Abstract
Beppo Levi, known as a pure mathematician because of his fundamental contributions to analysis and algebraic geometry, was also deeply interested in the development of theoretical physics of his times. His contributions to quantum mechanics are here reviewed.
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Notes
- 1.
On the equations of wave mechanics. It appeared on the Memoirs of this Accademia, Serie IX, Tomo I, Sezione Fisiche e Matematiche, 1933–1934.
- 2.
- 3.
A standard reference for the origins of quantum mechanics is [7].
- 4.
- 5.
The initials stand for G. Wentzel, H.A. Kramers and L. Brillouin, who introduced in wave mechanics the well known Liouville–Green method to obtain approximate solutions of linear ODE with a small parameter.
- 6.
New theories of Quantum mechanics.
- 7.
Foundations of logic and foundations of quantum mechanics.
- 8.
Lectures on wave mechanics.
- 9.
Enrico Persico (Rome 1900–Rome 1969), Professor of Theoretical Physics in Florence (1927–1930), Turin (1930–1947), and in Rome since 1947.
- 10.
Giulio Racah (Florence 1909–Florence 1965) and Bruno Rossi (Venice 1905–Boston 1993) became world famous physicists. As well as Beppo Levi, both had to emigrate in 1938 because of the racial legislation. Racah became Professor at the Hebrew University in Jerusalem, and Rossi at the MIT in Boston.
- 11.
[..a..] fact, extraordinary at first sight, that remarkable agreements could be obtained between physical experiments and a theory closer to divination than to deduction.
- 12.
Introduce a non-formal mathematical procedure aimed at reconcile wave and corpuscolar aspects of the matter.
- 13.
Look for deterministic equation of motion for the quantum corpuscle such that it is possible to associate with them a wave propagation motion satisfying Schrödinger’s equation.
- 14.
In the present case we deal with associating to the ordinary differential equations, which characterize the study of the motion according to classical mechanics, some partial differential equations relative to the behaviour of functions smeared in space which admit the wave appearance.
- 15.
To this purpose an inversion seems to be suited of the Hugoniot and Hadamard point of view in the study of wave motion in a continuous medium.
- 16.
The notation used today is obviously
$$\frac{\mathrm{d}{q}_{i}} {\mathrm{d}t} = \frac{\partial \mathcal{H}} {\partial {p}_{i}},\qquad \frac{\mathrm{d}{p}_{i}} {\mathrm{d}t} = -\frac{\partial \mathcal{H}} {\partial {q}_{i}}.$$ - 17.
Let us introduce a new coordinate Q proportional to the action, in such a way that this differential system can be written also as.
- 18.
We have thus written the system of the characteristic curves of the first order partial differential equation in the unknown function Q.
- 19.
under the interpretation.
- 20.
Denote then by \(\mathcal{T}\) the operator obtained from T replacing p i by the operator symbol \(\frac{\partial } {\partial {q}_{i}}\), \(\Psi \) being an arbitrary first order differential operator with respect to the variables Q,t,q i .
- 21.
the function \(f(Q,t,{q}_{i})\) being also arbitrary; let finally \(\Phi \) be an unknown function of these variables. The second order partial differential equation.
- 22.
has (6) as equation of the characteristics and hence (5) as differential system of the bicharacteristics. The equations (4) are then the differential equations for the space-time projections of these bicharacteristics.Since on the other hand the equations (4) define the possible trajectories} (that is, still undetermined through the initial conditions) of the motion characterized by the Hamiltonian function H, the conclusion is that the connection between classical and wave mechanics, which as only approximate in the Schrödinger equation, with degree of approximation impossible to estimate, becomes exact by means of equation (7).
- 23.
Setting.
- 24.
This equation becomes.
- 25.
and the operator \(\frac{1} {{a}^{2}}\mathcal{T}\) is obtained from T replacing the the variables p i by the differential operators \(\frac{1} {a} \frac{\partial } {\partial {q}_{i}}\) . Taking finally.
- 26.
(10) becomes exactly Schrödinger’s equation.
- 27.
It cannot be stated that (7) is nothing else than a generalization of the Schrödinger equation from which one goes to it through a convenient choice of constants and undetermined functions (choice which could also be the one suggested by physical experience); this is because it is essential in(7), by its correlation to (4), the occurrence of the variable Q which in (10) disappears only because of the position (8), unavaoidable link between (10), (7), (4).
- 28.
The physical meaning of the functions \(\Phi \) and \(\varphi \) , as well as of f and of the linear operator \(\Psi \) , remains so far completely undecided; now this has happened still happens for the wave function} of Schrödinger; but the present discussion shows that the determination of the physical meaning of these functions and operators actually is the essential part of the problem, and it seems to me that it accounts for the fact, extraordinary at first sight, that remarkable agreements could be obtained between physical experiments and a theory closer to divination than to deduction.
References
Dirac, P.A.M. 1930. Quantum mechanics, Oxford University Press: Oxford (London/Melbourne).
Fedoryuk, M.V., and V.P. Maslov. 1981. Semiclassical approximations in quantum mechanics. D. Reidel Kluwer Publishing, Dordercht.
Jackiw, R., and D. Kleppner. 2000. One hundred years of quantum physics. Science (2 August 2000) 289: 893–924.
Kato, T. 1966–1976. Perturbation theory of linear operators. Springer: Berlin.
Reed, M., and B. Simon. 1972–1978. Methods of modern mathematical physics. Voll. I-IV, Academic Press: New York.
Von Neumann, J. 1955. Mathematical foundations of quantum mechanics. Princeton University Press: Princeton, NJ.
Van der Waerden, B.L. 1966. Sources of quantum mechanics. Dover, New York.
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Graffi, S. (2012). Beppo Levi and Quantum Mechanics. In: Coen, S. (eds) Mathematicians in Bologna 1861–1960. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0227-7_10
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