
In one of them, a mathematical entity called the wave function provides information, in the form of probability amplitudes, about what measurements of a particle’s energy,
momentum, and other physical properties may yield. 
[3] Quantum mechanics differs from classical physics in that energy, momentum, angular momentum, and other quantities of a bound system are restricted to discrete values (quantization),
objects have characteristics of both particles and waves (wave–particle duality), and there are limits to how accurately the value of a physical quantity can be predicted prior to its measurement, given a complete set of initial conditions
(the uncertainty principle). 
One method, called perturbation theory, uses the analytic result for a simple quantum mechanical model to create a result for a related but more complicated model by (for
example) the addition of a weak potential energy. 
[13][14] It is not possible to present these concepts in more than a superficial way without introducing the actual mathematics involved; understanding quantum mechanics requires
not only manipulating complex numbers, but also linear algebra, differential equations, group theory, and other more advanced subjects. 
The constant is introduced so that the Hamiltonian is reduced to the classical Hamiltonian in cases where the quantum system can be approximated by a classical system; the
ability to make such an approximation in certain limits is called the correspondence principle. 
[30] Applications Quantum mechanics has had enormous success in explaining many of the features of our universe, with regards to smallscale and discrete quantities and interactions
which cannot be explained by classical methods. 
[10] When quantum systems interact, the result can be the creation of quantum entanglement: their properties become so intertwined that a description of the whole solely in
terms of the individual parts is no longer possible. 
Another method is called “semiclassical equation of motion”, which applies to systems for which quantum mechanics produces only small deviations from classical behavior.

The most famous form of this uncertainty principle says that no matter how a quantum particle is prepared or how carefully experiments upon it are arranged, it is impossible
to have a precise prediction for a measurement of its position and also at the same time for a measurement of its momentum. 
In its most familiar form, this states that no preparation of a quantum particle can imply simultaneously precise predictions both for a measurement of its position and for
a measurement of its momentum. 
[19]: 77–78 For the onedimensional case in the direction, the timeindependent Schrödinger equation may be written With the differential operator defined by the previous
equation is evocative of the classic kinetic energy analogue, with state in this case having energy coincident with the kinetic energy of the particle. 
A simpler approach, one that has been used since the inception of quantum mechanics, is to treat charged particles as quantum mechanical objects being acted on by a classical
electromagnetic field. 
[12] Another possibility opened by entanglement is testing for “hidden variables”, hypothetical properties more fundamental than the quantities addressed in quantum theory
itself, knowledge of which would allow more exact predictions than quantum theory can provide. 
A fully relativistic quantum theory required the development of quantum field theory, which applies quantization to a field (rather than a fixed set of particles).

An important guide for making these choices is the correspondence principle, a heuristic which states that the predictions of quantum mechanics reduce to those of classical
mechanics in the regime of large quantum numbers. 
The simplest example of quantum system with a position degree of freedom is a free particle in a single spatial dimension.

Another counterintuitive phenomenon predicted by quantum mechanics is quantum tunnelling: a particle that goes up against a potential barrier can cross it, even if its kinetic
energy is smaller than the maximum of the potential. 
A free particle is one which is not subject to external influences, so that its Hamiltonian consists only of its kinetic energy: The general solution of the Schrödinger equation
is given by which is a superposition of all possible plane waves , which are eigenstates of the momentum operator with momentum . 
As in the classical case, the potential for the quantum harmonic oscillator is given by This problem can either be treated by directly solving the Schrödinger equation, which
is not trivial, or by using the more elegant “ladder method” first proposed by Paul Dirac. 
[note 5] Many macroscopic properties of a classical system are a direct consequence of the quantum behavior of its parts.

The commutator of these two operators is and this provides the lower bound on the product of standard deviations: Another consequence of the canonical commutation relation
is that the position and momentum operators are Fourier transforms of each other, so that a description of an object according to its momentum is the Fourier transform of its description according to its position. 
For example, a quantum particle like an electron can be described by a wave function, which associates to each point in space a probability amplitude.

Newer interpretations of quantum mechanics have been formulated that do away with the concept of “wave function collapse” (see, for example, the manyworlds interpretation).

[26] An alternative formulation of quantum mechanics is Feynman’s path integral formulation, in which a quantummechanical amplitude is considered as a sum over all possible
classical and nonclassical paths between the initial and final states. 
A collection of results, most significantly Bell’s theorem, have demonstrated that broad classes of such hiddenvariable theories are in fact incompatible with quantum physics.

This time evolution is deterministic in the sense that – given an initial quantum state – it makes a definite prediction of what the quantum state will be at any later time.

The basic idea is that when a quantum system interacts with a measuring apparatus, their respective wave functions become entangled so that the original quantum system ceases
to exist as an independent entity. 
The angular momentum and energy are quantized and take only discrete values like those shown (as is the case for resonant frequencies in acoustics) Some wave functions produce
probability distributions that are independent of time, such as eigenstates of the Hamiltonian. 
Another consequence of the mathematical rules of quantum mechanics is the phenomenon of quantum interference, which is often illustrated with the doubleslit experiment.

[35][36] The full apparatus of quantum field theory is often unnecessary for describing electrodynamic systems.

This vector is postulated to be normalized under the Hilbert space inner product, that is, it obeys , and it is welldefined up to a complex number of modulus 1 (the global
phase), that is, and represent the same physical system. 
For example, the stability of bulk matter (consisting of atoms and molecules which would quickly collapse under electric forces alone), the rigidity of solids, and the mechanical,
thermal, chemical, optical and magnetic properties of matter are all results of the interaction of electric charges under the rules of quantum mechanics. 
Examples Free particle Main article: Free particle Position space probability density of a Gaussian wave packet moving in one dimension in free space.

According to Bell’s theorem, if nature actually operates in accord with any theory of local hidden variables, then the results of a Bell test will be constrained in a particular,
quantifiable way. 
One can instead define reduced density matrices that describe the statistics that can be obtained by making measurements on either component system alone.

Instead, the wave function must satisfy more complicated mathematical boundary conditions as it is nonzero in regions outside the well.

The finite potential well problem is mathematically more complicated than the infinite particleinabox problem as the wave function is not pinned to zero at the walls of
the well. 
Quantum decoherence is a mechanism through which quantum systems lose coherence, and thus become incapable of displaying many typically quantum effects: quantum superpositions
become simply probabilistic mixtures, and quantum entanglement becomes simply classical correlations. 
Applying the Born rule to these amplitudes gives a probability density function for the position that the electron will be found to have when an experiment is performed to
measure it. 
This is one of the most difficult aspects of quantum systems to understand.

Complications arise with chaotic systems, which do not have good quantum numbers, and quantum chaos studies the relationship between classical and quantum descriptions in
these systems. 
[33] One can also start from an established classical model of a particular system, and then try to guess the underlying quantum model that would give rise to the classical
model in the correspondence limit. 
[19] Composite systems and entanglement When two different quantum systems are considered together, the Hilbert space of the combined system is the tensor product of the Hilbert
spaces of the two components. 
The Hilbert space of the composite system is then If the state for the first system is the vector and the state for the second system is , then the state of the composite
system is Not all states in the joint Hilbert space can be written in this form, however, because the superposition principle implies that linear combinations of these “separable” or “product states” are also valid. 
[27] Particle in a box 1dimensional potential energy box (or infinite potential well) Main article: Particle in a box The particle in a onedimensional potential energy box
is the most mathematically simple example where restraints lead to the quantization of energy levels. 
Quantum electrodynamics is, along with general relativity, one of the most accurate physical theories ever devised.

This can explain why, in practice, quantum effects are difficult to observe in systems larger than microscopic.

This “semiclassical” approach fails if quantum fluctuations in the electromagnetic field play an important role, such as in the emission of photons by charged particles.

Therefore, since , must be an integer multiple of , This constraint on implies a constraint on the energy levels, yielding A finite potential well is the generalization of
the infinite potential well problem to potential wells having finite depth. 
Another related problem is that of the rectangular potential barrier, which furnishes a model for the quantum tunneling effect that plays an important role in the performance
of modern technologies such as flash memory and scanning tunneling microscopy. 

The fact that dependence in momentum is the Fourier transform of the dependence in position means that the momentum operator is equivalent (up to an factor) to taking the
derivative according to the position, since in Fourier analysis differentiation corresponds to multiplication in the dual space. 
This is why in quantum equations in position space, the momentum is replaced by , and in particular in the nonrelativistic Schrödinger equation in position space the momentumsquared
term is replaced with a Laplacian times . 
Quantum mechanics arose gradually from theories to explain observations which could not be reconciled with classical physics, such as Max Planck’s solution in 1900 to the
blackbody radiation problem, and the correspondence between energy and frequency in Albert Einstein’s 1905 paper which explained the photoelectric effect. 
It has been demonstrated to hold for complex molecules with thousands of atoms,[4] but its application to human beings raises philosophical problems, such as Wigner’s friend,
and its application to the universe as a whole remains speculative. 
The exact nature of this Hilbert space is dependent on the system – for example, for describing position and momentum the Hilbert space is the space of complex squareintegrable
functions , while the Hilbert space for the spin of a single proton is simply the space of twodimensional complex vectors with the usual inner product. 
These can be chosen appropriately in order to obtain a quantitative description of a quantum system, a necessary step in making physical predictions.

[6] However, the light is always found to be absorbed at the screen at discrete points, as individual particles rather than waves; the interference pattern appears via the
varying density of these particle hits on the screen. 
This is the best the theory can do; it cannot say for certain where the electron will be found.

Defining the uncertainty for an observable by a standard deviation, we have and likewise for the momentum: The uncertainty principle states that Either standard deviation
can in principle be made arbitrarily small, but not both simultaneously. 
Relation to other scientific theories Classical mechanics The rules of quantum mechanics assert that the state space of a system is a Hilbert space and that observables of
the system are Hermitian operators acting on vectors in that space – although they do not tell us which Hilbert space or which operators. 
In the continuous case, these formulas give instead the probability density.

Classical physics, the collection of theories that existed before the advent of quantum mechanics, describes many aspects of nature at an ordinary (macroscopic) scale, but
is not sufficient for describing them at small (atomic and subatomic) scales. 
This approach is particularly important in the field of quantum chaos.

As we let the Gaussian wave packet evolve in time, we see that its center moves through space at a constant velocity (like a classical particle with no forces acting on it).

Many systems that are treated dynamically in classical mechanics are described by such “static” wave functions.

The first complete quantum field theory, quantum electrodynamics, provides a fully quantum description of the electromagnetic interaction.

[22][24] As described above, entanglement is a key feature of models of measurement processes in which an apparatus becomes entangled with the system being measured.

Such wave functions are directly comparable to Chladni’s figures of acoustic modes of vibration in classical physics and are modes of oscillation as well, possessing a sharp
energy and thus, a definite frequency.
Works Cited
[‘1. See, for example, Precision tests of QED. The relativistic refinement of quantum mechanics known as quantum electrodynamics (QED) has been shown to agree with experiment to within 1 part in 108 for some atomic properties.
2. ^ Physicist John
C. Baez cautions, “there’s no way to understand the interpretation of quantum mechanics without also being able to solve quantum mechanics problems – to understand the theory, you need to be able to use it (and vice versa)”.[15] Carl Sagan outlined
the “mathematical underpinning” of quantum mechanics and wrote, “For most physics students, this might occupy them from, say, third grade to early graduate school – roughly 15 years. […] The job of the popularizer of science, trying to get across
some idea of quantum mechanics to a general audience that has not gone through these initiation rites, is daunting. Indeed, there are no successful popularizations of quantum mechanics in my opinion – partly for this reason.”[16]
3. ^ A momentum
eigenstate would be a perfectly monochromatic wave of infinite extent, which is not squareintegrable. Likewise, a position eigenstate would be a Dirac delta distribution, not squareintegrable and technically not a function at all. Consequently,
neither can belong to the particle’s Hilbert space. Physicists sometimes introduce fictitious “bases” for a Hilbert space comprising elements outside that space. These are invented for calculational convenience and do not represent physical states.[19]: 100–105
4. ^
See, for example, the Feynman Lectures on Physics for some of the technological applications which use quantum mechanics, e.g., transistors (vol III, pp. 14–11 ff), integrated circuits, which are followon technology in solidstate physics (vol II,
pp. 8–6), and lasers (vol III, pp. 9–13).
5. ^ see macroscopic quantum phenomena, Bose–Einstein condensate, and Quantum machine
6. ^ The published form of the EPR argument was due to Podolsky, and Einstein himself was not satisfied with it. In
his own publications and correspondence, Einstein used a different argument to insist that quantum mechanics is an incomplete theory.[46][47][48][49]
7. Born, M. (1926). “Zur Quantenmechanik der Stoßvorgänge” [On the Quantum Mechanics of Collision
Processes]. Zeitschrift für Physik. 37 (12): 863–867. Bibcode:1926ZPhy…37..863B. doi:10.1007/BF01397477. S2CID 119896026.
8. ^ Jump up to:a b c Feynman, Richard; Leighton, Robert; Sands, Matthew (1964). The Feynman Lectures on Physics. Vol. 3.
California Institute of Technology. ISBN 9780201500646. Retrieved 19 December 2020.
9. ^ Jaeger, Gregg (September 2014). “What in the (quantum) world is macroscopic?”. American Journal of Physics. 82 (9): 896–905. Bibcode:2014AmJPh..82..896J. doi:10.1119/1.4878358.
10. ^
Yaakov Y. Fein; Philipp Geyer; Patrick Zwick; Filip Kiałka; Sebastian Pedalino; Marcel Mayor; Stefan Gerlich; Markus Arndt (September 2019). “Quantum superposition of molecules beyond 25 kDa”. Nature Physics. 15 (12): 1242–1245. Bibcode:2019NatPh..15.1242F.
doi:10.1038/s4156701906639. S2CID 203638258.
11. ^ Bojowald, Martin (2015). “Quantum cosmology: a review”. Reports on Progress in Physics. 78 (2): 023901. arXiv:1501.04899. Bibcode:2015RPPh…78b3901B. doi:10.1088/00344885/78/2/023901. PMID
25582917. S2CID 18463042.
12. ^ Jump up to:a b c Lederman, Leon M.; Hill, Christopher T. (2011). Quantum Physics for Poets. US: Prometheus Books. ISBN 9781616142810.
13. ^ MüllerKirsten, H. J. W. (2006). Introduction to Quantum Mechanics: Schrödinger
Equation and Path Integral. US: World Scientific. p. 14. ISBN 9789812566911.
14. ^ Plotnitsky, Arkady (2012). Niels Bohr and Complementarity: An Introduction. US: Springer. pp. 75–76. ISBN 9781461445173.
15. ^ Griffiths, David J. (1995). Introduction
to Quantum Mechanics. Prentice Hall. ISBN 0131244051.
16. ^ Trixler, F. (2013). “Quantum tunnelling to the origin and evolution of life”. Current Organic Chemistry. 17 (16): 1758–1770. doi:10.2174/13852728113179990083. PMC 3768233. PMID 24039543.
17. ^
Bub, Jeffrey (2019). “Quantum entanglement”. In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University.
18. ^ Jump up to:a b Caves, Carlton M. (2015). “Quantum Information Science: Emerging No
More”. In Kelley, Paul; Agrawal, Govind; Bass, Mike; Hecht, Jeff; Stroud, Carlos (eds.). OSA Century of Optics. The Optical Society. pp. 320–323. arXiv:1302.1864. Bibcode:2013arXiv1302.1864C. ISBN 9781943580040.
19. ^ Jump up to:a b Wiseman,
Howard (October 2015). “Death by experiment for local realism”. Nature. 526 (7575): 649–650. doi:10.1038/nature15631. ISSN 00280836. PMID 26503054.
20. ^ Jump up to:a b Wolchover, Natalie (7 February 2017). “Experiment Reaffirms Quantum Weirdness”.
Quanta Magazine. Retrieved 8 February 2020.
21. ^ Baez, John C. (20 March 2020). “How to Learn Math and Physics”. University of California, Riverside. Retrieved 19 December 2020.
22. ^ Sagan, Carl (1996). The DemonHaunted World: Science as a
Candle in the Dark. Ballantine Books. p. 249. ISBN 0345409469.
23. ^ Greenstein, George; Zajonc, Arthur (2006). The Quantum Challenge: Modern Research on the Foundations of Quantum Mechanics (2nd ed.). Jones and Bartlett Publishers, Inc. p. 215.
ISBN 9780763724702., Chapter 8, p. 215
24. ^ Weinberg, Steven (2010). Dreams Of A Final Theory: The Search for The Fundamental Laws of Nature. Random House. p. 82. ISBN 9781407063966.
25. ^ Jump up to:a b c d CohenTannoudji, Claude; Diu,
Bernard; Laloë, Franck (2005). Quantum Mechanics. Translated by Hemley, Susan Reid; Ostrowsky, Nicole; Ostrowsky, Dan. John Wiley & Sons. ISBN 047116433X.
26. ^ Landau, L.D.; Lifschitz, E.M. (1977). Quantum Mechanics: NonRelativistic Theory.
Vol. 3 (3rd ed.). Pergamon Press. ISBN 9780080209401. OCLC 2284121.
27. ^ Section 3.2 of Ballentine, Leslie E. (1970), “The Statistical Interpretation of Quantum Mechanics”, Reviews of Modern Physics, 42 (4): 358–381, Bibcode:1970RvMP…42..358B,
doi:10.1103/RevModPhys.42.358. This fact is experimentally wellknown for example in quantum optics; see e.g. chap. 2 and Fig. 2.1 Leonhardt, Ulf (1997), Measuring the Quantum State of Light, Cambridge: Cambridge University Press, ISBN 0521497302
28. ^
Jump up to:a b c Nielsen, Michael A.; Chuang, Isaac L. (2010). Quantum Computation and Quantum Information (2nd ed.). Cambridge: Cambridge University Press. ISBN 9781107002173. OCLC 844974180.
29. ^ Jump up to:a b Rieffel, Eleanor G.; Polak,
Wolfgang H. (2011). Quantum Computing: A Gentle Introduction. MIT Press. ISBN 9780262015066.
30. ^ Wilde, Mark M. (2017). Quantum Information Theory (2nd ed.). Cambridge University Press. arXiv:1106.1445. doi:10.1017/9781316809976.001. ISBN
9781107176164. OCLC 973404322. S2CID 2515538.
31. ^ Schlosshauer, Maximilian (October 2019). “Quantum decoherence”. Physics Reports. 831: 1–57. arXiv:1911.06282. Bibcode:2019PhR…831….1S. doi:10.1016/j.physrep.2019.10.001. S2CID 208006050.
32. ^
Rechenberg, Helmut (1987). “Erwin Schrödinger and the creation of wave mechanics” (PDF). Acta Physica Polonica B. 19 (8): 683–695. Retrieved 13 June 2016.
33. ^ Mathews, Piravonu Mathews; Venkatesan, K. (1976). “The Schrödinger Equation and Stationary
States”. A Textbook of Quantum Mechanics. Tata McGrawHill. p. 36. ISBN 9780070965102.
34. ^ Paris, M. G. A. (1999). “Entanglement and visibility at the output of a Mach–Zehnder interferometer”. Physical Review A. 59 (2): 1615–1621. arXiv:quantph/9811078.
Bibcode:1999PhRvA..59.1615P. doi:10.1103/PhysRevA.59.1615. S2CID 13963928.
35. ^ Haack, G. R.; Förster, H.; Büttiker, M. (2010). “Parity detection and entanglement with a MachZehnder interferometer”. Physical Review B. 82 (15): 155303. arXiv:1005.3976.
Bibcode:2010PhRvB..82o5303H. doi:10.1103/PhysRevB.82.155303. S2CID 119261326.
36. ^ Vedral, Vlatko (2006). Introduction to Quantum Information Science. Oxford University Press. ISBN 9780199215706. OCLC 442351498.
37. ^ Cohen, Marvin L. (2008).
“Essay: Fifty Years of Condensed Matter Physics”. Physical Review Letters. 101 (25): 250001. Bibcode:2008PhRvL.101y0001C. doi:10.1103/PhysRevLett.101.250001. PMID 19113681. Retrieved 31 March 2012.
38. ^ Matson, John. “What Is Quantum Mechanics
Good for?”. Scientific American. Retrieved 18 May 2016.
39. ^ Tipler, Paul; Llewellyn, Ralph (2008). Modern Physics (5th ed.). W.H. Freeman and Company. pp. 160–161. ISBN 9780716775508.
40. ^ “Atomic Properties”. Academic.brooklyn.cuny.edu.
Retrieved 18 August 2012.
41. ^ Hawking, Stephen; Penrose, Roger (2010). The Nature of Space and Time. ISBN 9781400834747.
42. ^ Tatsumi Aoyama; Masashi Hayakawa; Toichiro Kinoshita; Makiko Nio (2012). “TenthOrder QED Contribution to the Electron
g2 and an Improved Value of the Fine Structure Constant”. Physical Review Letters. 109 (11): 111807. arXiv:1205.5368. Bibcode:2012PhRvL.109k1807A. doi:10.1103/PhysRevLett.109.111807. PMID 23005618. S2CID 14712017.
43. ^ “The Nobel Prize in Physics
1979”. Nobel Foundation. Retrieved 16 December 2020.
44. ^ Becker, Katrin; Becker, Melanie; Schwarz, John (2007). String theory and Mtheory: A modern introduction. Cambridge University Press. ISBN 9780521860697.
45. ^ Zwiebach, Barton (2009).
A First Course in String Theory. Cambridge University Press. ISBN 9780521880329.
46. ^ Rovelli, Carlo; Vidotto, Francesca (13 November 2014). Covariant Loop Quantum Gravity: An Elementary Introduction to Quantum Gravity and Spinfoam Theory.
Cambridge University Press. ISBN 9781316148112.
47. ^ Feynman, Richard (1967). The Character of Physical Law. MIT Press. p. 129. ISBN 0262560038.
48. ^ Weinberg, Steven (2012). “Collapse of the state vector”. Physical Review A. 85 (6):
062116. arXiv:1109.6462. Bibcode:2012PhRvA..85f2116W. doi:10.1103/PhysRevA.85.062116. S2CID 119273840.
49. ^ Howard, Don (December 2004). “Who Invented the ‘Copenhagen Interpretation’? A Study in Mythology”. Philosophy of Science. 71 (5): 669–682.
doi:10.1086/425941. ISSN 00318248. S2CID 9454552.
50. ^ Camilleri, Kristian (May 2009). “Constructing the Myth of the Copenhagen Interpretation”. Perspectives on Science. 17 (1): 26–57. doi:10.1162/posc.2009.17.1.26. ISSN 10636145. S2CID 57559199.
51. ^
Schlosshauer, Maximilian; Kofler, Johannes; Zeilinger, Anton (1 August 2013). “A snapshot of foundational attitudes toward quantum mechanics”. Studies in History and Philosophy of Science Part B. 44 (3): 222–230. arXiv:1301.1069. Bibcode:2013SHPMP..44..222S.
doi:10.1016/j.shpsb.2013.04.004. S2CID 55537196.
52. ^ Harrigan, Nicholas; Spekkens, Robert W. (2010). “Einstein, incompleteness, and the epistemic view of quantum states”. Foundations of Physics. 40 (2): 125. arXiv:0706.2661. Bibcode:2010FoPh…40..125H.
doi:10.1007/s1070100993470. S2CID 32755624.
53. ^ Howard, D. (1985). “Einstein on locality and separability”. Studies in History and Philosophy of Science Part A. 16 (3): 171–201. Bibcode:1985SHPSA..16..171H. doi:10.1016/00393681(85)900019.
54. ^
Sauer, Tilman (1 December 2007). “An Einstein manuscript on the EPR paradox for spin observables”. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics. 38 (4): 879–887. Bibcode:2007SHPMP..38..879S.
CiteSeerX 10.1.1.571.6089. doi:10.1016/j.shpsb.2007.03.002. ISSN 13552198.
55. ^ Einstein, Albert (1949). “Autobiographical Notes”. In Schilpp, Paul Arthur (ed.). Albert Einstein: PhilosopherScientist. Open Court Publishing Company.
56. ^ Bell,
J. S. (1 November 1964). “On the Einstein Podolsky Rosen paradox”. Physics Physique Fizika. 1 (3): 195–200. doi:10.1103/PhysicsPhysiqueFizika.1.195.
57. ^ Goldstein, Sheldon (2017). “Bohmian Mechanics”. Stanford Encyclopedia of Philosophy. Metaphysics
Research Lab, Stanford University.
58. ^ Barrett, Jeffrey (2018). “Everett’s RelativeState Formulation of Quantum Mechanics”. In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University.
59. ^
Everett, Hugh; Wheeler, J. A.; DeWitt, B. S.; Cooper, L. N.; Van Vechten, D.; Graham, N. (1973). DeWitt, Bryce; Graham, R. Neill (eds.). The ManyWorlds Interpretation of Quantum Mechanics. Princeton Series in Physics. Princeton, NJ: Princeton University
Press. p. v. ISBN 069108131X.
60. ^ Wallace, David (2003). “Everettian Rationality: defending Deutsch’s approach to probability in the Everett interpretation”. Stud. Hist. Phil. Mod. Phys. 34 (3): 415–438. arXiv:quantph/0303050. Bibcode:2003SHPMP..34..415W.
doi:10.1016/S13552198(03)000364. S2CID 1921913.
61. ^ Ballentine, L. E. (1973). “Can the statistical postulate of quantum theory be derived? – A critique of the manyuniverses interpretation”. Foundations of Physics. 3 (2): 229–240. Bibcode:1973FoPh….3..229B.
doi:10.1007/BF00708440. S2CID 121747282.
62. ^ Landsman, N. P. (2008). “The Born rule and its interpretation” (PDF). In Weinert, F.; Hentschel, K.; Greenberger, D.; Falkenburg, B. (eds.). Compendium of Quantum Physics. Springer. ISBN 9783540706229.
The conclusion seems to be that no generally accepted derivation of the Born rule has been given to date, but this does not imply that such a derivation is impossible in principle.
63. ^ Kent, Adrian (2010). “One world versus many: The inadequacy
of Everettian accounts of evolution, probability, and scientific confirmation”. In S. Saunders; J. Barrett; A. Kent; D. Wallace (eds.). Many Worlds? Everett, Quantum Theory and Reality. Oxford University Press. arXiv:0905.0624. Bibcode:2009arXiv0905.0624K.
64. ^
Van Fraassen, Bas C. (April 2010). “Rovelli’s World”. Foundations of Physics. 40 (4): 390–417. Bibcode:2010FoPh…40..390V. doi:10.1007/s1070100993265. ISSN 00159018. S2CID 17217776.
65. ^ Healey, Richard (2016). “QuantumBayesian and Pragmatist
Views of Quantum Theory”. In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University.
66. ^ Born, Max; Wolf, Emil (1999). Principles of Optics. Cambridge University Press. ISBN 0521642221. OCLC
1151058062.
67. ^ Scheider, Walter (April 1986). “Bringing one of the great moments of science to the classroom”. The Physics Teacher. 24 (4): 217–219. Bibcode:1986PhTea..24..217S. doi:10.1119/1.2341987. ISSN 0031921X.
68. ^ Feynman, Richard;
Leighton, Robert; Sands, Matthew (1964). The Feynman Lectures on Physics. Vol. 1. California Institute of Technology. ISBN 9780201500646. Retrieved 30 September 2021.
69. ^ Martin, Andre (1986), “Cathode Ray Tubes for Industrial and Military Applications”,
in Hawkes, Peter (ed.), Advances in Electronics and Electron Physics, Volume 67, Academic Press, p. 183, ISBN 9780080577333, Evidence for the existence of “cathoderays” was first found by Plücker and Hittorf …
70. ^ Dahl, Per F. (1997). Flash
of the Cathode Rays: A History of J J Thomson’s Electron. CRC Press. pp. 47–57. ISBN 9780750304535.
71. ^ Mehra, J.; Rechenberg, H. (1982). The Historical Development of Quantum Theory, Vol. 1: The Quantum Theory of Planck, Einstein, Bohr and
Sommerfeld. Its Foundation and the Rise of Its Difficulties (1900–1925). New York: SpringerVerlag. ISBN 9780387906423.
72. ^ “Quantum – Definition and More from the Free MerriamWebster Dictionary”. Merriamwebster.com. Retrieved 18 August 2012.
73. ^
Kuhn, T. S. (1978). Blackbody theory and the quantum discontinuity 1894–1912. Oxford: Clarendon Press. ISBN 9780195023831.
74. ^ Kragh, Helge (1 December 2000). “Max Planck: the reluctant revolutionary”. Physics World. Retrieved 12 December 2020.
75. ^
Stachel, John (2009). “Bohr and the Photon”. Quantum Reality, Relativistic Causality and the Closing of the Epistemic Circle. The Western Ontario Series in Philosophy of Science. Vol. 73. Dordrecht: Springer. pp. 69–83. doi:10.1007/9781402091070_5.
ISBN 9781402091063.
76. ^ Einstein, A. (1905). “Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt” [On a heuristic point of view concerning the production and transformation of light]. Annalen der
Physik. 17 (6): 132–148. Bibcode:1905AnP…322..132E. doi:10.1002/andp.19053220607. Reprinted in Stachel, John, ed. (1989). The Collected Papers of Albert Einstein (in German). Vol. 2. Princeton University Press. pp. 149–166. See also “Einstein’s
early work on the quantum hypothesis”, ibid. pp. 134–148.
77. ^ Einstein, Albert (1917). “Zur Quantentheorie der Strahlung” [On the Quantum Theory of Radiation]. Physikalische Zeitschrift (in German). 18: 121–128. Bibcode:1917PhyZ…18..121E. Translated
in Einstein, A. (1967). “On the Quantum Theory of Radiation”. The Old Quantum Theory. Elsevier. pp. 167–183. doi:10.1016/b9780080121024.500188. ISBN 9780080121024.
78. ^ ter Haar, D. (1967). The Old Quantum Theory. Pergamon Press. pp. 206.
ISBN 9780080121017.
79. ^ “Semiclassical approximation”. Encyclopedia of Mathematics. Retrieved 1 February 2020.
80. ^ Sakurai, J. J.; Napolitano, J. (2014). “Quantum Dynamics”. Modern Quantum Mechanics. Pearson. ISBN 9781292024103. OCLC
929609283.
81. ^ David Edwards,”The Mathematical Foundations of Quantum Mechanics”, Synthese, Volume 42, Number 1/September, 1979, pp. 1–70.
82. ^ D. Edwards, “The Mathematical Foundations of Quantum Field Theory: Fermions, Gauge Fields, and Supersymmetry,
Part I: Lattice Field Theories”, International J. of Theor. Phys., Vol. 20, No. 7 (1981).
83. ^ Bernstein, Jeremy (November 2005). “Max Born and the quantum theory”. American Journal of Physics. 73 (11): 999–1008. Bibcode:2005AmJPh..73..999B. doi:10.1119/1.2060717.
ISSN 00029505.
84. ^ Pais, Abraham (1997). A Tale of Two Continents: A Physicist’s Life in a Turbulent World. Princeton, New Jersey: Princeton University Press. ISBN 0691012431.
85. ^ Van Hove, Leon (1958). “Von Neumann’s contributions to
quantum mechanics” (PDF). Bulletin of the American Mathematical Society. 64 (3): Part 2:95–99. doi:10.1090/s000299041958102062.
86. ^ Feynman, Richard. “The Feynman Lectures on Physics III 214”. California Institute of Technology. Retrieved
24 November 2015. …it was long believed that the wave function of the Schrödinger equation would never have a macroscopic representation analogous to the macroscopic representation of the amplitude for photons. On the other hand, it is now realized
that the phenomena of superconductivity presents us with just this situation.
87. ^ Packard, Richard (2006). “Berkeley Experiments on Superfluid Macroscopic Quantum Effects” (PDF). Archived from the original (PDF) on 25 November 2015. Retrieved
24 November 2015.
‘]