Abstract
In 1929 P.A.M. Dirac, one of the fathers of quantum mechanics, wrote the following celebrated sentence, which is often quoted in discussions of the reduction of chemistry to physics
Physics is mathematical, not because we know so much about the physical world, but because we know so little: it is only its mathematical properties that we can discover. For the rest our knowledge is negative.
B. Russell.
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Notes
- 1.
Basically the solution of the Schrödinger equation for the hydrogen atom is the starting point of approximations for heavier atoms, e.g. using the Hartree-Fock or Thomas-Fermi methods.
- 2.
One can say that for Heisenberg, in the context of quantum mechanics, the classical mechanics is a Kantian “a priori”.
- 3.
Quite similar to the “metaphysical nomological pluralism” of N. Cartwright: different domains of nature are ruled by different systems of laws.
- 4.
Just for notation simplicity we write the Wigner function \(W(q,p,t)\) for one-dimensional systems:
$$\begin{aligned} W(q,p,t)={1 \over h} \int dq' \psi ^{*}(q+q',t) \psi (q-q',t) e^{i p q'/ \hbar } \end{aligned}$$where \(\psi (q,t)\) is the wave function which obeys to the Schrödinger equation. By integrating \(W(q,p,t)\) over either \(q\) or \(p\), one obtains the probability density for the other variable. Note that \(W(q,p,t)\) is not positive definite, sometimes it is called a “quasi-probability” distribution, however the expectation value of an operator \(G({\hat{q}}, {\hat{p}})\) can be written as
$$\begin{aligned} \langle \hat{G} \rangle =\int \, dq \,dp G( q, p) W(q,p,t) . \end{aligned}$$ - 5.
These are the exponential rates at which nearby trajctories separate, and at which information on the initial state is lost (which are strictly related concepts).
- 6.
For instance, if we consider the passive advection of a scalar field \(\theta (\mathbf{x}, t)\) in a given velocity field \(\mathbf{u}(\mathbf{x}, t)\)
$$\begin{aligned} \partial _t \theta +\mathbf{u}\cdot \nabla \theta = D \varDelta \theta , \end{aligned}$$it is easy to show that an initial uncertainty \(\delta \theta (\mathbf{x}, 0)\) cannot grow forever in time. On the other hand, the solution of this equation can be nontrivial (e.g. the spatial gradient of \(\theta (\mathbf{x}, t)\) can grow quickly) if the equation
$$\begin{aligned} {d\mathbf{x} \over dt}= \mathbf{u}(\mathbf{x}, t) \end{aligned}$$ - 7.
Cited by Schweber (1990).
- 8.
In a semi-serious fashion, we may claim that the symmetry breaking is somehow a solution of Buridan’s ass paradox. In a (large) magnetic system, the spins “choose” a preferred orientation, and, in an analogous way, the ass chooses one of the two haystacks, if they are large enough (formally infinite).
- 9.
For some aspects, DFT is similar to the Boltzmann equation, which describes the one-particle distribution.
- 10.
For instance, the Hohenberg-Kohn theorem states that the properties of the electronic ground state depend only on the density \(n(\mathbf{r})\) Jones and Gunnarsson (1989). But the determination of \(n(\mathbf{r})\) remains exceedingly difficult.
References
Bell, J.S.: Speakable and Unspeakable in Quantum Mechanics: Collected Papers on Quantum Philosophy. Cambridge University Press, Cambridge (2004)
Berry, M.V.: Chaos and the Semiclassical Limit of Quantum Mechanics is the Moon There When Somebody Looks? In: Russell, R.J., Clayton, P., Wegter-McNelly, K., Polkinghorne, J. (eds.) Quantum Mechanics: Scientific Perspectives on Divine Action, pp. 41. Vatican Observatory CTNS Publications, (2001)
Berry, M.V.: Classical limits. In: Gross, D., Henneaux, M., Sevrin, A. (eds.) The Theory of the Quantum World Proceedings of the 25th Solvay Conference on Physics, pp. 52 (2013)
Berry, M.V.: True Quantum Chaos? an Instructive Example. In: Abe, Y., Horiuchi, H., Matsuyanagi, K. (eds), New Trends in Nuclear Collective Dynamics, pp. 183. Springer (1992)
Berry, M.V.: Alisa Bokulich, Reexamining the quantum-classical relation: beyond reductionism and pluralism. Br. J. Philos. Sci. 61, 889 (2010)
Bohr, N.: On the Application of the Quantum Theory to Atomic Structure. Cambridge University Press, Cambridge (1924)
Bohr, N.: Atomic Theory and the Description of Nature. Cambridge University Press, Cambridge (1929)
Bokulich, A.: Reexamining the Quantum-Classical Relation. Cambridge University Press, Cambridge (2008)
Bouguerra, M.L.: Linus Pauling: Génie Rebelle et Humaniste. Belin, Paris (2002)
Car, R., Parrinello, M.: Unified approach for molecular dynamics and density-functional theory. Phys. Rev. Lett. 55(22), 2471 (1985)
Claverie, P., Jona-Lasinio, G.: Instability of tunneling and the concept of molecular structure in quantum mechanics. Phy. Rev. A 33(4), 2245 (1986)
Coste, J., Hénon, M.: Invariant cycles in the random mapping of \(n\) integers onto themselves. Comparison with Kauffman binary network. In Bienestock, et al., (eds.), Disordered Systems and Biological Organization, pp. 361. Springer, Berlin, (1986)
Coulson, C.A.: Valence. Oxford University Press, Oxford (1960)
Crisanti, A., Falcioni, M., Mantica, G., Vulpiani, A.: Applying algorithmic complexity to define chaos in the motion of complex systems. Phys. Rev. E 50(3), 1959 (1994)
Dirac, P.A.M.: Oral interviews with P.A.M. Dirac by Thomas S. Kuhn. In Archive for the History of Quantum Physics. Deposit at Harvard University, Cambridge (1962)
Dirac P.A.M.: The relation of classical to quantum mechanics. In: Proceedings of the Second Canadian mathematical Congress, Vancouver, pp. 10. University of Toronto (1951)
Dirac, P.A.M.: Quantum mechanics of many-electron systems. Proc. Roy. Soc. London. Ser. A 123(792), 714 (1929)
Dirac, P.A.M.: The Principles of Quantum Mechanics. Clarendon Press Oxford, Oxford (1958)
Ezra, G.S.: Classical-quantum correspondence and the analysis of highly excited states: Periodic orbits, rational tori, and beyond. In Hase, W.L. (ed.) Advances in Classical Trajectory Methods, vol. 3, pp. 35. JAI Press (1998)
Falcioni, M., Vulpiani, A., Mantica, G., Pigolotti, S.: Coarse-grained probabilistic automata mimicking chaotic systems. Phys. Rev. Lett. 91(4), 44101 (2003)
Ford, J., Mantica, G., Ristow, G.H.: The Arnol’d cat: failure of the correspondence principle. Phys. D: Nonlinear Phenomena 50(3), 493 (1991)
Heisenberg W.: Oral history interview of W. Heisenberg by Thomas Kuhn. In Archive for the History of Quantum Physics. Deposit at Harvard University, Cambridge, (1963)
Heisenberg W.: The notion of a Closed Theory in Modern Science. In Heisenberg, W. (ed.) Across the Frontiers, pp. 39, Harper & Row, New York (1948)
Jona-Lasinio, G.: Spontaneous symmetry breaking—variations on a theme. Prog. Theretical Phy. 124(5), 731 (2010)
Jones, R.O., Gunnarsson, O.: The density functional formalism, its applications and prospects. Rev. Mod. Phys. 61(3), 689 (1989)
Landau, L.D., Lifshitz, E.M.: Quantum Mechanics Non-Relativistic Theory. Pergamon Press Oxford, Oxford (1978)
Mantica, G.: Quantum algorithmic integrability: The metaphor of classical polygonal billiards. Phy. Rev. E 61(6), 6434 (2000)
Pauling, L.: Nature Chemical Bond. Cornell University Press, New York (1928)
Pauling, L.: The nature of the theory of resonance. In: Todd, A. (ed.) Perspectives in Organic Chemistry, p. 1. Interscience, London (1956)
Pople, J.A.: Nobel lecture: Quantum chemical models. Rev. Mod. Phys. 71(5), 1267 (1999)
Porter, M.A., Cvitanovic, P.: Ground control to Niels Bohr: Exploring outer space with atomic physics. Not. AMS 52(9), 1020 (2005)
Primas, H.: Chemistry Quantum Mechanics and Reductionism Perspectives in Theoretical Chemistry. Springer, Berlin (1981)
Scerri, E.R.: Collected Papers on Philosophy of Chemistry. World Scientific, Singapore (2008)
Schweber, S.S.: The young John Clarke Slater and the development of quantum chemistry. Hist. Stud. Phy. Biol. Sci. 20(2), 339 (1990)
Thijssen, J.M.: Computational Physics. Cambridge Univ Press, Cambridge (2007)
Woolley, R.G.: Must a molecule have a shape? J. Am. Chem. Soc. 100(4), 1073 (1978)
Woolley, R.G.: Molecular shapes and molecular structures. Chem. phy. lett. 125(2), 200 (1986)
Zurek, W.H.: Decoherence and the transition from quantum to classical. Phys. Today, 44(10), (1991)
Zurek, W.H.: Decoherence, einselection, and the quantum origins of the classical. Rev. Mod. Phy. 75(3), 715 (2003)
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Chibbaro, S., Rondoni, L., Vulpiani, A. (2014). Quantum Mechanics, Its Classical Limit and Its Relation to Chemistry. In: Reductionism, Emergence and Levels of Reality. Springer, Cham. https://doi.org/10.1007/978-3-319-06361-4_6
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