Ettore Majorana: Unpublished Research Notes on Theoretical Physics

Ettore Majorana: Unpublished Research Notes on Theoretical Physics

Erasmo Recami

2009

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AI-generated Abstract

Ettore Majorana's unpublished research notes delve into advanced topics in theoretical physics, demonstrating his original insights and methodologies in fields such as quantum mechanics and nuclear forces. The manuscript presents a series of complex equations and theoretical constructs related to electron scattering, wave functions, and Riemannian geometry, revealing Majorana's unique approach and contributions to the scientific discourse of his time.

Figures (495)

1@ This German word means “tentatively”, and refers to the successive assumptions. Note, however, that in the original paper the cited word is written as “versucherweiser”.

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

2@ Note that, more appropriately, it should be written d4r = d?V dt, since dr denotes the 4-dimensional volume element, while drmV is the 3-dimensional space volume element.

4@ Note that some things in the last three square brackets (the x,y, z-derivatives and the indices 1,2,3,4 of the w components) should be slightly corrected. However, at variance with what is usually done by us, we choose to leave unchanged the expressions appearing in the original manuscript.

The canonical quantization of a Dirac field w is here considered (start ing from a Lagrangian density L), by introducing the field variables P, F conjugate to w,w. After imposing the commutation rules, the Hamilto: nian H was deduced, and an expression for the energy W was obtaine in terms of the annthilation and creation operators a,b. The quantitie. n; are number operators.

5@ This German word means “gauge invariance”; the author uses this property in order to set the potential ~ to zero.

Particular representations of Dirac operators.

8@ In the original manuscript, the author neglect to equate the following expression to zero. 7@ Here, the following insert appears in the original manuscript, reporting what follows:

8@ Here, the following insert appears in the original manuscript, reporting what follows:

Following the Dirac method, the eigenvalues of the Maxwell equation are obtained from:

4@ In the original manuscript, the author considered in what follows the role of the operators V? =L? and L = VV?. He denoted with q the vector (x,y,z). 5@ A bar over a quantity denotes complex conjugation.

6@ Probably, the author proceeded in analogy with the Dirac field .

T is the angular momentum in units h/2r.

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS CONTINUATION II: INCLUDING THE MATTER FIELDS

Without matter fields, the conjugate Hamiltonian variables are:

11@ In the following, the author looked for the variable conjugate to wy.

21@ The page ended with an attempt to generalize the above results to arbitrary linear com- binations of the E and H fields (with space-time dependent coefficients), in the case of Mazwell equations without sources:

Variational Method i.e., the minimum value assumed by the mean value of the energy with respect to any wavefunction y. If we consider only a given set of func- tions y, the minimum will correspond to an approximate value. The given approximation improves when the set is enlarged. When this set reduces to the only unperturbed wavefunction considered in the pertur- bation method, we obtain the same result given by that method. If the set is composed also of further wavefunctions besides the unperturbed wavefunction, in general we will have a better approximation.

7@ The original manuscript then continues with some calculations aimed at evaluating the derivatives of p. In the following we report only the final results.

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS Preliminaries For The X And Y Terms

10@ In the original manuscript it is noted that:

11@ Remember that the X term is a superposition of the 2s2s!S and 2p2p1S' ones.

12@ Remember that the 2s2s'S and 2p2p'S terms are superpositions of the terms called X and Y by the author. The notation used here is the same as in the previous subsection.

13@ The author evaluates the matrix elements of operators L, between p states, in terms of those between y states, already considered on the previous pages. In the following, we do not report the mere arithmetic calculations aimed at obtaining the numbers given in the tables.

14@ For the new states considered by the author, see the previous footnote.

16@ In the original manuscript, in the last line of the previous expression, the first two terms are missing.

1°@ The energy W is measured in rydbergs. 20@ In the following tables, Majorana gave the numerical values of the Schrodinger radial wavefunction x (and its derivatives) for some values of r. They should have been obtained by solving the differential equation reported above. However, it is interesting to note that the quoted numerical values do not come out neither by using the series expansion method out- lined in the notes (method I), nor by solving numerically the equation with the approximate expression for the potential quoted just above (method II). Probably, the numerical values given by the author were obtained by considering the complete potential considered at the end of the previous subsection (method III). To give an idea of the departure from the Majorana tables, in the following we give some values of x and its derivatives for W = 0.32, obtained by using the mentioned three methods. Notice that the series solution can be applied only for r<i:

21@ Observe that the author apparently uses « or r to denote the same quantity. However, below, itisr=k-+ a2, quantity k being the distance from the last node of the eigenfunction.

22@ In the following table the author puts for some approximated expressions for the poly- nomial P for some maximum values n of the index r. For a given n, the first one of the coefficient an is equal to 1, while the other non-vanishing coefficients (with decreasing r) are obtained from the formula r(r+1)

Denoting with D the distance from the last node: E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

24@ This Section was left incomplete by the author.

ve obtain the values reported in the following tables:

By using the Houston formula (Goudsmith method), for the terms *po12, ‘o we have:

33@ In the following expression, as reported in the original manuscript, the factor E in the second term and the equating to zero is lacking. 34@ In the following, E denotes the electric field, gac,qpc the electric dipole moments and vac,V’Bc the frequencies of the given transitions.

RESONANCE BETWEEN A p (£ = 1) ELECTRON AND AN ELECTRON WITH AZIMUTHAL QUANTUM NUMBER ¢’

Only the coefficients by_; and by 4, are non vanishing. Resonance Between A d Electron And A p Shell I

Eigenfunctions Of ds, ds, ps And pi Electrons

44@ See the previous footnote. Notice also that S = S; +82, and analogously for the T terms. Here the manuscript is corrupted and we have represented by dots the expressions we cannot easily interpret.

46The wavelengths of the following spectral lines are expressed in angstroms.

HYPERFINE STRUCTURES AND MAGNETIC MOMENTS: FORMULAE AND TABLES

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS Hyperfine structure formula:

On using electronic units:?! 51@ In the original manuscript, an unidentified reference (see pages 15 and 25) appears here. First Method

1@ In the original manuscript some scratch calculations are reported, leading to the follow- ing expressions for b and c (obtained by substituting the expansions for P2, P3, P3' into the differential equation for Pz written above).

Probably, the numerical values for the second derivative of P3' were deduced in some manner from the following formula (which appears in the manuscript): 2@ The table reported in the original manuscript contains slightly different numerical values with respect to those one can evaluate from the formulae given by the author, namely:

3@ In the original manuscript the upper limit of the following integrals is not explicitly indicated.

By substitution into the above equation, from the vanishing of first-order terms, one has

6@ This Section was probably left incomplete. The corresponding page in the original manuscript reported the following table with practically no entry, pointing out the inten- tion of the author to evaluate P, and its derivatives for some values of W and X, in analogy with what was already done for P2:

whose physical meaning probably, was not clear.

1@ In the original manuscript, the unidentified Ref. 8.47 appears here.

2@ In the original manuscript some numerical calculations appear here, that probably rep- resent an attempt to evaluate the magnetic susceptibility of sodium Na (considered as an In the following notes the author reported a (preliminary?) study or Pauli paramagnetism. He considered an ensemble of N degenerate fer- mions (so that N is proportional to the third power of the Fermi mo- mentum, or V3/2 where V is the electrostatic potential) interacting with a magnetic field H by means of the Pauli term uoH, and obtained ar expression for the magnetic susceptibility x. The number of spin-up anc spin-down fermions was denoted with n’ and n", respectively.

In the ferromagnetic case, if each atom has n neighbor atoms:

For any given value of N, y solutions exist:

4@ We find the original text quite obscure.

7@ We find the original teat to be quite obscure, and our own interpretation is only a probable one.

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS FERROMAGNETISM: APPLICATIONS

By imposing the matching conditions for the wavefunction and its deriva tive, one obtains:

2@ In the original manuscript, there appear here the following calculations:

[The footnote continues on the next page]. 3@ In the original manuscript, the following calculations appear at this point:

The author applied the perturbation theory to the problem of the scat tering of a particle of momentum p = hy from a potential V; the free particle wavefunction is denoted with dy. Some approximated expres sions for the transition probability were obtained within the framewor. of the Dirac method, which are subsequently applied to the particula case of Coulomb scattering.

8@ This equation is a particular case of the more general one reported just after it, and is also considered by the author in another place; see Appendix 6.10.

Now, replace ¢ with ¢ — eg; the phase advancement becomes then

11@ Jn the original manuscript, the factor 3 is lacking.

Let us now consider a fixed a and take the limit G — 0. At zeroth orde approximation, i.e., exactly for 6 = 0, we can use the Wentzel methoc By choosing as mass unit M, wavelength unit \/27 and velocity unit for the incident particles, from A = h/Mv it follows that h = 27 in ov units. Moreover, the kinetic energy of the incident particle is 1/2. From Eqs. (4) and (2) it follows that b = 3, k = @/2 anda=a. B substituting these into Eq. (1), we get the expression for the potenti energy, and the Schrddinger equation corresponding to the eigenvalu 1/2 will be:

COULOMB SCATTERING: ANOTHER REGULARIZATION METHOD E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

The author considered the transition from a discrete (unperturbed) state wo with energy Eo to a continuum (perturbed) state ~, assuming that the unperturbed system has two continuum spectra dw and wy with energy Eo + W. What here reported are the scratch calculations which prepared the Sect. 28 of Volumetto IV, to which we refer the reader for notations and further explanations of the arguments treated by the author. However, a further generalization is present here with respect to what considered after Eq. (4.499) of Volumetto IV.

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS NUCLEAR POTENTIAL

2@ The numerical values 2500/81 and 1250/81 seem to have been written by the author after he deduced the numerical value for p (see below).

Explicit expressions for a particular form of the interaction potential between Z protons and Y neutrons are worked out. See also the paper published by Majorana in Z. Phys. 82 (1933) 137, or in La Ricerca Scientifica 4 (1933) 559. 4@ Notice that here the author refers to the “ordinary” exchange energy depending on the electrostatic interaction among protons. Nucleon Interaction I

12@ In the original manuscript, the unidentified Ref. 5.25 appears here.

13@ In the original manuscript, the unidentified Ref. 11.59 appears here.

15@ In the original manuscript, the unidentified Ref. 3.43 appears here.

7.3.5.2 Zeroth approximation C=0; P?Vp = P®Vy; P = constant, P’ = constant. k= PUP, t= Py] P-

Let us consider three fields In the following pages the author apparently elaborated a relativistic field theory for nuclei composed of scalar particles of two different kinds (one

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS SYSTEMS WITH TWO FERMIONS AND ONE BOSON

24@ Note that, in subsequent pages, the author denotes with Z the following operator corre- sponding, effectively, to the net charge rather than to the total number N of particles.

THOMSON’S METHOD FOR THE DETERMINATION OF e/m

1N is the total number of charged particles, while 7 is the atmospheric pressure (see below).

From this, the Richardson formula for the electron current i follow (Richardson effect):

Integrals Of An Ordinary Differential System And The Partial Differential Equation Which Determines Them

Integrals Of A Total Differential System And The Associated System Of Partial Differential Equation That Determines Them

ALGEBRAIC FOUNDATIONS OF THE TENSOR CALCULUS

Euclidean Manifolds. Any V, Can Always Be Considered As Immersed In A Euclidean Space

or a definite form a, and taking x; and y; not proportional, it follows:

GEOMETRICAL INTRODUCTION TO THE THEORY OF DIFFERENTIAL QUADRATIC FORMS II 1@ In the original manuscript, a reference appears (p. 154) of a unspecified text.

Divergence Of A Double (Contravariant) ‘Tensor

Geodesic Coordinates Along A Given Line 2@ The symbol G is introduced by the author in reference to the initial of the Italian word “giacitura”, which means “section”.

From (3), by summing over every value of i, we obtain the quantities 5;, (with n? arbitrary constants; for example they are given by the initial

3@ In the original manuscript, the following note appears: Change the sign of Riemann’s symbols. Also, the following is pointed out, referring to equations reported in Levi-Civita I: Notes on the Tallis formulae: Eq. (3), p.201 is correct; Eq. (4), p.201, change the sign; Eq. (26), p.219 is correct.

) Number of the independent symbols of first kind.

Erasmo Recami

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ETTORE MAJORANA: UNPUBLISHED RESEARCH NOTES ON THEORETICAL PHYSICS (Book of 487 pages, by SPRINGER, 2009)

Alwyn Van Der Merwe, Erasmo Recami

see the Contents and the Preface in the attached file

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Ettore Majorana's course on theoretical physics: A recent discovery

antonino drago

2007

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The Preface to the Book << Unpublished Research Notes on Theoretical Physics by Ettore Majorana >> .

Erasmo Recami

This file contains Contents and Preface of the SPRINGER's 2009 Book << Ettore Majorana - Unpublished Reserach Notes on Theoretical Physics >> . The book's editors are S.Esposito, E.Recami, A.van der Merwe, and R.Battiston

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Ettore Majorana: Notes on Theoretical Physics

Erasmo Recami

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The scientific manuscripts left unpublished by Ettore Majorana: with outlines of his work

Erasmo Recami

We present here a panoramic view of the main scientific manuscripts left unpublished by the brightest Italian theoretical physicist of the XX century, Ettore Majorana. We deal in particular: (i) with his very original "study" notes (the socalled Volumetti), already published by us in English, in

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ETTORE MAJORANA: HIS WORK AND HIS LIFE April 2014

Erasmo Recami

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Following Weyl on Quantum Mechanics: The Contribution of Ettore Majorana

antonino drago

Foundations of Physics, 2000

After a quick historical account of the introduction of the group-theoretical description of Quantum Mechanics in terms of symmetries, as proposed by Weyl, we examine some unpublished papers by Ettore Majorana. Remarkable results achieved by him in frontier research topics as well as in physics teaching point out that the Italian physicist can be well considered as a follower of Weyl in his reformulation of Quantum Mechanics.

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Ettore Majorana as a guide in Quirino Majorana's experiments. Original letters and documents on an experimental and theoretical collaboration

Mago Chines

It is offered a preview of the lengthy, intense and rich exchange of scientific correspondence (currently in press) which took place in the period from 1931 to 1937 between Ettore and Quirino Majorana. This collection of letters provides insight into the specifically experimental and theoretical aspects tied to the study of changes in photoresistance in thin metal films exposed to light in suitable conditions and the possible identification of new effects of a photoelectric type. It also leads us to realise that many theoretical parts of the most mature papers by Quirino Majorana are attributable in fact to Ettore, and moreover allows us to identify Ettore as being, in a certain sense, the co-author of those publications and studies, thanks also to the constant role he played vis-à-vis his uncle Quirino as a methodologically astute critic and epistemological guide. It is pointed out, furthermore, that this correspondence allows us to recognise Ettore as the true author of the Lecture on Galvani that Quirino Majorana presented in Bologna in 1937, on the occasion of a great international Congress organised to celebrate the second centenary of the birth of Luigi Galvani. Finally, the article refutes the widespread and common belief that after 1933 Ettore no longer concerned himself with physics.

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A forgotten publication of Ettore Majorana on the improvement of the Thomas-Fermi statistical model

Francesco Guerra

2005

Ettore Majorana proposed an important improvement to the Thomas-Fermi statistical model for atoms, with a communication at the general meeting of the Italian Physical Society, held in Rome, on December 29th, 1928, regularly published on Nuovo Cimento. This communication did not receive any mention, neither in the numerous publications of Fermi and his associates on the subject, nor in the further reconstructions of the life and activity of Ettore Majorana. The Majorana proposal was not accepted by Enrico Fermi for years, and forgotten. However, it was finally exploited, without reference, in the 1934 conclusive paper by Fermi and Amaldi on the statistical model for atoms. We analyze the improved model proposed by Majorana, with the help of unpublished notes, kept in the Majorana Archives in Pisa. Moreover, we trace the path leading Fermi to the eventual late acceptance of Majorana improvement, also with the help of the material in Fermi Archives in Pisa. The conclusion is that Ettor...

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