However, Mulder is still puzzled, because he is swayed by Tim Maudlin’s verypersuasive arguments that the violation of the Bell Inequalities in experiments such as thiscan only be accounted for by the assumption that there is some sort of superluminalcausation, in apparent violation of the theory of relativity. [9]

Trying to find some more general explanation of the Bohr atom, de Broglie proposed

Planck noted that both these phenomena

Schrödinger took the absurd implications of this thought experiment (a cat simultaneously dead and alive) as an argument against the Copenhagen interpretation. However, this interpretation remains the most commonly taught view of quantum mechanics.

Third, if a matter wave is given by the wave function Ψ(x,t)Ψ(x,t)Ψ(x,t)<math display="inline"><semantics><mrow><mrow><mtext>Ψ</mtext><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><mtext>Ψ</mtext><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></annotation-xml></semantics></math>, where exactly is the particle? Two answers exist: (1) when the observer is not looking (or the particle is not being otherwise detected), the particle is everywhere (x=−∞,+∞)(x=−∞,+∞)(x=−∞,+∞)<math display="inline"><semantics><mrow><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>=</mo><mtext>−</mtext><mi>∞</mi><mo>,</mo><mtext>+</mtext><mi>∞</mi><mo stretchy="false">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>=</mo><mtext>−</mtext><mi>∞</mi><mo>,</mo><mtext>+</mtext><mi>∞</mi><mo stretchy="false">)</mo></mrow></annotation-xml></semantics></math>; and (2) when the observer is looking (the particle is being detected), the particle “jumps into” a particular position state (x,x+dx)(x,x+dx)(x,x+dx)<math display="inline"><semantics><mrow><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>x</mi><mo>+</mo><mi>d</mi><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>x</mi><mo>+</mo><mi>d</mi><mi>x</mi><mo stretchy="false">)</mo></mrow></annotation-xml></semantics></math> with a probability given by P(x,x+dx)=|Ψ(x,t)|2dxP(x,x+dx)=|Ψ(x,t)|2dxP(x,x+dx)=|Ψ(x,t)|2dx<math display="inline"><semantics><mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>x</mi><mo>+</mo><mi>d</mi><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo>|</mo><mtext>Ψ</mtext><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><msup><mo>|</mo><mn>2</mn></msup><mi>d</mi><mi>x</mi></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>x</mi><mo>+</mo><mi>d</mi><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo>|</mo><mtext>Ψ</mtext><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><msup><mo>|</mo><mn>2</mn></msup><mi>d</mi><mi>x</mi></mrow></annotation-xml></semantics></math> —a process called state reduction or wave function collapse. This answer is called the Copenhagen interpretation of the wave function, or of quantum mechanics.

acabó interpretándose por los astrónomos en el sentido de que eso (predecir correctamente las observaciones) es lo único que hay que pedirle a una teoría, no que sea una descripción de la realidad

El mundo existe independientemente de nosotros como conocedores y es como es independientemente de nuestro conocimiento teórico sobre él.

It was just at this time, during the summer of 1900, that Curlbaum and Rubens in Berlin had made very accurate new measurements of the spectrum of heat radiation. When Planck heard of these results he tried to represent them by simple mathematical formulas which looked plausible from his research on the general connection between heat and radiation. One day Planck and Rubens met for tea in Planck's home and compared Rubens' latest results with a new formula suggested by Planck. The comparison showed a complete agreement. This was the discovery of Planck's law of heat radiation.

Notice the sequence - possibilities > probabilities > actuality: thewave function gives us the possibilities, for which we can calculateprobabilities. Each experiment gives us one actuality. A verylarge number of identical experiments confirms our probabilisticpredictions. Confirmations are always only statistics,

The most appropriate basis set is one in which the eigenfunction-eigenvalue pairs match up with the natural states of the measure-ment apparatus. In the case of polarizers, one basis is the two statesof horizontal and vertical polarization

All of quantum mechanics rests on the Schrōdinger equation ofmotion that deterministically describes the time evolution of theprobabilistic wave function, plus Dirac’s three basic assumptions,the principle of superposition (of wave functions), the axiom ofmeasurement (of expectation values for observables), and theprojection postulate (the “collapse” of the wave function thatintroduces indeterminism or chance during interactions)

Einstein’s objectively real view that aparticle has a position, a continuous path, and various propertiesthat are conserved as long as the particle suffers no interaction thatcould change any of those properties

Dirac’s “transformation theory” allows us to “represent” theinitial wave function (before an interaction) in terms of a “basis set”of “eigenfunctions” appropriate for the possible quantum states ofour measuring instruments that will describe the interaction

One cannot picture in detail a photon being partly in each oftwo states; still less can one see how this can be equivalent to itsbeing partly in each of two other different states or wholly in asingle state

It is impossible to predict in which of the two beams thephoton will be found

can

Dirac’s “manner of speaking” has given the false impression that asingle particle can actually be in two states at the same time. This isseriously misleading. Dirac expresses the concern that some wouldbe misled - don’t “give too much meaning to it.”

s Einstein’s “objective reality”sees it, an individual photon is always in a single quantum state!

And he nowintroduces what he calls a “manner of speaking” which is today thesource of much confusion interpreting quantum mechanics

If one did the experiment a large number of times, onewould find in a fraction cos2α of the total number of timesthat the whole of the energy is in the α-component and in afraction sin2α that the whole of the energy is in the (α + π/2)-component.

He says“This question cannot be answered without the help of anentirely new concept which is quite foreign to classical ideas...The result predicted by quantum mechanics is that sometimesone would find the whole of the energy in one componentand the other times one would find the whole in the othercomponent. One would never find part of the energy in oneand part in the other. Experiment can never reveal a fractionof a photon.” 3

How do photons in the original state change into photonsat the right-angle states

To explain his fundamental principle of superposition, Diracconsiders a photon which is plane-polarized at a certain angle αand then gets resolved into two components at right angles to oneanother.

Decoherence theorists (chapter 35) simply deny quantumjumps and even the existence of particles!

David Bohm’s “pilot-wave” theory (chapter 30) introduceshidden variables moving at speeds faster than light to restoredeterminism to quantum physics

Manyfield-theoretic physicists believe that individual quantum systemscan be in a superposition (e.g., a particle in two places at the sametime, or going through both slits, a cat “both dead and alive.”)This is the source of much of the “quantum nonsense” in today’spopular science literature

Einstein’s reference to photons passing through an obliquepolarizer is taken straight from chapter 1 of Dirac’s classic 1930text, The Principles of Quantum Mechanics. Dirac uses the passageof a photon through an oblique polarizer to explain his principleof superposition, which he says “forms the fundamental new ideaof quantum mechanics and the basis of the departure from theclassical theory.” 2

This is to remind us that Einstein had long accepted thecontroversial idea that quantum mechanics is a statistical theory,despite the claims of some of his colleagues, notably Born, thatEinstein’s criticisms of quantum mechanics were all intended torestore determinism and eliminate chance and probabilities.

What is happening here quantum mechanically? If A crossedwith B blocks all light, how can adding another polarization filteradd light?

These weights are just the probabilities (actually the complexsquare roots of the probabilities).

t follows I shall explain brie fly and in an elementary way why I considerthe methods of quantum mechanics fundamentally unsatisfactory

herefore inclined to believe that the description of quantum mechanics inthe sense of Ia has to be regarded as an incomplete and indirect description of reality,to be replaced at some later date by a more complete and direct one.

It follows that every statement about S2 which we arriveat as a result of a complete measurement of S1 has to be valid for the system S2, evenif no measurement whatsoever is carried out on S1. This would mean that allstatements which can be deduced from the settlement of ψ2 or ψ2' mustsimultaneously be valid for S2

This is, of course, impossible, if ψ2, ψ2', etc. shouldrepresent different real states of affairs for S2, that is, one comes into conflict with theIb interpretation of the ψ-function

an operation, however, can have no direct in fluence on the physical realityin a remote part R2 of space

Einstein cannot accept the fundamental fact of "entangled" systems explained to himby Schrödinger, that they cannot be separated

principle II, i.e. the independent existence of the real stateof affairs existing in two separate parts of space R1 and R2

from the point of view of quantum mechanics alone, this does not presentany difficulty. For, according to the choice of measurement to be carried out on S1, adifferent real situation is created, and the necessity of having to attach two or moredifferent ψ-functions ψ2, ψ2', ... to one and the same system S1 cannot arise.

sulting ψ2 depends on this choice, so that di fferent kinds of (statistical)predictions regarding measurements to be carried out later on S2 are obtained,according to the choice of measurement carried out on S1

This means, from the pointof view of the interpretations of Ib, that according to the choice of completemeasurement of S1 a different real situation is being created in regard to S2, which canbe described variously by ψ2, ψ2', ψ2'', etc

Any "measurement" instantaneously collapses the two-particle wave function ψ12.There is no "later" collapse when measuring the "other" system S2

-functions of the single part-systems S1 and S2 are then unknown tobegin with, that is, they do not exist at all

make the assertion that the interpretation of quantum mechanics(according to Ib) is not consistent with principle II

llowing idea characterizes the relative independence of objects far apart inspace (A and B): external influence on A has no direct influence on B; this is known asthe 'principle of contiguity', which is used consistently only in the field theory. If thisaxiom were to be completely abolished, the idea of the existence of (quasi-) enclosedsystems, and thereby the postulation of laws which can be checked empirically in theaccepted sense, would become impossible

Anessential aspect of this arrangement of things in physics is that they lay claim, at acertain time, to an existence independent of one another, provided these objects 'aresituated in different parts of space'. Unless one makes this kind of assumption aboutthe independence of the existence (the 'being-thus') of objects which are far apart fromone another in space which stems in the first place from everyday thinking - physicalthinking in the familiar sense would not be possible. It is also hard to see any way offormulating and testing the laws of physics unless one makes a clear distinction of thiskind

the concepts ofphysics relate to a real outside world, that is, ideas are established relating to thingssuch as bodies, fields, etc., which claim a 'real existence' that is independent of theperceiving subject

we assume (in the sense of interpretation Ib) that the ψ-function completely describes a real state of affairs, and that two (essentially) differentψ-functions describe two different real states of affairs, even if they could lead toidentical results when a complete measurement is made. If the results of themeasurement tally, it is put down to the influence, partly unknown, of the measurementarrangements

According to this point of view, two ψ-functions which differ in more thantrivialities always describe two different real situations

italone does justice in a natural way to the empirical state of affairs expressed inHeisenberg's principle within the framework of quantum mechanics

(b) In reality the particle has neither a definite momentum nor a definite position;the description by ψ-function is in principle a complete description. The sharply-defined position of the particle, obtained by measuring the position, cannot beinterpreted as the position of the particle prior to the measurement. The sharplocalisation which appears as a result of the measurement is brought about onlyas a result of the unavoidable (but not unimportant) operation of measurement.The result of the measurement depends not only on the real particle situationbut also on the nature of the measuring mechanism, which in principle isincompletely known. An analogous situation arises when the momentum or anyother observable relating to the particle is being measured. This is presumablythe interpretation preferred by physicists at present

According to this point of view, the ψ-function represents an incompletedescription of the real state of affairs. This point of view is not the one physicistsaccept

(a) The (free) particle really has a definite position and a definite momentum,even if they cannot both be ascertained by measurement in the same individualcase

der a free particle described at a certain time by a spatially restricted ψ-function (completely described - in the sense of quantum mechanics). According tothis, the particle possesses neither a sharply defined momentum nor a sharply definedposition. In which sense shall I imagine that this representation describes a real,individual state of affairs?

I imagine that this theory maywell become a part of a subsequent one, in the same way as geometrical optics is nowincorporated in wave optics: the inter-relationships will remain, but the foundation willbe deepened or replaced by a more comprehensive one.

one trivial model to explain the zero probability of an | V V 〉 <math display="inline"><semantics> <mrow> <mo stretchy="false">|</mo> <mrow> <mi>V</mi> <mi>V</mi> </mrow> <mo stretchy="false">〉</mo> </mrow> </semantics></math> or | H H 〉 <math display="inline"><semantics> <mrow> <mo stretchy="false">|</mo> <mrow> <mi>H</mi> <mi>H</mi> </mrow> <mo stretchy="false">〉</mo> </mrow> </semantics></math> outcome would be to suppose that classical EM waves were always sent to the two detectors such that one wave was horizontally polarized and the other was vertically polarized. However, such an ad hoc model would not explain why such a preparation would correspond to this particular state, and it would not be generalizable to other evident anti-correlations from the very same state. The above quantum example also would exhibit anti-correlations if each photon were measured in the same diagonally polarized basis, while the ad hoc classical model would not.

We believe that this step has already beentaken, but not fully acknowledged, by a substantial part of the quantum-opticscommunity. For example, three review articles (2−4) on light squeezing makeextensive use of phase-space diagrams, and one of them(3) states explicitly thatthe photon description of the light field is not helpful in the understanding ofthe phenomenon

The quantum harmonic oscillator is the quantum analog of the classical harmonic oscillator and is one of the most important model systems in quantum mechanics. This is due in partially to the fact that an arbitrary potential curve V(x)V(x)V(x) can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point. Furthermore, it is one of the few quantum-mechanical systems for which an exact, analytical solution exists. Solving other potentials typically require either approximations or numerical approaches to identify the corresponding eigenstates and eigenvalues (i.e., wavefunctions and energies).

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