The 1925 paper “On quantum mechanics” by M. Born and P. Jordan, and the sequel “On quantum mechanics II” by M. Born, W. Heisenberg, and P. Jordan, developed Heisenberg’s pioneering theory into the first complete formulation of quantum mechanics. The Born and Jordan paper is the subject of the present article. This paper introduced matrices to physicists. We discuss the original postulates of quantum mechanics, present the two-part discovery of the law of commutation, and clarify the origin of Heisenberg’s equation. We show how the 1925 proof of energy conservation and Bohr’s frequency condition served as the gold standard with which to measure the validity of the new quantum mechanics.

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The name “matrix mechanics” did not appear in the original papers of 1925 and 1926. The new mechanics was most often called “quantum mechanics.” At Göttingen, some began to call it “matrix physics.” Heisenberg disliked this terminology and tried to eliminate the mathematical term “matrix” from the subject in favor of the physical expression “quantum-theoretical magnitude.” [Ref. 17, p.

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In 1928 Einstein nominated Heisenberg, Born, and Jordan for the Nobel Prize [See

A.

Pais

, Subtle Is the Lord: The Science and the Life of Albert Einstein

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, New York

, 1982

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]. Possible explanations of why Born and Jordan did not receive the Nobel Prize are given in Ref. 10 and Ref. 22, pp. 191

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Liboff

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Shankar

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Cohen-Tannoudji

, B.

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, Quantum Mechanics

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In Heisenberg’s paper (see Ref. 2) the connection between $\u2223q(nm)\u22232$ and transition probability is implied but not discussed. See Ref. 3, pp.

30

–32

, for a discussion of Heisenberg’s assertion that the transition amplitudes determine the transition probabilibities. The relation between a squared amplitude and a transition probability originated with Bohr who conjectured that the squared Fourier amplitude of the classical electron motion provides a measure of the transition probability (see Refs. 8 and 26). The correspondence between classical intensities and quantum probabilities was studied by several physicists including H.

Kramers

, Intensities of Spectral Lines

(A. F. Host and Sons

, Kobenhaven, 1919

);R.

Ladenburg

in Ref. 3, paper 4, and J. H.

Van Vleck

, “Quantum principles and line spectra

,” Bulletin of the National Research Council

, Washington, DC, 1926

, pp. 118

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Reference 3, p.

287

, paper 13.40.

W.

Ritz

, “On a new law of series spectra

,” Astrophys. J.

28

, 237

–243

(1908

). The Ritz combination principle was crucial in making sense of the regularities in the line spectra of atoms. It was a key principle that guided Bohr in constructing a quantum theory of line spectra. Observations of spectral lines revealed that pairs of line frequencies combine (add) to give the frequency of another line in the spectrum. The Ritz combination rule is $\nu (nk)+\nu (km)=\nu (nm)$, which follows from Eqs. (4) and (7). As a universal, exact law of spectroscopy, the Ritz rule provided a powerful tool to analyze spectra and to discover new lines. Given the measured frequencies $\nu 1$ and $\nu 2$ of two known lines in a spectrum, the Ritz rule told spectroscopists to look for new lines at the frequencies $\nu 1+\nu 2$ or $\nu 1\u2212\nu 2$.41.

In the letter dated 18 September 1925 Heisenberg explained to Pauli that the frequencies $\nu ik$ in the Born–Jordan theory obey the “

combination relation $\nu ik+\nu kl=\nu il$ or $\nu ik=(Wi\u2212Wk)\u2215h$ but naturally it is not to be assumed that $W$ is the energy

.” See Ref. 3, p. 45

.42.

Reference 3, p.

287

, paper 13.43.

Reference 3, p.

289

, paper 13.44.

Born and Jordan devote a large portion of Chap. 1 to developing a matrix calculus to give meaning to matrix derivatives such as $dq\u2215dt$ and $\u2202H\u2215\u2202p$. They introduce the process of “

symbolic differentiation

” for constructing the derivative of a matrix with respect to another matrix. For a discussion of Born and Jordan’s matrix calculus, see Ref. 13, pp. 68

–71

. To deal with arbitrary Hamiltonian functions, Born and Jordan formulated a more general dynamical law by converting the classical action principal, $\u222bLdt=extremum$, into a quantal action principal, $D(pq\u0307\u2212H(pq))=extremum$, where $D$ denotes the trace (diagonal sum) of the Lagrangian matrix, $pq\u0307\u2212H$. See Ref. 3, pp. 289

–290

.46.

In contemporary language the states labeled $n=0,1,2,3,\u2026$ in Heisenberg’s paper and the Born–Jordan paper are exact stationary states (eigenstates of $H$). The Hamiltonian matrix is automatically a diagonal matrix with respect to this basis.

47.

Although Heisenberg, Born, and Jordan made the “

energy of the state

” and the “transition between states

” rigorous concepts, it was Schrödinger who formalized the concept of the “state” itself. It is interesting to note that “On quantum mechanics II” by Born, Heisenberg, and Jordan was published before Schrödinger and implicitly contains the first mathematical notion of a quantum state. In this paper (Ref. 3, pp. 348

–353

), each Hermitian matrix $a$ is associated with a “bilinear form” $\u2211nma(nm)xnxm*$. Furthermore, they identified the “energy spectrum” of a system with the set of “eigenvalues” $W$ in the equation $Wxk\u2212\u2211lH(kl)xl=0$. In present-day symbolic language the bilinear form and eigenvalue problem are $\u27e8\Psi \u2223a\u2223\Psi \u27e9$ and $H\u2223\Psi \u27e9=W\u2223\Psi \u27e9$, respectively, where the variables $xn$ are the expansion coefficients of the quantum state $\u2223\Psi \u27e9$. At the time, they did not realize the physical significance of their eigenvector $(x1,x2,\u2026)$ as representing a stationary state.48.

51.

52.

The exact relation between the orbital frequency and the optical frequency is derived as follows. Consider the transition from state $n$ of energy $E(n)$ to state $n\u2212\tau $ of energy $E(n\u2212\tau )$. In the limit $n\u2aa2\tau $, that is, large “orbit” and small “jump,” the difference $E(n)\u2212E(n\u2212\tau )$ is equal to the derivative $\tau dE\u2215dn$. Given the old quantum condition $J=nh$, it follows that $dE\u2215dn=hdE\u2215dJ$. Thus for $n\u2aa2\tau $ and $J=nh$, we have the relation $[E(n)\u2212E(n\u2212\tau )]\u2215h=\tau dE\u2215dJ$, or equivalently, $\nu (n,n\u2212\tau )=\tau \nu (n)$. This relation proves an important correspondence theorem: In the limit $n\u2aa2\tau $, the frequency $\nu (n,n\u2212\tau )$ associated with the quantum jump $n\u2192n\u2212\tau $ is equal to the frequency $\tau \nu (n)$ associated with the $\tau th$ harmonic of the classical motion in the state $n$. See Refs. 8 and 26.

53.

Reference 3, p.

191

, paper 7.54.

Suppose that the number of states is finite and equal to the integer $N$. Then, according to Eq. (20), the diagonal sum (trace) of $pq\u2212qp$ would be $D(pq\u2212qp)=Nh\u22152\pi i$. This nonzero value of the trace contradicts the purely mathematical relation $D(pq\u2212qp)=0$, which must be obeyed by all finite matrices.

55.

Heisenberg interview quoted in Ref. 12, p.

281

, footnote 45.56.

Reference 17, p.

361

.57.

Reference 3, p.

288

, paper 13. The name “Diagonality theorem” is ours. The condition $\nu (nm)\u22600$ when $n\u2260m$ implies that the system is nondegenerate.58.

In contemporary language a conserved quantity is an operator that commutes with the Hamiltonian operator $H$. For such commuting operators there exists a common set of eigenvectors. In the energy eigenbasis that underlies the Born–Jordan formulation, the matrices representing $H$ and all conserved quantities are automatically diagonal.

59.

Born and Jordan’s proof that Eq. (24) vanishes is based on a purely mathematical property of “symbolic differentiation” discussed in Sec. II of their paper (See Ref. 4). For a separable Hamiltonian of the form $H=p2\u22152m+U(q)$, the proof is simpler. For this case Eq. (24) becomes $d\u0307=q(\u2202U\u2215\u2202q)\u2212(\u2202U\u2215\u2202q)q+p(p\u2215m)\u2212(p\u2215m)p$. Because $p$ and $q$ are separated in this expression, we do not have to consider the inequality $pq\u2260qp$. The expression reduces to $d\u0307=0$.

60.

Reference 3, p.

292

, paper 13. In Ref. 4, Born and Jordan refer to $pq\u2212qp=(h\u22152\pi i)1$ as the “vershärfte Quantenbedingung,” which has been translated as “sharpened quantum condition” (Ref. 13, p. 77

), “stronger quantum condition” (Ref. 3, p. 292

), and “exact quantum condition” (Ref. 12, p. 220

).61.

J. J.

Sakurai

, Modern Quantum Mechanics

(Addison-Wesley

, San Francisco

, 1994

), pp. 83

–84

;62.

Reference 3, p.

293

, paper 13.63.

Proving the frequency condition—the second general principle of Bohr—was especially important because this purely quantal condition was generally regarded as a safely established part of physics. Prior to Born and Jordan’s mechanical proof of the frequency condition, there existed a “thermal proof” given by Einstein in his historic paper, “

On the quantum theory of radiation

,” Phys. Z.

18

, 121

(1917

), translated in Ref. 3, pp, 63–77. In this paper Einstein provides a completely new derivation of Planck’s thermal radiation law by introducing the notion of transition probabilities (A and B coefficients). Bohr’s frequency condition emerges as the condition necessary to reduce the Boltzmann factor $exp[(En\u2212Em)\u2215kT]$ in Einstein’s formula to the “Wien factor” $exp(h\nu \u2215kT)$ in Planck’s formula.64.

Reference 3, p.

291

, paper 13.65.

Reference 3, pp.

291

–292

, paper 13. Born and Jordan do not refer to the consequences in Eqs. (37) and (38) as theorems. The label “Energy theorems” is ours.66.

Instead of postulating the equations of motion and deriving the energy theorems, we could invert the proof and postulate the energy theorems and derive the equations of motion. This alternate logic is mentioned in Ref. 3, p.

296

Also see

J. H.

Van Vleck

, “Note on the postulates of the matrix quantum dynamics

,” Proc. Natl. Acad. Sci. U.S.A.

12

, 385

–388

(1926

).
[PubMed]

67.

Reference 3, pp.

293

–294

, paper 13. The proof of the energy theorems was based on separable Hamiltonians defined in Eq. (29). To generalize the proof Born and Jordan consider more general Hamiltonian functions $H(pq)$ and discover the need to symmetrize the functions. For example, for $H*=p2q$, it does not follow that $H\u0307*=0$. However, they note that $H=(p2q+qp2)\u22152$ yields the same equations of motion as $H*$ and also conserves energy, $H\u0307=0$. The symmetrization rule reflects the noncommutativity of $p$ and $q$.68.

In the Heisenberg, Born–Jordan approach the transition components of the “matter variables” $q$ and $p$ are simply assumed to oscillate in time with the radiation frequencies. In contemporary texts a rigorous proof of Bohr’s frequency condition involves an analysis of the interaction between matter and radiation (radiative transitions) using time-dependent perturbation theory. See Ref. 24, Chap. 19.

69.

Reference 3, p.

292

, paper 13.70.

Using the language of state vectors and bra-kets, the matrix element of an operator $g$ is $gnm(t)=\u27e8\Psi n(t)\u2223g\u2223\Psi m(t)\u27e9$, where the energy eigenstate is $\u2223\Psi n(t)\u27e9=exp(\u22122\pi iEnt\u2215h)\u2223\Psi n(0)\u27e9$. This Schrödinger element is equivalent to the Born–Jordan element in Eq. (44).

71.

Reference 3, p.

279

, paper 13. Heisenberg was able to demonstrate energy conservation and Bohr’s frequency condition for two systems (anharmonic oscillator and rotator). The anharmonic oscillator analysis was limited to second-order perturbation theory.72.

Reference 3. Born and Jordan do not pursue this direct method of proof noting that for the most general Hamiltonians the calculation “becomes so exceedingly involved that it seems hardly feasible.” (Ref. 3, p.

296

). In a footnote on p. 296, they note that for the special case $H=p2\u22152m+U(q)$, the proof can be carried out immediately. The details of this proof can be found in Ref. 73.73.

J. J.

Prentis

and W. A.

Fedak

, “Energy conservation in quantum mechanics

,” Am. J. Phys.

72

, 580

–590

(2004

).74.

Reference 3, p.

296

, paper 13.75.

76.

77.

M.

Born

, “Zur Quantenmechanik der Stoßvorgänge

,” Z. Phys.

37

, 863

–867

(1926

).78.

Heisenberg’s “classical” quantity $x(n,t)$ is the classical solution $x(t)$ of Newton’s equation of motion subject to the old quantum condition $\u222emx\u0307dx=nh$. For example, given the purely classical position function $x(t)=acos\omega t$ of a harmonic oscillator, the condition $\u222emx\u03072dt=nh$ quantizes the amplitude, making $a$ depend on $n$ as follows: $a(n)=nh\u2215\pi m\omega $. Thus, the motion of the harmonic oscillator in the stationary state $n$ is described by $x(n,t)=nh\u2215\pi m\omega cos\omega t$.

79.

The introduction of transition components $a(n,n\u2212\tau )ei\omega (n,n\u2212\alpha )t$ into the formalism was a milestone in the development of quantum theory. The one-line abstract of Heisenberg’s paper reads “The present paper seeks to establish a basis for theoretical quantum mechanics founded exclusively upon relationships between quantities which in principle are observable” (Ref. 3, p.

261

). For Heisenberg, the observable quantities were $a(n,n\u2212\tau )$ and $\omega (n,n\u2212\tau )$, that is, the amplitudes and the frequencies of the spectral lines. Prior to 1925, little was known about transition amplitudes. There was a sense that Einstein’s transition probabilities were related to the squares of the transition amplitudes. Heisenberg made the transition amplitudes (and frequencies) the central quantities of his theory. He discovered how to manipulate them, relate them, and calculate their values.80.

81.

Heisenberg notes (Ref. 3, p.

268

, paper 12) that Eq. (A11) is equivalent to the sum rule of Kuhn and Thomas (Ref. 3, paper 11). For a discussion of Heisenberg’s development of the quantum condition, see Mehra

and H.

Rechenberg

, The Historical Development of Quantum Theory

(Springer

, New York

, 1982

), Vol. 2

, pp. 243

–245

, and Ref. 14.© 2009 American Association of Physics Teachers.

2009

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