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APPENDIX v Examination of an Objection. The Two-Slit Experiment (cf. section 76)* 1 The imaginary experiment described below under (a) is intended to refute my assertion that arbitrarily exact simultaneous (non-predictive) measurements of the position and momentum of a particle are compatible with the quantum theory. (a) Let A be a radiating atom, and let light from it fall on a screen S after passing through two slits, Sl 1 and Sl 2. According to Heisenberg we can in this case measure exactly either the position of A or the momentum of the radiation (but not both). If we measure the position exactly (an operation that ‘blurs’ or ‘smears’ the momentum) then we can assume that light is emitted from A in spherical waves. But if we measure the momentum exactly, for example by measuring the recoils due to the emission of photons (thereby ‘blurring’ or ‘smearing’ the * 1 See also appendix *xi and my Postscript, chapter *v, section *110. My present view is that the two-slit experiment is to be treated differently, but that the interpretation proposed in this appendix is still of some interest. The remarks under (e) seem to me still to contain a valid criticism of the attempt to explain the dualism of particle and wave in terms of ‘complementarity’—an attempt which, it seems, has been abandoned by some physicists more recently.
298 appendices position), then we are able to calculate the exact direction and the momentum of the emitted photons. In this case we shall have to regard the radiation as corpuscular (‘needle-radiation’). Thus to the two measuring operations there correspond two different kinds of radiation, so that we obtain two different experimental results. For if we measure the position exactly we obtain an interference-pattern on the screen: a point-like source of light—and one whose position can be exactly measured is point-like—emits coherent light. If on the other hand we measure the momentum exactly, we get no interference pattern. (Flashes of light, or scintillations, without interference pattern, appear on the screen after the photons have passed through the slits, consonantly with the fact that the position is ‘blurred’ or ‘smeared’ and that a non-point-like source of light does not emit coherent light.) If we were to suppose that we could measure both the position and the momentum exactly, then the atom would have to emit, on the one hand, according to the wave theory, continuous spherical waves that would produce interference patterns; and it would have to emit, on the other hand, an incoherent corpuscular beam of photons. (If we were able to calculate the path of each photon we should never get anything like ‘interference’, in view of the fact that photons neither destroy one another nor otherwise interact.) The assumption of exact measurements of position and momentum taken simultaneously leads thus to two mutually contradictory predictions. For on the one hand it leads to the prediction that interference patterns will appear, and on the other hand to the prediction that no interference patterns will appear. (b) I shall now re-interpret this imaginary experiment statistically. I shall deal first with the attempt to measure position exactly. I replace the single radiating atom by a cluster of atoms in such a way that coherent light is emitted by them, and propagated in the form of spherical waves. This result is obtained by the use of a second screen pierced by a very small aperture A so placed between the atomcluster and the first screen that the aperture A is in exactly the place previously occupied by the single radiating atom A. The atomcluster emits light which, after undergoing selection according to a given position by passing through the aperture A, spreads in the form of continuous spherical waves. Thus we replace the single atom, whose position is exactly determined, by a statistical case of pure positional selection.
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The Logic of Scientific Discovery
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Logik der Forschung first published
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CONTENTS Translators’ Note xii Pr
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contents ix 7 37 Logical Ranges. No
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iii Derivation of the First Form of
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translators’ note xiii In these n
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PREFACE TO THE FIRST EDITION, 1934
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There is nothing more necessary to
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preface, 1959 xix Language analysts
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xxii preface, 1959 perception or kn
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preface, 1959 xxiii essence of phil
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preface, 1959 xxv elaborate and fam
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ACKNOWLEDGMENTS, 1960 and 1968 I wi
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1 A SURVEY OF SOME FUNDAMENTAL PROB
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a survey of some fundamental proble
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trivial; for metaphysics has usuall
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non-contradictory, a possible world
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2 ON THE PROBLEM OF A THEORY OF SCI
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on the problem of a theory of scien
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Part II Some Structural Components
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38 some structural components of a
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5 THE PROBLEM OF THE EMPIRICAL BASI
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Page 275 and 276: 246 some structural components of a
Page 277 and 278: 10 CORROBORATION, OR HOW A THEORY S
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Page 315 and 316: APPENDIX ii The General Calculus of
Page 317 and 318: 288 appendices —we obtain from (2
Page 319 and 320: APPENDIX iii Derivation of the Firs
Page 321 and 322: 292 appendices (6,1) αF″(σ´ m.
Page 323 and 324: 294 appendices after effect) in the
Page 325: 296 appendices (b) An analogous met
Page 329 and 330: 300 appendices incoherent photons.
Page 331 and 332: 302 appendices interposition of a f
Page 333 and 334: 304 appendices wave-packets (obtain
Page 335 and 336: 306 appendices slit. This angle φ
Page 337 and 338: 308 appendices latter, if we use hi
Page 339 and 340: 310 new appendices The first two of
Page 341 and 342: APPENDIX *i Two Notes on Induction
Page 343 and 344: 314 new appendices statements’;*
Page 345 and 346: 316 new appendices be the source fr
Page 347 and 348: 318 new appendices with the help of
Page 349 and 350: 320 new appendices logical interpre
Page 351 and 352: 322 new appendices It is desirable
Page 353 and 354: 324 new appendices operations—con
Page 355 and 356: 326 new appendices (3) It will be a
Page 357 and 358: 328 new appendices (7) The derivati
Page 359 and 360: 330 new appendices There are three
Page 361 and 362: 332 new appendices that is to say,
Page 363 and 364: 334 new appendices invalid, and sho
Page 365 and 366: 336 new appendices simplified syste
Page 367 and 368: 338 new appendices The following co
Page 369 and 370: 340 new appendices able to show tha
Page 371 and 372: 342 new appendices obtain p(a, c) =
Page 373 and 374: 344 new appendices The second examp
Page 375 and 376: 346 new appendices first consistenc
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348 new appendices S′ ={0, 1, 2,
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350 new appendices B1 is violated b
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352 new appendices always fill the
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354 new appendices It should be men
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APPENDIX *v Derivations in the Form
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358 new appendices (26) (Eb) (Ea) p
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360 new appendices (Applying B2 twi
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362 new appendices form: it is unco
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364 new appendices In order to prov
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366 new appendices which is neverth
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368 new appendices the other axioms
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370 new appendices between rain and
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372 new appendices in the widest po
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APPENDIX *vii Zero Probability and
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376 new appendices favourable possi
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378 new appendices (4) p(a) = lim p
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380 new appendices The same point m
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382 new appendices inferences from
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384 new appendices the required a a
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386 new appendices ‘universe’
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388 new appendices All this does no
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390 new appendices The fine-structu
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APPENDIX *viii Content, Simplicity,
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394 new appendices each entry repre
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396 new appendices But formula (*)
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398 new appendices (−) d(a 1)p(a
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400 new appendices thus becomes som
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APPENDIX *ix Corroboration, the Wei
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404 new appendices I will now brief
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406 new appendices of the brevity o
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408 new appendices who identify, ex
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410 new appendices Kemeny.) Therefo
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412 new appendices DEGREE OF CONFIR
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414 new appendices (4.5) C(x 1x 2 o
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416 new appendices strongly upon P(
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418 new appendices (vii) If C(x) =
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420 new appendices to be slightly a
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422 new appendices length, to take
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424 new appendices A THIRD NOTE ON
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426 new appendices (3) P(a) = P(a,
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428 new appendices behaviour of E a
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430 new appendices either if δ is
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432 new appendices statistical inte
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434 new appendices them, there is a
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436 new appendices statements e, f,
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438 new appendices simply deceive o
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APPENDIX *x Universals, Disposition
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442 new appendices One can easily s
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444 new appendices statement such a
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446 new appendices more abstract. T
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448 new appendices natural laws see
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450 new appendices indirect proof.
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452 new appendices physical theory
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454 new appendices is deducible fro
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456 new appendices A little reflect
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458 new appendices differ (if at al
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460 new appendices the world. And a
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462 new appendices the phenomenalis
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APPENDIX *xi On the Use and Misuse
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466 new appendices a gravitational
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468 new appendices with, its co-ord
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470 new appendices based upon his f
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472 new appendices moving freely be
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474 new appendices of insuperable b
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476 new appendices Einstein.) The m
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478 new appendices photographic pla
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480 new appendices My impression is
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482 new appendices letter were to b
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484 new appendices statistical desc
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486 new appendices
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488 new appendices
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490 name index Carnap, R. 7n, 13n,
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492 name index Mises, R. 133, 135n,
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SUBJECT INDEX Italicized page numbe
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496 subject index Binomial formula
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498 subject index Cosmology, its pr
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500 subject index Empirical basis s
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502 subject index and ad hoc hypoth
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504 subject index 150&nt, see also
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506 subject index Modus Ponens 71n,
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508 subject index 319-20; indutive
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510 subject index 142&n, 154n, 174&
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512 subject index experiment 474-5,
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