with the magnet switched on, this central intensity maximum will become a minimum. The central trace will disappear and two clearly separated traces will appear, deflected upwards and downwards respectively (Figure I). With the magnet switched on, the magnet will cause the atom beam to split exactly into two halves (under appropriate conditions). This shift will happen only if the magnetic gradient is large enough to cause the displacement of the magnetic moments.





On the modern theory, an electron has orbital angular momentum, L, and spin angular momentum, S. The total angular momentum, J, is the sum of L and S:



Generally, the magnetic moment, m, is related to J through the expression



The SG experiment detected two traces, in violation of equation (2). The silver atoms were in their ground state (orbital angular momentum l=0, ml=0 and hence no deflection is expected; spectroscopic notion 2S½) but the splitting was due to the magnetic moment of the spin angular momentum of the electron (ms = ± ½h) in the z-direction (direction of the magnetic field). When l = 0, it follows from expression (1) that we are left with the value for S = ½h for intrinsic spin, so that the beam splits into two and leaves two traces.

The historical situation was more complicated than this textbook account. [13] Strictly speaking, Stern and Gerlach believed that they had found Sommerfeld’s quantization of angular momentum, L. They did not realize that the observed space quantization was due to the magnetic moment of the spinning electron (hence S). The two experimenters believed that their experiments had decidedly disproved the classical Lamor theory, which was based on continuous values for magnetic moments. They thought their experiments confirmed Sommerfeld’s old quantum theory (1916), which postulated, in addition to the usual quantum numbers for the size and shape of orbits, a spatial orientation of the ‘Keplerian’ orbits of the electrons around the nucleus. The discovery of spin angular momentum of the electron came in 1925, when George Eugene Uhlenbeck (1900-1988) and Samuel Abraham Goudsmit (1902-1978) proposed the concept of ®spin. Contrary to frequently made claims in modern physics textbooks, Stern and Gerlach were not surprised by their results (splitting of beam into two traces) because this is just what Sommerfeld’s theory told them to expect. Today many features of the Stern-Gerlach and the double-slit experiments reappear in so-called ®which-way experiments.


The Stern-Gerlach experiment is also interesting from a philosophical point of view.  First, they demonstrate the relative robustness of experimental results and their relative independence from the theoretical conceptions, on which they are based. Secondly, they tell us that the often-quoted acausality of quantum mechanical processes is not supported by the SG experiments. It is not difficult to apply Mill’s ‘method of difference’, a form of eliminative induction, to this situation to establish its causal nature. The only difference between otherwise two identical situations, including the preparation of the atoms in identical atomic states, specified by the quantum numbers, lies in the behaviour of the magnet. If it is not switched on and there is no magnetic field, one central trace appears; if it is switched on and a magnetic field is applied to the passing atoms, two traces appear in the simplest case (l=0). The set of causal conditions is closed. There are no other interfering factors to be considered. We are therefore justified in concluding that the creation of the non-uniform magnetic field is the cause, given the initial state of the atoms, of the splitting of the atomic beam into two parts. As is customary in quantum mechanics, no claim is made about the behaviour of the individual atoms making up the beam. Since the initial orientation of their magnetic moment is random it is not possible to predict, which way they will turn under the influence of the magnet. But statistical predictions can be made about the behaviour of the whole beam. The rules of quantum mechanics specify how atom beams in different states behave. For instance, if l¹0 an odd number of traces will appear on the screen.  The SG experiments show that causal relations obtain in the quantum domain but they are not deterministic. Hence causality and the pair ®determinism-indeterminism must be distinguished.




[1] O. Stern: Ein Weg zur experimentellen Prüfung der Richtungsquantelung im Magnetfeld. Zeitschrift für Physik 7, 249-53 (1921)

[2] W. Gerlach, O. Stern: Der experimentelle Nachweis des magnetischen Moments des Silberatoms. Zeitschrift für Physik 8, 110-1 (1921)

[3] W. Gerlach, O. Stern: Der experimentelle Nachweis der Richtungsquantelung im Magnetfeld. Zeitschrift für Physik 9, 349-52 (1922)

[4] W. Gerlach, O. Stern: Das magnetische Moment des Silberatoms. Zeitschrift für Physik 9, 353-5 (1922)

[5] W. Gerlach, O. Stern: Über die Richtungsquantelung im Magnetfeld. Annalen der Physik 74 (1924)

[6] W. Gerlach, O. Stern: Über die Richtungsquantelung im Magnetfeld II. Annalen der Physik 76 (1925)

[7] A. Einstein, P. Ehrenfest: Quantentheoretische Bemerkungen zum Experiment von Stern und Gerlach. Zeitschrift für Physik 11, 31-4 (1922); reprinted in P. Ehrenfest: Collected Scientific Papers. Ed. M. Klein (North Holland, Amsterdam 1959, 452-55)



[8] R. G. Griffiths: Consistent Quantum Theory (Cambridge University Press, Cambridge 2003, Ch. 17-17)

[9] R. Harré: Great Scientific Experiments (Oxford University Press, Oxford/New York 1983, Part III, B)

[10] R. I. G. Hughes: The Structure and Interpretation of Quantum Mechanics. (Harvard University Press, Cambridge, Massachusetts/London 1989, 1-8)

[11] M. Jammer: The Conceptual Development of Quantum Mechanics (McGraw Hill, New York 1966)

[12] M. Morrison: Spin: All is not what it seems. Studies in History and Philosophy of Modern Physics 2007

[13] F. Weinert: Wrong Theory – Right Experiment: The Significance of the Stern-Gerlach Experiments. Studies in History and Philosophy of Modern Physics 26, 75-86 (1995)

[14] W. Walter: Otto Stern: Leistung und Schicksal. Gesellschaft Deutscher Chemiker, Fachgruppe Geschichte der Chemie: Mitteilungen 3, 69-82 (1989)

[15] R. Heinrich, H.-R. Bachmann: Walther Gerlach (Deutsches Museum, Munich1989, esp. 48-54)





·        The distinction between pure case (reiner Fall) and mixed case (Gemenge) was introduced by Hermann Weyl (1885-1955). Mathematically, a pure case is represented by a vector or wave function. The representation of a mixed case requires a density operator. Today a uniform representation of pure and mixed cases is achieved through the use of statistical operators.

Physically a pure case occurs when some operator, like spin, of a quantum system possesses an eigenvalue with certainty. For instance, once the inhomogeneous magnet in the Stern-Gerlach experiment has split the atom beam into two parts, it is certain that atoms in the upper beam possess, say, the value +½, while those in the lower beam have the value ¬½. The most important characteristic in the present context is that pure cases lead to coherent superpositions, with their unmistakable interference fringes. When interference patterns are observed, say between photons, their paths are indistinguishable. The experimenter cannot know in principle through which arm of the interferometer the respective photons have travelled. This fact is well known from the double-slit experiments. The degree of indistinguishability is directly related to the degree of coherence.

Mixed cases are sometimes called incoherent superpositions. That is they do not display interference patterns, which means that the paths of their respective components are distinguishable. The experimenter can therefore tell through which arm a particular photon travelled. It is the distinguishability, which destroys the interference. Again this is well known from double-slit experiments, in which any attempt to determine through which slit a photon travelled will destroy the interference patterns. According to Heisenberg’s indeterminacy relations, the position of the photon and its momentum are incommensurable operators. More generally, if the experimenter gains some information about the state of the system, the coherence of the quantum system is disturbed. The coherent superposition is destroyed. Mixed states can be formed out of any other states; for instance two electron beams, pointed in slightly different x-directions, or a two-system entangled spin state, measured with polarizers tipped at different angles. It is the distinguishing mark of mixed states that their component states (for instance, polarization states) are only known probabilistically. If we adopt Dirac's symbol for a quantum state, , and  and  are two different states of an ensemble of electrons, then there is a state , where are probability coefficients, with .




[1] H. Weyl: Quantenmechanik und Gruppentheorie. Zeitschrift für Physik 46, 1-46 (1927)

[2] B. D’Espagnat: Conceptual Foundations of Quantum Mechanics (Addison-Wesley-Benjamin, Reading, Mass., 1976, Ch. 6)



[3] R. I. G. Hughes: The Structure and Interpretation of Quantum Mechanics (Harvard University Press, Cambridge, Mass., London 1989, passim)

[4] P. Mittelstaedt: Philosophische Probleme der modernen Physik (Bibliographisches Institut, Mannheim, Wien, Zürich  41972)  [Engl. Philosophical Problems of Modern Physics (D. Reidel 1976, Ch. III)]

[5] B. van Fraassen: Quantum Mechanics (Clarendon Press, Oxford 1991, passim)

[6] R. Torretti: The Philosophy of Physics (Cambridge University Press, Cambridge 1999, 345-8)