The Rise of Quantum Mechanics 3

Home , Gustav Ludwig Hertz, Jagdish Mehra, Walther Ritz

arXiv:0806.4515v1 [physics.hist-ph] 27 Jun 2008 netgto fntr.Aon the classi- Around so-called nowadays the 1900 nature. year intense of of centuries af- investigation three than emerged more that ter at looking phenomena of physical way the reoriented pletely [1]. of theory quantum development historical the Rechenberg describing Helmuth with enter- together cyclopical words prise a emphatic starts such Mehra Jagdish With forces nature.” fundamental of the of radia- many and and matter tion of worldview structure new the a about with fashioned break and con- past complete its the a in made it man: ceptions of and science history of history intellectual the in unique the- is quantum ory the mechanics, edifice classical the of completed which special general, both and theories, relativity the more than Much theory. constitute quantum of to fabric came the prob- together of which multitude lems analyz- the detail without some glory in in ing its told of be fullness cannot appropri- the story This demands and telling. story ate epic an century is twentieth the in theory quantum Introduction 1. 2008 June - 0 N. Università deg Teorica, e Nucleare Fisica di Dipartimento Boffi Sigfrido Mechanics Quantum of Rise The FISICA DELLA STORIA DI QUADERNI nfc,qatmmcaishscom- has mechanics quantum fact, In of development and discovery “The econsidered. to be have effects oc- non-locality and of probability currence, of terms in ac- be complished only can phenomena of the description phenomenon, the determining in pic- essential role an a plays observer by the where substituted ture were determinism locality new Objectivity, and for was over. accounting paradigm turned of this discoveries, result and facts a as ics in- mutual constituents. the of of lead teraction knowledge be the can to behaviour back global the law, causality where the by objec- governed an world of tive ori- paradigm scientific the the at of because were gin laws role Newton’s privileged three a the play case in would any mechanics Analytical nomena. phe- physical of description a unified towards global road the on physics of differentbranches only were thermodynam- electromagnetism ics, mechanics, sug- that were matter, gesting of theory kinetic phenomena the magnetic or and electric unifica- of the tion as such syntheses, markable re- Some the phenomenology. for account corresponding satisfactorily to able laws was and closed concepts a of system sector coherent each and differ- Within in sectors. organized ent well was physics cal ihteavn fqatmmechan- quantum of advent the With iSuid Pavia di Studi li 2 Sigfrido Boffi

Fig. 1. – In the first quarter of the twentieth century the crisis produced when trying to unify the different sectors of physics, such as macrophysics (described in terms of temperature T and entropy S), mechanics (with its Lagrangian L and Hamiltonian H) and electromagnetism (with its electric and magnetic fields E~ and B~ , respectively), was overcome by introducing new concepts and a new way of thinking of reality as a consequence of the development of relativity theory (with its equivalence between energy E and mass m and the invariance of the light velocity c) and quantum mechanics (that associates, through the Planck’s constant h, a wave with wavelength λ and frequency ν to the motion of a particle with momentum p and energy E, respectively).

This was achieved in the first quarter a scheme suffered a big attack when of the twentieth century, especially be- physicists realized that the mechanical tween June 1925 and October 1927, as a equations of motions are not compatible consequence of an extraordinary develop- with the Maxwell’s equations for the elec- ment of new data, ideas, formalisms, in- tromagnetic phenomena. The solution terpretations, within a polyphonic frame- found in 1905 by Albert Einstein (1879– work where very young researchers and 1955) with his revision of the concept more experienced scientists were chal- of simultaneity and the space-time struc- lenging each other in a cooperative and ture made it possible to reconcile me- unique effort. chanics and electromagnetism in a unified and objective picture. Thus, though revolutionary, relativity theory, even with 2. Crisis towards unification its extension to general relativity, still obeys the principle of objectivity and In analytical mechanics observers are lives within the paradigm of classical simulated by inertial frames of reference physics. and time is assumed to be an absolute evolution parameter. Then the ob- In contrast, in the attempt to es- jective description of phenomena means tablish a connection between the macro- that any physical law is translated into scopic behaviour of a complex system and one and the same equation when pass- the microscopic motion of its constituent ing from one observer to another. Such particles or to account for the thermody- The Rise of Quantum Mechanics 3

namic effects of radiation, one meets dif- On a different side, the discoveries of ficulties that are unsurmountable within radioactivity by Wilhelm Conrad Rönt- the classical framework (Fig. 1). For ex- gen (1845–1923) and of the electron in ample, the frequency distribution of the the study of cathode rays by Joseph radiation energy density cannot be pre- John Thomson (1856–1940) added im- dicted invoking the classical thermody- portant insights into the constitution of namics of radiation. The formula pro- matter. In atomic physics by the end of posed on heuristic arguments by Max the 19th century a large amount of ac- Planck (1858–1947) in 1900 could only cumulating data on the line spectra were be explained by Einstein under the as- organized according to the combination sumption that the energy of the har- principle emerging from the studies of monic oscillator associated to each fre- Johann Jakob Balmer (1858–1898), Jo- quency takes discrete values or, alterna- hannes Robert Rydberg (1854–1919) and tively, the action corresponding to a com- Walther Ritz (1878–1909). In the case of plete oscillation is an integer multiple of the hydrogen atom, for example, in the an elementary value h, the Planck’s con- Balmer’s formula the inverse wavelength stant. Similarly, the temperature depen- of every spectral line could be expressed dence of specific heat of solids cannot be as the difference of two terms, each of explained assuming a classical motion of which depending on an integer number. atoms within the solid and violates the The discrete nature of the line spectra is classical equipartition principle of energy, incompatible with the stable atom gov- unless again one assumes with Einstein erned by the laws of classical physics, and and Peter Debye (1884–1966) the possi- their classification in terms of the inter- bility of a discrete energy spectrum for nal atomic dynamics was a big puzzle. the oscillating atoms in solids. The discovery of the effect of a mag- The discrete nature of the electro- netic field on the spectral lines by Pieter magnetic field interacting with matter Zeeman (1865–1943) and its explana- and Einstein’s idea of a light quantum tion by Hendrik Antoon Lorentz (1853– with energy hν and momentum hν/c 1928) and Joseph Larmor (1857–1942) were not accepted by the physics commu- were a great success of the electron the- nity without a long discussion. Even af- ory of matter. However, in some cases ter the successful test of Einstein’s equa- an anomalous line splitting was observed tion for the photoelectric effect predict- such as that occurring for the two sodium ing a linear relation between the maxi- D-lines, with the D1-line splitting into mal kinetic energy of the ejected photo- a quartet and the D2-line into a sextet. electron and the frequency ν of the inci- Within the classical theory one could not dent radiation, Robert Andrews Millikan explain such an anomalous Zeeman ef- (1868–1953) remarked that “the semi- fect. corpuscolar theory by which Einstein ar- According to the model put forward rived at this equation seems at present to in 1911 by Ernest Rutherford of Nelson be wholly untenable” [2]. It took other (1871–1937) electrons revolving about ten years to look at the light quantum the positively charged atomic nucleus fol- as the “photon” responsible, e.g., of the low a periodic motion. Quantization Compton effect [3]. rules for such periodic systems were pro- 4 Sigfrido Boffi

posed in 1913 by Niels Hendrik David could also provide the necessary founda- Bohr (1885–1962) and implemented in tion for atomic mechanics. 1916 by Arnold Sommerfeld (1868–1951). The Bohr-Sommerfeld rules were With such rules one defines azimuthal soon applied to a variety of problems and radial quantum numbers describ- such as quantum theory of radiation, ing the Kepler’s orbit of the electron in atoms with one electron and with sev- a plane, and the Balmer’s formula for eral electrons, quantum theory of solids spectral lines can be easily recovered. and gases, atomic magnetism. They were Also the normal Zeeman effect could be so successful describing the constitution described by Sommerfeld introducing a of atoms and the periodic table of ele- third quantum number, whose values de- ments that the predicted element with termine the discrete positions of the elec- atomic number 72 was just discovered by tron orbit with respect to the external Dirk Coster (1889–1950) and George de magnetic field. Hevesy (1985–1966) in Bohr’s Institute in The Bohr-Sommerfeld rules are de- Copenhagen and called hafnium after the rived from two postulates, i.e. the ex- Latin name of Copenhagen (Hafnia), in istence of stable stationary states of the time for Bohr to mention it in his Nobel atom and the definition of the emitted or lecture in 1922. absorbed radiation frequency in terms of However, there were also some fail- the energy difference between initial and ures, such as the calculation of the en- final stationary states. These two pos- ergy states of the helium atom and simi- tulates are consequences of the adiabatic lar many-electron atoms, the description principle and the correspondence prin- of the anomalous Zeeman effect, and the ciple. According to the adiabatic prin- difficulty to describe time dependent pro- ciple the quantized action remains con- cesses such as the interaction between ra- stant during the electron motion also in diation and matter. In the attempt to the case of transitions between stationary overcome the difficulty of explaining the states induced by an external perturba- dispersion of light by atoms Niels Bohr, tion. The correspondence principle im- Hendrik Antoon Kramers (1894–1952) plies that under suitable conditions, i.e. and John Clarke Slater (1900–1976) as- when the action per cycle is large com- sumed that a given atom in a certain pared to Planck’s constant h, one must stationary state communicates continu- recover the classical limit of the radiation ally with other atoms through a mecha- frequency. nism which is equivalent with the classi- The existence of stable stationary cal field of radiation originating from the states was confirmed in a series of ex- virtual oscillators corresponding to the periments by James Franck (1882–1964) various possibile transitions to other sta- and Gustav Ludwig Hertz (1887–1975). tionary states. However, the communi- The correspondence principle was a fruit- cation between atoms, i.e. the absorp- ful guideline in the development of the tion and emission processes, were con- theory. Thus, scientists were confi- nected by probability laws implying that dent that classical mechanics, imple- energy and momentum were conserved mented by the two Bohr’s postulates and only on the average [4]. This conclu- the Bohr-Sommerfeld quantization rules, sion was disproved by the result of an The Rise of Quantum Mechanics 5

experiment performed by Hans Wilhelm In about one year from mid 1925 Geiger (1882–1945) and Walter Wilhelm to mid 1926 as a result of an intensive Georg Bothe (1891–1957) [5] where the work in Göttingen, Zürich, Cambridge scattered X-ray was detected in coinci- and Copenhagen the situation changed dence with the recoiling electron. This dramatically. New formalisms were pro- result shows that energy and momen- posed and successfully applied to solve tum are conserved in individual elemen- the problems left open by the old quan- tary processes and confirms the parti- tum theory. The different approaches cle behaviour of radiation, already intro- were soon found to be equivalent, so duced by Einstein with the light quantum that a complete and consistent formalism hypothesis and clearly demonstrated by could be developed. Arthur Holly Compton (1892–1962) [6]. On the other hand, new facts and 3.1. Matrix mechanics productive ideas were emerging. In 1924 in his doctoral thesis Louis-Victor A breakthrough came with the paper de Broglie (1892–1987) suggested that conceived in June 1925 by Werner Karl waves could also be associated to parti- Heisenberg (1901–1976) on the rocky is- cles such as electrons, a bold assumption land in the North Sea called Helgoland, that was most surprisingly confirmed by where he spent a two-week vacation to the findings by Clinton Joseph Davisson recover from a hay fever attack [9]. He (1881–1958) in his studies on secondary noticed that the formal rules in quan- cathode rays together with Lester Hal- tum theory make use of relationships be- bert Germer (1896–1971) and by George tween unobservable quantities, such as, Paget Thomson (1892–1975) working on e.g., the position and time of revolution the diffraction of cathode rays by a thin of the electron. One has rather to fo- film together with Alexander Reid. In cus on the observable quantities during Göttingen Walter Elsasser (1904–1987) emission and absorption, i.e. the radia- showed that also the so-called Ramsauer tion frequency and intensity. According effect involving the scattering of low- to Bohr the radiation frequency ν is iden- energy electrons by atoms of a noble gas tified by assigning the energy of the ini- can be interpreted on the basis of de tial and final stationary states, W (n) and Broglie’s idea. This called for a new W (n − α), respectively: electromagnetism suitable to describe the 1 wave-particle duality and its name, i.e. ν(n,n − α)= [W (n) − W (n − α)] . h quantum mechanics, already appeared in the title of a paper by Max Born (1882– Just in the same way one can asso- 1970) [7]. Thus, the systematic presenta- ciate a two-dimensional pattern also to tion of the results of what is now called the amplitude of the emitted light wave, the old quantum theory, based on the A(n,n − α), with rows and columns or- Bohr-Sommerfeld rules, just stopped af- dered according to the different initial ter the first paper [8]. and final states involved in the transition. The radiation intensity is then obtained 3. New formalisms by the squared transition amplitude. Making use of the Bohr-Sommerfeld rules 6 Sigfrido Boffi

and the correspondence principle Heisen- tems having arbitrarily many degrees berg then arrived at the following quan- of freedom and developed a quantum- tum condition: mechanical perturbation theory. Their ∞ paper was soon cited as the Dreimänner- 2 h =4πm |A(n + α, n)| ω(n + α, n) arbeit (three-men’s paper), and the the- αX=1 ory they developed was called matrix me- 2 − |A(n,n − α)| ω(n,n − α) , chanics [10]. In particular, in complete where ω(n,n′)=2πν(n,n′) and m is the analogy with the classical Hamilton- electron mass. Jacobi equation they found that the en- In such a reformulation scheme an es- ergy levels could be derived by transform- sential mathematical difficulty occurred, ing the Hamiltonian matrix H to its diag- namely the factors in product of two pat- onal form W by using a (unitary) trans- terns did not commute in general. In formation matrix S: Göttingen Born and Ernst Pascual Jor- −1 dan (1902–1980) recognized that Heisen- (4) H(q, p)= SH(q0, p0)S = W. berg’s patterns are nothing else than matrices obeying the noncommutative law When the transformation with S is ap- of the matrix product. By sharpening plied to the matrices q and p, the quan- the idea of the correspondence principle tum condition (2) is left invariant. This they adopted the classical equations of justifies the name of canonical transfor- motion considering them as relations be- mation given to it by Born, Heisenberg tween matrices representing classical ob- and Jordan. servables. That is, for a one-dimensional quantum system described by the Hamil- Canonical transformation and diag- tonian matrix H = H(q, p) the equations onalization of the Hamiltonian matrix H q p of motion assume the canonical form, ( , ) are the central ideas of the three men’s paper. They help to find all the ∂H ∂H (1) q˙ = , p˙ = − , conserved quantities for a given quantum ∂p ∂q system as those represented by matrices where the dynamical matrices q and p that have a vanishing commutator with representing the system position and mo- H(q, p). This is in turn in close analogy mentum, respectively, satisfy the quan- with classical mechanics where an inte- tum condition grable system with f degrees of freedom h has exactly f − 1 independent constants (2) pq − qp = I, 2πi of motion besides the Hamiltonian. with I being the identity matrix. By re- The first application to a physical peated application of the quantum condi- problem was successfully accomplished tion (2) the equations of motion (1) could by Wolfgang Pauli (1900–1958) who was equivalently be rewritten as able to derive the correct spectrum of the 2πi 2πi hydrogen atom after a brilliant and labo- q˙ = (Hq−qH), p˙ = (Hp−pH). rious calculation [11], where he showed h h (3) that also in the quantum-mechanical case Together with Heisenberg they ex- both angular momentum and the Runge- tended the scheme to include sys- Lenz vector are constants of motion. The Rise of Quantum Mechanics 7

3.2. Wave mechanics also de Broglie did, Schrödinger first derived an equation which is nowadays known as the Klein-Gordon equation, but On a different side, the new elec- he gave it up because it did not yield tromagnetism wished by De Broglie was the right fine-structure of the hydrogen formulated by Erwin Schrödinger (1887– atom. In fact, the equation entitled after 1961) in four papers produced between Oskar Benjamin Klein (1894–1977) and January and June 1926 where quantiza- Walter Gordon (1893–c.1940) has many tion was developed as an eigenvalue prob- fathers [13] and was recovered a few years lem [12]. Schrödinger took advantage of later by Pauli and Victor Frederick Weis- the analogy between the wave propaga- skopf (1908–2002) who gave it the correct tion and the motion of a particle already interpretation within a newly developing explored by de Broglie in his thesis and quantum field theory [14]. even earlier by William Rowan Hamil- Confining himself to nonrelativistic ton (1805-1865) in 1824. Wave propaga- kinematics, in the first paper of the se- tion of light can be visualized according ries [12] Schrödinger considered an elec- to Christian Huyghens (1629–1695) by tron bound in the hydrogen atom. In this looking at the motion of the wave front, case the Bohr-Sommerfeld quantization i.e. the surface with constant wave phase rules imply stationary waves with wave- perpendicular to the wave vector; alter- length tuned to the orbit length. Then natively, according to Pierre de Fermat the Hamilton-Jacobi equation of analyti- (1601–1665) one can describe the prop- cal mechanics becomes agation in terms of a light ray always tangent to the wave vector. The bend- 2m e2 (5) ∇2ψ + E + ψ =0, ing of the ray and the distortion of the K2 r phase wave are both due to local varia- tions of the refractive index. In quite a where K has the dimension of an ac- similar way the motion of a particle in tion (nowadays K ≡ ¯h = h/2π) and terms of its trajectory, always tangent to ψ describes the wave to be associated the particle momentum, can also be visu- with the electron motion with energy E. alized in terms of an action wave, always Therefore the Hamilton-Jacobi equation perpendicular to the particle momentum. has turned into an eigenvalue equation Modulations of the potential affect the for the electron Hamiltonian. Its solu- momentum just like the refractive index tions correspond to the stationary states modifies the wavelength. of Bohr’s atomic theory and provide the These similarities were already sum- discrete spectrum of the bound electron, marized in the Einstein-Planck formula in good agreement with data. This was a E = hν and the de Broglie’s hypothe- much more direct approach to the prob- sis p = h/λ, where particle quantities, lem than the quite complicated calcu- such as the energy E and momentum lation of Pauli with matrix mechanics. p, were connected through Planck’s con- In modern textbooks the nonrelativis- stant with wave quantities, such as the tic quantum hydrogen problem is solved frequency ν and wavelength λ. according to the procedure followed by Assuming relativistic kinematics, as Schrödinger. 8 Sigfrido Boffi

The analogy between particle motion Physically, it is a wave function reflecting and geometrical optics was further exam- the linear superposition principle typical ined in the second communication, where of a wave behaviour. the eigenvalue equation was derived from a variational principle and applied to 3.3. Equivalence between matrix and wave other soluble cases such as the Planck’s mechanics oscillators and the rigid rotator. In the third paper the method could also be During a stay at MIT in the winter applied in perturbation theory to cases semester 1925–1926 to take advantage of where exact analytical solutions are im- the collaboration with Norbert Wiener possible, such as the Stark effect. Only in (1894–1964), the future father of cyber- the fourth communication the process of netics, Born realized that also the Hermi- building the new wave mechanics was ac- tian matrices of matrix mechanics could complished with the introduction of the be regarded as operators acting on vec- time-dependent Schrödinger equation: tors in a multidimensional space. As- suming the Hamiltonian to be an oper- 8π2 4πi ∂ψ (6) ∇2ψ − V ψ + =0 ator function of the dynamical variables, h2 h ∂t having the same functional dependence (with m = 1). This was obtained by on the operators p and q as the classical replacing the energy E with the opera- Hamiltonian has on its dynamical vari- tor i(h/2π)∂/∂t acting on the wave func- ables, one could reformulate the laws of tion ψ and assuming that the potential matrix mechanics for any system [15]. energy V works on ψ as a multiplica- In March 1926, just after his second tive operator. Similarly, the kinetic en- communication and before his third one, ergy is responsible for the Laplace op- Schrödinger was able to show the link erator ∇2 coming from the replacement between wave mechanics and matrix me- p → −i(h/2π)∇. Thus, as we do it to- chanics [16] claiming that “from the for- day, eq. (6) can equivalently be written mal mathematical standpoint one may as even say that the two theories are iden- ∂ψ (7) i¯h = Hψ, tical”. In fact the matrix elements of ∂t the Hermitian matrices representing op- where H is the Hamilton operator. For erators in matrix mechanics are just the an isolated system its most general solu- same elements obtained using the wave tion is given by a linear superposition of functions and the operators in wave me- particular solutions, chanics. −iEnt/h¯ (8) ψ = cnun e . Also Pauli, in a letter to Jordan dated Xn April 12, established the connection between wave and matrix mechanics show- where u and E are the eigenfunctions n n ing that “the energy values resulting from and eigenvalues, respectively, of H, i.e. Schrödinger’s approach are always the same as those of the Göttingen Mechan- (9) Hun = Enun. ics, and that from Schrödinger’s function Mathematically, Eq. (8) is dictated ψ, which describes the eigenvibrations, by linearity of Schrödinger’s equation. one can in a quite simple and general way The Rise of Quantum Mechanics 9

construct matrices satisfying the equa- ables from the quantum noncommuting tions of the Göttingen Mechanics”. He objects “we shall call the quantum vari- never published the content of his letter ables q-numbers and the numbers of clas- that was discussed in public only many sical mathematics which satisfy the com- years later [17]. mutative law c-numbers, while the word It may be of some interest to re- number will be used to denote either a call that already at the end of December q-number or a c-number” [21]. Thus, re- 1925 also Cornelius Lanczos (1893–1974) placing the Poisson’s bracket {A, B} of arrived at an integral equation equiva- two classical observables by the commu- lent to Schrödinger’s equation starting tator [A, B] = AB − BA of the two cor- from the matrix mechanics of Heisenberg, responding q-numbers, Born and Jordan and applying Hamil- ton’s variational principle [18]. His pa- i (10) {A, B}→− [A, B], per, however, was not appreciated by ¯h Schrödinger and Pauli and remained isolated in Lanczos’ production. Dirac could extend the Hamilton for- For completeness, one should men- malism to quantum equations of motion that also Carl Henry Eckart (1902– tions [20]. 1973), after attending Born’s lectures “The new quantum mechanics con- during his tour in U.S.A. in winter 1926, sists of a scheme of equations which are was able to prove the equivalence be- very closely analogous to the equations of tween matrix and wave mechanics [19] classical mechanics, with the fundamen- just before publication of Schrödinger’s tal difference that the dynamical vari- paper. ables do not obey the commutative law of multiplication, but satisfy instead the 3.4. Dirac’s q-numbers well-known quantum conditions. It follows that one cannot suppose the dynam- After reading the first article of ical variables to be ordinary numbers (c- Heisenberg [9] Paul Adrien Maurice numbers), but may call them numbers Dirac (1902–1984) realized that the new of a special type (q-numbers). The the- theory was suggesting “that it is not the ory shows that these q-numbers can in equations of classical mechanics that are general be represented by matrices whose in any way at fault, but that the mathe- elements are c-numbers (functions of a matical operations by which physical re- time parameter)” [22]. sults are deduced from them require mod- In a series of eleven papers published ification. All the information supplied in twenty months, without particularly by the classical theory can thus be made new results, Dirac was able to give an use of in the new theory” [20]. This extraordinary new perspective in the for- statement followed from the fact that mal and conceptual development of the Dirac recognized the commutation rela- new quantum mechanics. tions between quantities representing ob- In particular, for a multiply periodic servables to have similar properties as system action and angle variables, Jk and the Poisson’s brackets of classical me- wk respectively, could be introduced sat- chanics. To distinguish classical vari- isfying equations of motion formally iden- 10 Sigfrido Boffi

tical to the classical equations, i.e. tions. It also provides a unified formalism for the new quantum theory, because ∂H J˙ = [J , H]=0, w˙ = [w , H]= . the q-numbers can be related to the op- k k k k ∂J k erators of the Born-Wiener approach and An application to the hydrogen atom [21] Schrödinger’s wave mechanics by build- immediately followed that given by Pauli ing their matrix representation [27]. with the matrix mechanics [11] confirm- Dirac’s quantum algebra makes use of ing his results. Then Dirac extended the what is now called the abstract Hilbert action-angle scheme to investigate many- space. This is a linear manifold of vec- electron atoms; he recovered the Landé tors (i.e. closed under vector addition formula for the anomalous Zeeman ef- and multiplication by scalars, and with fect as well as the relative intensities a strictly positive inner product over the of the spectral lines in a multiplet and field of complex numbers) which is com- their components in the presence of a plete with respect to the metric generated weak magnetic field. The only new re- by the inner product and separable. Its sult was an application to the Compton elements are legitimate objects to repre- effect, where his “theory gives the cor- sent physical states, and observables correct law of variation of intensity with an- respond to suitable linear (self-adjoint) gle, and suggests that in absolute mag- operators acting on such elements. A ba- nitude Compton’s values are 25 per cent sis in the Hilbert space is given by the too small” [23]. Indeed, a few months complete set of eigenvectors of one of later in a letter to Dirac [24], Compton these self-adjoint operators, so that the announced that new observations were vector representing the state of the sys- then in quite good agreement with the- tem can be written as a linear superpo- ory! sition of the basis elements. This fact In Dirac’s opinion a good notation reflects the linear superposition principle and a clear nomenclature are essen- which has a central role in quantum me- tial tools. Therefore he invented new chanics to describe the wave-particle du- terms and symbols, some of them still ality. in common use today, such as ‘commu- This scheme was brought to a pre- tator’, ‘q-numbers’, ‘eigenfunction’, the cise formulation by the Göttingen math- ‘δ-function’. The famous bra and ket ematical school flourished around David notation appeared only much later [25] Hilbert (1862–1943), in particular by and was used in the third edition of the Johannes (John) von Neumann (1903– celebrated book [26], originally published 1957) [28]. in 1930 and practically unmodified in ten over twelve chapters up to the last (fourth) edition in 1958, remaining a fun- 4. The wave function and transforma- damental reference for any beginner also tion theory today. Dirac’s algebraic approach in terms The existence of a continuity equa- of q-numbers may be considered a gen- tion for ρ = |ψ|2 associated with his eralization of the matrix mechanics suit- wave equation, in quite analogy with hy- able for both periodic and aperiodic mo- drodynamics, induced Schrödinger in his The Rise of Quantum Mechanics 11

fourth communication to assume ρ to de- ber 1927 that de Broglie abandoned it for scribe the matter distribution of the par- many years and resurrected it only when ticle (an electron of total charge e) and David Bohm (1917–1992) proposed his eρ its electric charge distribution. This approach to hidden variables in 1952 [33]. idea was strengthened by finding that a “As a matter of fact, the new mechan- suitable wave packet built as a superpo- ics does not answer, as the old one, to sition of harmonic oscillator eigenfunc- the question how does the particle move, tions could remain concentrated during but rather to the question how proba- its motion through space with a back- ble is that a particle moves in a given and-forth time behaviour just as in classi- way” [34]. This revolutionary statement cal case [29] (1). This interpretation was follows Born’s discovery that in scatter- immediately rejected by Erwin Madelung ing processes in particle collisions the (1881-1972) [31] because in the Hamil- wave function ψ plays the same role as tonian driving the Schrödinger’s equa- the electric field E in light diffraction. tion there is no mutual interaction be- What matters to describe the angular tween different parts of the particles dis- distribution of particles and light is nei- tributed over the whole space. For both ther ψ nor E, but rather their squared Schrödinger and Madelung the quantity modulus. The arrival of a particle (or ρ is in any case a weight function nec- a photon) at some point on a screen is essary to calculate average values just as not predictable, only its probability can we do it today. be calculated with |ψ|2. Thus the wave In contrast, de Broglie gave a realis- function acquires a statistical interpreta- tic interpretation assuming that the wave tion and is well recognized to have only function ψ is a physically real wave, a a pure auxiliary role in the calculation of pilot wave driving the particle during observable quantities [35]. its motion. Its velocity is derived by On the other hand, within the formal- the guidance formula, v = ∇S/m, and ism the wave function provides a com- is always perpendicular to a surface of plete and exhaustive description of the constant action S determined by ψ [32]. system under consideration, as shown by Through the definition of a quantum po- Dirac in his transformation theory [22]. tential to be added to the usual poten- Originally, this approach is an elegant so- tial in the classical equation of motion, lution to the general problem of solving de Broglie intended to recover the deter- the quantum equations of motion either ministic behaviour of classical mechanics in matrix or in wave mechanics. Simi- in terms of variables that remain hidden lar ideas were proposed independently at in the theory. Such an interpretation was the same time by Jordan [36,37] and Fritz so strongly confuted by Pauli at the Fifth Wolfgang London (1900–1954) [38]. Solvay Conference in Bruxelles in Octo- Equations of motion in quantum mechanics, in the form of either Eqs. (3) or Eq. (7), involve the Hamiltonian, i.e. a Hermitian (or, better, self-adjoint) oper- 1 ( ) Schrödinger’s wave packet is the mini- ator. In both cases one has to construct mum uncertainty (coherent) state introduced by Roy Jay Glauber (b. 1925) many years later to a matrix representation of the Hamilto- describe the laser radiation field [30]. nian and apply a suitable transformation 12 Sigfrido Boffi

to bring it to a diagonal form by solv- was developing, a new degree of freedom ing either Eq. (4) or Eq. (9). Already entered the scene of atomic physics. It in the three-men’s paper such a transfor- immediately appeared to be the last re- mation was found to be unitary. Dirac maining building element in the puzzle to extended the procedure by showing that fit the data. in a scheme of matrices representing the dynamical variables unitary transforma- 5.1. Spin tions preserve all algebraic relations such as the commutation relations, the equa- In order to account for the periodic tions of motion and the expectation val- system of elements, something was still ues. Therefore, the transformed set of missing. At the time, electrons in an matrices is just equivalent to the original atom were divided in two groups, one of one, so that it is a free choice to adopt a which living in the passive atomic core scheme where, e.g., the position (the (q) and the other defining the position of scheme) or the energy (the Hamiltonian) that atom in a series of the periodic sys- are diagonal. “The eigenfunctions of tem. Such series electrons were consid- Schrödinger’s wave equation are just the ered responsible for all atomic proper- transformation functions . . . that enable ties including magneto-mechanical effects one to transform from the (q) scheme and radiative processes. Their distribu- of matrix representation to a scheme in tion among atomic levels could explain, which the Hamiltonian is a diagonal ma- e.g., the ground state structure of noble trix” [22]. gases [39]. However, in the attempt to Commuting matrices can be put in di- explain the multiplet structure and the agonal form in the same scheme. In the selection rules of the anomalous Zeeman scheme where the Hamiltonian is diago- effect by refining Sommerfeld’s approach, nal only a few set of matrices can also Alfred Landé(1888–1975) and Werner be brought to diagonal form: the cor- Heisenberg already in 1921-1922 realized responding dynamical variables are the that half-integral quantum numbers had only possible constants of motion. At to be used for the series electrons. Also any time the state of the physical sys- Pauli during his stay in Copenhagen by tem with a precise value of energy is then Bohr in 1923 convinced himself that for fully characterized by assigning also the elements that follow each other in the pe- values of such constants. For other vari- riodic table the values of the magnetic ables, not commuting with the Hamilto- quantum number are alternatively half- nian, one can only calculate their aver- integral and integral. He then speculated age value in that state: “this information that this was caused “by a peculiar, clas- appears to be all that one can hope to sically not describable kind of duplicity get” [22]. of quantum-theoretical properties of the series electron” [40] demanding the introduction of a fourth quantum number in 5. A new degree of freedom the classification of electron orbits. In the following discussion of the problem During the same couple of years when of equivalent electrons, i.e. electrons hav- the basic formalism of quantum theory ing the same binding energy (or the same The Rise of Quantum Mechanics 13

principal quantum number), he arrived 2π2e4m/h3), gives the shift of the en- at the conclusion that “there can never ergy levels characterized by the princi- exist two or more equivalence electrons pal quantum number n and total spin in the atom for which . . . the values of j. Still the first excited state of the hy- all [four] quantum numbers . . . coincide”. drogen atom remains degenerate: today The Pauli exclusion principle was imme- we know that the splitting between the 2s+1 2 2 diately accepted by the physicists work- n Lj =2 S1/2 and 2 P1/2 states, the ing in the field and became clearer when so-called Lamb shift [44], can only be ex- George Eugene Uhlenbeck (1900–1988) plained within a completely relativistic and Samuel Abraham Goudsmit (1902– quantum field theory approach. 1978) formulated the spin hypothesis as- The spin was soon incorporated in sociating Pauli’s fourth quantum number the emerging formalism of wave mechan- with “an intrinsic rotation of the elec- ics by Pauli, who represented the electron” [41]. tron spin as an operator with the same According to Uhlenbeck and Goudsmit formal properties of angular momentum. an intrinsic magnetic moment has to be Introducing the famous Pauli matrices, associated with each electron, he derived the Schrödinger’s equation for e a particle interacting with an external µ = − s, S mc magnetic field [45]. However, spin has a relativistic origin, as realized by Dirac in where s is the angular momentum of connection with his equation for the rela- the intrinsic rotation, the spin vector of tivistic electron [46], and Pauli’s equation the electron, and the gyromagnetic ra- can be derived as the nonrelativistic limit tio µS/s is assumed to be twice as large of Dirac’s equation. as the one for the orbital motion. Tak- ing into account this new degree of free- 5.2. Spin statistics dom Heisenberg and Jordan were able to finally describe the anomalous Zeeman A derivation of Planck’s formula was effect correctly [42] including the spin- given by Satyendra Nath Bose (1894– orbit interaction due to the internal mag- 1974) in 1924 using only the corpusco- netic field felt by the electron moving lar picture without any reference to wave- in the Coulomb field of the nucleus, as theoretical concepts [47]. The paper, sent explained by Llewellyn Hilleth Thomas to Einstein with the request to sponsor its (1903–1992) [43]. In addition, the fine publication, was enthusiastically trans- structure of the one-electron atom could lated into German by Einstein himself be reproduced considering the effect of and submitted to Zeitschrift für Physik. the spin-orbit interaction together with It also inspired Einstein to give an anal- the relativistic correction of the kinetic ogous application to the theory of the 4 energy to order p . The result for an so-called degeneration of ideal gases [48], atom with atomic number Z, now known to describe the thermody- 2R2h2Z4 3 1 namical properties of a system of par- ∆E = 2 3 − 1 ticles with symmetrical wave functions mc n 4n j + 2 according to what is called the Bose- (with the Rydberg’s constant R = Einstein statistics. 14 Sigfrido Boffi

The spin hypothesis, together with many-electron atom proposed by Douglas Pauli exclusion principle, was also of Raynes Hartree (1897–1958) [54]. The great help in solving the problem of the Hartree-Fock method opened the road two-electron atom and the stability of or- towards understanding atomic structure tohelium. This is an excited state of from the basic quantum theoretical prin- the helium atom in a triplet configura- ciples [55]. tion where the spin of the two electrons The spin hypothesis together with are aligned in a symmetrical spin config- Fermi-Dirac statistics found one of its uration with total spin S = 1. With- first applications when Pauli explained out considering spin orthohelium would the paramagnetism of the electrons in a be degenerate with parahelium, where metal [56]. The general form of the con- the spins of two electrons are antipar- nection between spin and statistics was allel and live in a singlet (antisymmet- proven by Pauli some years later within ric) state with S = 0. Degeneration is the frame of quantum field theory [57]. removed and orthohelium becomes more It states that particles with integer or bound when antisymmetry of the total half-integer spin must be quantized ac- wave function under interchange of the cording to Bose-Einstein or Fermi-Dirac two electrons is taken into account [49], statistics, respectively. in agreement with the rule formulated by Friedrich Hund (1896–1997) concern- ing the lower energy of states with higher 6. Indeterminacy principle spin value [50]. In addition, for symmetry reasons ortohelium can hardly decay to By the end of 1926 and beginning of the ground (singlet) state, thus explain- 1927 a critical analysis of the formalism ing its stability. lead to the discovery of an in principle The Pauli exclusion principle was also difficulty to determine the values of all applied by Enrico Fermi (1901–1954) to the independent dynamical variables re- molecules in a quantum gas [51]. A few quired by the system degrees of freedom months later, Dirac found that symmetri- to specify its state. cal wave functions of a system of identical In classical mechanics it is assumed particles lead to the Bose-Einstein sta- that the position qr of the r-th particle tistical mechanics, and antisymmetrical in a system can be determined together wave functions satisy Pauli principle [27]. with its momentum pr at a specific in- 1 Therefore, spin- 2 particles, like electrons stant of time. Such a knowledge allows in an atom, are said to obey Fermi-Dirac one to follow the particle motion by look- statistics. ing at its trajectory trough space accord- Incidentally, the determinantal form ing to the classical equations of motion. of the antisymmetric wave function In contrast, by the end of 1926 Dirac of several independent electrons, now found that, as a consequence of the com- known as the Slater determinant [52], was mutative law of multiplication existing used for the first time just by Dirac in among q-numbers, in the quantum the- Ref. [27]. It was also adopted by Vladimir ory it is impossible to specify the value of Alexsandrovich Fock (1898–1974) [53] to any “constant of integration” by numer- refine the mean field approach to the ical values of the initial coordinates and The Rise of Quantum Mechanics 15

momenta qr0 and pr0. “One cannot an- formation can be derived from the Dirac- swer any question on the quantum theory Jordan’s transformation theory he dis- which refers to numerical values for both covered that canonically conjugate vari- qr0 and pr0. One would expect, however, ables such as the position and momen- to be able to answer questions in which tum of a particle cannot be exactly deter- only the qr0 or only the pr0 are given nu- mined simultaneously. There is rather an merical values” [22] (2). indeterminacy relation between the pre- When writing this paper Dirac was cision ∆q in position and the precision in Copenhagen where he presented his ∆p in momentum involving the Planck’s ideas in a seminar. Three months later, constant [58], i.e. Heisenberg, who attended the seminar, delivered his famous paper on the intu- h (11) ∆p ∆q ≥ . itive (anschaulich) content of kinematics 4π and mechanics (3). Inquiring what in- Relation (11) found its counterpart in a careful scrutiny of the measurement pro- (2) A similar conclusion was reached also by cess of position and momentum of an Jordan: “with a given value of q all possible val- electron that unavoidably has to involve ues of p are equally probable” [37]. physical phenomena such as Compton (3) According to Immanuel Kant (1724– 1804) in his Kritik der reinen Vernunft, the Ger- effect and wave diffraction. Therefore, man word Anschauung is the intuition, or knowl- relation (11) reflects an indeterminacy edge, that results from the immediate apprehen- principle characterizing physics. “The sion of an independently real object. Ethimo- logically, the word corresponds to English “to more accurately the position is deter- look at”, an active process to grasp the mean- mined, the less accurately the momen- ing of some observed fact. It corresponds to the tum is known and conversely” [58] (4). Latin “intuere”, therefore anschaulich is better The classical concept of trajectory with translated as “intuitive”. In contrast, the often used word “evident”, like the Latin “e-videre”, its sharp definition of position and mo- is more suited to describe a passive role of the mentum at any time becomes meaning- observer who acquires his knowledge as emerg- less, and “in the strong formulation of ing to his consciousness from the phenomenon itself. Heisenberg, who held Kant in high esteem, the causal law ‘If we know exactly the was particularly sensitive to Anschaulichkeit, a present, we can predict the future’ it is necessary property for him to describe the phys- not the conclusion but rather the premise ical world. However, for him Anschaulichkeit could not refer to the lost classical and causal space-time description. “We believe that we in- tuitively understand the physical theory when we can think qualitatively about individual ex- (4) Throughout the paper Heisenberg used perimental consequences and at the same time the word Ungenauigkeit (imprecision) rather we know that application of the theory never than Unbestimmtheit (indeterminacy) or Un- contains internal contradictions” [58]. sicherheit (uncertainty) that were later on also As a comment, we have to admit that the intu- used by him. As a matter of principle, relation itive description in quantum mechanical terms (11) states that it is impossible to simultane- is not always evident. Schrödinger, for example, ously and precisely determine position and mo- felt abgeschreckt (discouraged) by the abstract mentum. Therefore, the word ‘indeterminacy’ approach of matrix mechanics [16], whereas should be preferred. The word ‘uncertainty’, Heisenberg found Schrödinger’s approach ab- in current use in English written textbooks, re- scheulich (disgusting) and his claim about its minds us of our feeling rather than of the result Anschaulichkeit a mist (rubbish) [59]. of an observation. 16 Sigfrido Boffi

which is false. We cannot know, as a mat- belonging to α. Another measurement ter of principle, the present in all its de- of the variable B, immediately performed tails”. As a consequence, “because all ex- after the first one, would project the new periments are subject to the laws of quan- state onto an eigenstate of B. When A tum mechanics and therefore to equation and B do not commute, the new state is [(11)], the invalidity of the causal law different and the information on A is lost. is definitely established in quantum me- Contrary to classical physics the second chanics”. measurement does not enrich our infor- For a quantum particle the minimum mation on the system. indeterminacy, corresponding to equal The irreversible projection pheno- sign in (11), is gained when it is de- menon, not predictable by Schrödinger’s scribed by a wave packet with Gaussian equation, is known as the collapse, or re- form [60], where ∆q (∆p) represents the duction, of the wave packet. It was pro- width of the packet in configuration (mo- moted to an explicit postulate of the for- mentum) space. malism by von Neumann [28]. In reply to an objection raised by Ed- Analyzing a Stern-Gerlach experi- ward Uhler Condon (1902–1974) suggest- ment, Heisenberg also showed that the ing that in some cases conjugate variables precise determination of energy is higher, could be determined simultaneously [61], the larger is the time spent by the atom relation (11) was shown by Howard Percy crossing the deviating magnetic field, i.e. Robertson (1903–1961) to be a particular case of a more general relation where the (13) ∆E ∆t ∼ h, state of the system also plays a role [62] (see also [63]), i.e. so that there is also an indeterminacy principle for the conjugate variables en- 1 (12) ∆A ∆B ≥ 2 |h[A, B]i|, ergy and time, although time is keeping in quantum mechanics the same paramet- where ∆A is the variance of the distribu- ric role as in classical physics. tion of the possible values of the dynamical variable A for the state of the system, and the nonvanishing average value 7. The physical interpretation of the commutator [A, B] indicates that the two observables associated with the The mathematical framework of self-adjoint operators A and B are not quantum theory was basically fixed in compatible, i.e. cannot simultaneously 1932 [28]. However, since its first ap- be determined with any accuracy. pearance and already during its devel- According to the picture emerging opment the new formalism was exten- from the Dirac’s and Heisenberg’s papers, sively and successfully applied to atomic before measurement the system could be physics. With the discovery of the in- in any eigenstate of A, in fact in a linear determinacy principle, imposing to defi- superposition of eigenstates. After mea- nitely abandon the space-time causal de- surement, among all possible outcomes scription, it soon became clear that also one value of A is selected, say α, and the epistemological and ontological problems system is ‘projected’ onto the eigenstate had to be addressed. The Rise of Quantum Mechanics 17

In quantum theory, as in any physical tion within the theory. In the history theory, one may distinguish three com- of physics for the first time interpreta- ponents. There is the formalism with its tion has acquired particular importance primitive notions and axioms from which when dealing with quantum mechanics. one logically derives a set of formulae. In The reason is that with it the classical order to be physically meaningful these picture of a real world to be described formulae have to be correlated with ob- objectively has been entirely turned over servable phenomena by a set of rules of by the linear superposition principle and correspondence. This correlation is ul- the indeterminacy principle. Morevover, timately intended to establish physical as Heisenberg and Bohr were claiming, laws referring to both describing what is quantum theory provides us for a com- observed and predicting new data. The plete scientific account of atomic phe- formalism and the correspondence rules nomena. “The essentially new feature make quantum theory first of all a proce- in the analysis of quantum phenomena dure: it is “a procedure by which scien- is . . . the introduction of a fundamental tists predict probabilities that measure- distinction between the measuring appa- ments of specified kinds will yield results ratus and the objects under investigation of specified kinds in situations of speci- . . . While within the scope of classical fied kinds”’ [64]. Accordingly, the quan- physics the interaction between the ob- tum theoretical formalism is to be inter- ject and apparatus can be neglected or, if preted pragmatically. “The task of sci- necessary, compensated for, in quantum ence is both to extend the range of our physics this interaction thus forms an in- experience and to reduce it to order” [65]. separable part of the phenomenon” [66]. Specifically, quantum theory does not de- Therefore, the phenomenon, i.e. what scribe a system in itself, but only deals appears to us, is not merely a manifes- with the results of actual observations tation to our senses (even powered by soon it. Thus, particular attention has to phisticated instruments) of an objective be paid to the measurement process be- reality, which is absolute and indepen- cause deciding the kind of measurement dent of the observer, with its determin- already means to emphasize one of the istic laws. It is rather the encounter be- complementary aspects of the quantum tween the observed and the observer: it system. In order to determine the posi- is the result of an autonomous decision of tion and momentum of a particle, mutu- the scientist to look at one of the comple- ally exclusive experimental arrangements mentary aspects with the kind of appara- must be made use of. While the measure- tus he has chosen. Consequently, the task ment device and its operation are spec- of science is no longer to explain an ob- ified by classical physics concepts, the jective reality, but rather to reduce obser- mathematical formalism of quantum me- vations to order finding connections be- chanics offers rules to calculate expecta- tween them and predicting the outcome tions about observations. of new measurements, being aware that The third component is the interpre- single individual events are subject to ca- tation of the theory. This involves philo- suality and only a statistical prediction sophical issues, such as the questions can be made. about the physical reality and its descrip- This kind of interpretation first 18 Sigfrido Boffi

emerged with Bohr’s contributions at the “We might call modern quantum the- International Physics Congress held in ory as ‘The Theory of Complementar- Como in September 1927 and in the ity’ (in analogy with the terminology next October at the Fifth Solvay Con- ‘Theory of Relativity’)” [70]. Gradually ference in Bruxelles. It is at the heart convinced that “it must never be for- of the so-called Copenhagen interpreta- gotten that we ourselves are both action developed by Bohr with the contri- tors and spectators in the drama of exis- bution of the physicists who visited him tence” [71], Bohr applied the complemen- in Copenhagen, in particular Heisenberg tarity concept not only to physics, but and Pauli. even in the search for a harmonious atti- According to Max Jammer [67] the tude towards life. writings of Charles Renouvier (1815– The idea that the end of the story 1903), Emile´ Boutroux (1845–1921), was achieved with complementarity and Søren Kierkegaard (1813–1855), and in completeness of quantum theory was not particular of Harald Høffding (1843– entirely convincing. Most scientists, also 1931), with their giving value to the role today, do not care about the philoso- of consciousness to appreciate the differ- phy of quantum mechanics. Pragmat- ent levels of reality, seem to have been ically, they accept the statistical inter- influential in shaping Bohr’s philosophi- pretation of the formalism and apply cal background. In fact, “Bohr was pri- it to make predictions. “Questions of marily a philosopher, not a physicist, but this type appear to be the only ones to he understood that natural philosophy in which the quantum theory can give a our day and age carries weight only if its definite answer, and they are probably every detail can be subjected to the in- the only ones to which the physicist re- exorable test of experiment” [68]. And quires an answer” [22]. Therefore, “or- with the wave-particle duality of atomic dinary quantum mechanics (as far as I phenomena experiment was unavoidably know) is just fine for all practical pur- showing complementary aspects of real- poses” [72]. Others, however, like Ein- ity. Certainly, the development of the stein and Schrödinger, tried to envisage complementarity idea was tuned with the paradoxical situations [73, 74] to show Zeitgeist of that time, where in any field that quantum theory is not complete of the culture between the end of the 18th and requires further study to understand and the beginning of the 19th century the measurement process and the conse- a subjective perspective replaced the ob- quences of quantum entanglement pro- jective positivistic one, and the human duced by the linear superposition prin- expression in the arts, literature, philos- ciple. After all, also the measurement ophy, and ultimately in science became more abstract (5).

Bohr’s and Born’s communications at the Como Congress are translated and discussed. Ref. [69] (5) For the interested Italian speaking is part of a series of booklets, available at reader a presentation of the cultural environ- www.pv.infn.it/˜boﬃ/quaderni.html, where the ment in Europe by the time of the rise of quan- original papers of those who fabricated quantum tum mechanics can be found in [69], where mechanics are presented and discussed. The Rise of Quantum Mechanics 19

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