Abstract
The aim of this chapter is to introduce some basic notions of quantum mechanics (=QM). The physicist or the scholar who already knows this background may skip it. After having introduced the basic historical reasons that led to quantum theory with the final result of the Schrödinger equation, I introduce the reader to the two basic principles of this theoretical building: the superposition and the quantisation principles. This will be the appropriate context for introducing the basic physical observables, like position and momentum as well as time and energy. It will be shown that all basic transformations in QM are unitary (reversible). After an examination of the non-commutativity among quantum observables (strictly related to quantisation) with the definition of angular momentum and the consequent uncertainty relations, some further principles are presented: Pauli’s exclusion principle with the notion of spin and the two statistics (Fermi–Dirac and Bose–Einstein) ruling the two families of quantum particles: fermions and bosons, respectively; a first and rough version of the complementarity principle; finally a brief presentation of the heuristic correspondence principle follows. In the final section of the chapter, we shall deal with the formalism of the density matrix, very helpful for the notion of quantum-state space allowing the distinction between pure states and mixtures. Moreover, this formalism will be particularly relevant for distinguishing between entangled and product states and for dealing with compound systems (total and marginal states).
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Notes
- 1.
Planck (1922).
- 2.
- 3.
- 4.
Throughout this book I shall use the SI system of units.
- 5.
Planck (1906).
- 6.
The basic physical units are length L, time T, mass M, electric current I, temperature \(\Theta \), quantity of substance N, and luminous intensity J.
- 7.
Letter of Planck to R. Williams, 7 October 1931.
- 8.
Planck (1931). Dynamical variables are momentum, angular momentum, energy, while kinematic ones are space, angle, time.
- 9.
But, as we shall see in the following, he also developed a robust philosophical reflection.
- 10.
Einstein (1905).
- 11.
See Kuhn (1978, Chaps. 3–4).
- 12.
As clearly pointed out in Born (1949, p. 80). As we shall see, Planck performed an inference that is called abduction.
- 13.
See also Bohr (1949, p. 202).
- 14.
See Kuhn (1978, Chap. 5).
- 15.
Bohr (1913).
- 16.
Auletta (2000, p. 17).
- 17.
Compton (1923).
- 18.
Note that Bohr was dismissive of the concept of a light’s particle up to the Compton’s experiment (Home and Whitaker 2007, p. 25).
- 19.
- 20.
- 21.
Auletta (2000, p. 23).
- 22.
Born (1949, Chaps. 6–7). “How small a fraction of a grain of millet must I demand is put on the first square of the chess board if after doubling up at every square I end up having to payout only a pound of millet? It would be a figure of such smallness as to have no meaning as a figure for a margin of error” (Anscombe and Elizabeth 1971, pp. 95–96).
- 23.
See also Margenau (1950, Sect. 6.5).
- 24.
- 25.
As recalled in Auletta (2000, pp. 567–68).
- 26.
For a summary of the historical background of the formula see Rindler (2001, p. 114).
- 27.
The physical continuum, according to Aristotle, cannot be made up of indivisibles. An indivisible object should not have parts. Continuum is always divisible in other divisible parts. See Aristotle (1950, VI, 1, 231a18–b19). The extension is a continuous one and, therefore, it is divisible. See Aristotle (1950, III, 6, 206b1-20).
- 28.
Descartes (1641). Descartes was the father of analytical geometry succeeding in unifying mathematical analysis and geometry thanks to the introduction of the Cartesian coordinate system. However, we shall deal here mainly with his metaphysical theses. It may be also noted that equating matter and extension deprives the former of its main causal resources of which we shall have to say more in the following.
- 29.
Leibniz (1712–14).
- 30.
Among the first interpretations of light, a considerable stress was made on its geometric properties of e.g. reflection (Newton 1704).
- 31.
Compton (1923).
- 32.
de Broglie (1924).
- 33.
Davisson and Germer (1927). Note that de Broglie did not agree with their statistical interpretation of the phenomenon.
- 34.
- 35.
Eddington (1929, p. 201).
- 36.
Dirac (1939). Thus, I do not follow here a strict historical reconstruction.
- 37.
- 38.
Dirac (1930, Chap. 1). For a long time, this has been a reference textbook on QM.
- 39.
On this see also Schrödinger (1926e).
- 40.
- 41.
- 42.
For a presentation of the Schrödinger equation the reader may have a look at Auletta et al. (2009, Chap. 3), Auletta and Wang (2014, Chap. 7). These two textbooks will be very often quoted since they can provide both formal understanding and further reference about several topics of the present book: the former gives an account of QM for physicists, the latter is addressed to a wider audience and therefore I suggest the inexpert reader have a look at it for a better understanding of what follows. Since they are very much quoted, the final index of authors does not report these entries.
- 43.
To the reader who needs a refresh in derivation and integration I strongly recommend (Auletta and Wang 2014, Sects. 6.2 and 6.4–6.5).
- 44.
- 45.
- 46.
Lagrange (1788–89).
- 47.
For this examination I consider still the classical text (Landau and Lifshitz 1976) as fundamental.
- 48.
The reason is that the Hamiltonian selects energy as a privileged quantity, but since energy is conjugate with time, it would also select a special time coordinate, which is in contrast with Einstein’s relativity (Kastner 2013, p. 51).
- 49.
Landau and Lifshitz (1976, Sects. 40 and 43).
- 50.
- 51.
See Dirac (1930, p. 9).
- 52.
- 53.
As e.g. formulated in Baumgarten (1739) , Kant (1763). Peirce understood that modern philosophy is characterised by this assumption (Peirce 1903, p. 180) and is likely to be the first philosopher to have considered that individual beings are not perfectly determined (Peirce 1877). He also acknowledged that the omnimoda determinatio is a necessary condition of classical determinism (Peirce 1892, p. 116). On this see also Auletta (2004).
- 54.
Schlosshauer (2007, pp. 116–17).
- 55.
- 56.
Auletta (2000, pp. 104–105).
- 57.
I follow here Auletta and Wang (2014, Chap. 3).
- 58.
In the following, when there is no lack of generality, I shall introduce the formalism for the 2D Hilbert space. Sometimes, I shall explicitly show the generalisation to the nD case.
- 59.
Byron and Fuller (1969–70, I, Sect. 4.2).
- 60.
- 61.
For what follows see also Jammer (1966, Chap. 5).
- 62.
Heisenberg (1925).
- 63.
For a technical account of this problem see Auletta et al. (2009, Sect. 1.3.2).
- 64.
- 65.
I follow Auletta and Wang (2014, Sect. 3.7).
- 66.
- 67.
Prugovec̆ki (1971, pp. 250–51). This is a very technical book but important for understanding the mathematics of QM.
- 68.
Byron and Fuller (1969–70, I, Sect. 4.4).
- 69.
Here I follow Auletta and Wang (2014, Sects. 4.5–4.6).
- 70.
By using unitary operators of this kind we can always diagonalize a matrix (Byron and Fuller 1969–70, I, Sect. 4.7).
- 71.
Here I follow Auletta and Wang (2014, Sect. 6.3).
- 72.
Byron and Fuller (1969–70, I, Sect. 5.1).
- 73.
Byron and Fuller (1969–70, I, Sect. 5.3).
- 74.
I follow Auletta and Wang (2014, Sect. 6.6).
- 75.
Note that Eq. (1.142) is formally identical to a classical diffusion equation.
- 76.
Byron and Fuller (1969–70, I, Sect. 1.7).
- 77.
Misner et al. (1970, pp. 53–59).
- 78.
Misner et al. (1970, p. 83).
- 79.
Byron and Fuller (1969–70, I, Sect. 5.7).
- 80.
Auletta et al. (2009, Sect. 2.2.2).
- 81.
On this subject see Auletta et al. (2009, Sect. 3.1.3).
- 82.
- 83.
In fact, the commutation relation can be considered as a particular case of Leibniz’s rule (Penrose 2004, pp. 494–95).
- 84.
Here I follow Auletta and Wang (2014, Sect. 6.8).
- 85.
Byron and Fuller (1969–70, I, Sect. 4.7).
- 86.
- 87.
- 88.
Byron and Fuller (1969–70, I, p. 17).
- 89.
This result can also be written in terms of determinants Byron and Fuller (1969–70, I, Sect. 3.8).
- 90.
Byron and Fuller (1969–70, I, Sect. 1.4).
- 91.
I follow for the Euler angles the \(x-y-z\)-convention.
- 92.
On these subjects see Auletta et al. (2009, Sects. 3.9, 6.5).
- 93.
Bohr (1913).
- 94.
The interested reader may have a look at Auletta et al. (2009, Chap. 6).
- 95.
See Auletta and Wang (2014, Sect. 8.4) for details about the transformation from Cartesian into spherical coordinates and vice versa.
- 96.
Byron and Fuller (1969–70, I, Sects. 1.8 and 5.5).
- 97.
Byron and Fuller (1969–70, I, Sect. 5.8).
- 98.
Auletta (2000, pp. 32, 35, 127, 216–17, 291–94).
- 99.
- 100.
Robertson (1929).
- 101.
Heisenberg (1927).
- 102.
See Auletta et al. (2009, Sects. 3.8, 4.4, 6.5, 13.3).
- 103.
- 104.
Auletta (2000, pp. 124–27).
- 105.
- 106.
See Auletta (2000, pp. 107–115, 129–31) and quoted literature.
- 107.
Renninger (1960).
- 108.
The reader interested in this kind of problems may have a look at Auletta (2004).
- 109.
- 110.
On unitary operators see Auletta et al. (2009, Chaps. 3 and 8).
- 111.
See Landsman (2017, Sect. 5.12).
- 112.
Also J. von Neumann contributed to prove this theorem.
- 113.
- 114.
Fuchs and Schweigert (1997, p. 11).
- 115.
Penrose (2004, Sect. 13.5).
- 116.
On what follows see Auletta et al. (2009, Sects. 3.5.3 and 8.3).
- 117.
Wigner (1959).
- 118.
On this subject see Auletta et al. (2009, Sect. 3.1.4).
- 119.
On the subject see Auletta et al. (2009, Sect. 8.1).
- 120.
- 121.
I follow here Auletta and Wang (2014, Sect. 8.6), which gives a quite comprehensive account of this subject.
- 122.
Pauli (1927).
- 123.
Auletta et al. (2009, Sect. 6.4.4).
- 124.
More precisely, we should consider the common eigenstates \(\left| \left. j_1,j_2,j,m\right\rangle \right. \) of \(\hat{\mathbf {J}}^2_1\), \(\hat{\mathbf {J}}^2_2\), \(\hat{\mathbf {J}}^2\), and \(\hat{J}_z\). This however does not change our discussion and conclusions. Therefore, \(\left| \left. j,m\right\rangle \right. \) can be thought of as a shorthand for \(\left| \left. j_1,j_2,j,m\right\rangle \right. \).
- 125.
On this subject see Auletta et al. (2009, Chap. 7).
- 126.
The reader interested to know more can have a look at Auletta et al. (2009, Sect. 7.2) in particular.
- 127.
Penrose (2004, Sect. 5.5).
- 128.
‘T Hooft (2016, p. 20).
- 129.
Reported in Wilczek (2006). An accessible book (not a textbook!) to many problems of fundamental physics.
- 130.
Pauli (1925).
- 131.
Planck (1906).
- 132.
- 133.
A summary in Jammer (1966, Sect. 3.2). See also Auletta (2000, Sect. 3.2). In Bokulich (2008, Sect. 4.2) it is argued that Bohr’s formulation of the principle was slightly different and concerned the correspondence between transition probabilities between stationary states like for different atomic levels, on the one hand, and amplitudes of the components of a classical harmonic oscillator. This should correspond to the early formulation of the principle but it is rather limited in scope.
- 134.
Dirac (1933, p. 111).
- 135.
See Kuhn (1978, p. 118).
- 136.
For all this issue, I recommend (Bokulich 2008, Sect. 1.4). Bokulich points out that also the Ehrenfest theorem (allowing to derive quantum analogues of the Hamilton’s equations (1.45): on this see Auletta et al. (2009, Sect. 3.7) cannot be taken to adequately characterise the classical limit. In fact, the related equation only holds under highly restricted circumstances due to the fact that quantum-mechanical expectation values do not follow classical equations of motion. Thus, in that limit, one must be able to replace the mean values of functions (say, of position) with a function of the means (of position). However, “this substitution is legitimate only if the Hamiltonian of the system is a polynomial of second degree or less, such as in the case of linear or harmonic oscillator potentials.”
- 137.
Home and Whitaker (2007, p. 54).
- 138.
It was made known to a large scientific audience in the historical paper (Bohr 1928). For historical reconstruction see Jammer (1966, Sect. 7.2). See also Auletta (2000, Chap. 8) for further examination and references. In fact, there is a huge amount of different interpretations of the meaning of this principle.
- 139.
A point of view still supported in Englert et al. (1994).
- 140.
‘T Hooft (2016, p. 30).
- 141.
Englert et al. (1994).
- 142.
- 143.
Byron and Fuller (1969–70, I, p. 123).
- 144.
See also Landsman (2017, Sects. 2.2–2.3).
- 145.
For the mathematical aspects see Byron and Fuller (1969–70, I, Sect. 3.11).
- 146.
- 147.
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Auletta, G. (2019). Summary of the Basic Elements of the Theory. In: The Quantum Mechanics Conundrum. Springer, Cham. https://doi.org/10.1007/978-3-030-16649-6_1
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