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Nobel prize to E. Schrödinger awarded in 1933. Co-winner P. A. M. Dirac "for the discovery of new productive forms of atomic theory'' |  |
SCHRÖDINGER 1926C
Schrödinger, E.;
Über das Verhältnis der Heisenberg Born Jordanischen Quantenmechanik zu der meinen / On the Relation Between the Quantum Mechanics of Heisenberg, Born, and Jordan, and that of Schrödinger
Annalen der Physik. Leipzig 79 (1926) 734;
Reprinted in
(translation into English) Collected Papers on Wave Mechanics by E. Schrödinger, Glazgow (1928) 45.
Abhandlungen zur Wellenmechanik, E. Schrödinger, (1927d) 62.
Gesammelte Abhandlungen / Collected Works, E. Schrödinger (1984c) 98.
Introduction
Considering the extraordinary differences between the starting-points and the concepts of Heisenberg's quantum mechanics and of the theory which has been designated "undulatory'' or "physical'' mechanics, and has lately been described here, it is very strange that these two new theories agree with one another with regard to the known facts, where they differ from the old quantum theory. I refer, in particular, to the peculiar "half-integralness'' which arises in connection with the oscillator
and the rotator. That is really very remarkable, because starting-points, presentations, methods, and in fact the whole mathematical apparatus, seem fundamentally different. Above all, however, the departure from classical mechanics in the two theories seems to occur in diametrically opposed directions. In Heisenberg's work the classical continuous variables are replaced by systems of discrete numerical quantities (matrices), which depend on a pair of integral indices, and are defined by algebraic
equations. The authors themselves describe the theory as a "true theory of a discontinuum''. On the other hand, wave mechanics shows just the reverse tendency; it is a step from classical point-mechanics towards a continuum-theory. In place of a process described in terms of a finite number of dependent variables occurring in a finite number of total differential equations, we have a continuous field-like process in configuration space, which is governed by a single partial
differential equation, derived from a principle of action. This principle and this differential equation replace the equations of motion and the quantum conditions of the older "classical quantum theory''.
In what follows the very intimate inner connection between Heisenberg's quantum mechanics and my wave mechanics will be disclosed. From the formal mathematical standpoint, one might well speak of the identity of the two theories. The train of thought in the proof
is as follows.
Heisenberg's theory connects the solution of a problem in quantum mechanics with the solution of a system of an infinite number of algebraic equations, in which the unknowns-infinite matrices-are allied to the classical position- and momentum-co-ordinates of the mechanical system, and functions of these, and obey peculiar calculating rules. (The relation is this: to one position-, one momentum-co-ordinate, or to one function of these corresponds
always one infinite matrix.)
I will first show (S S 2 and 3) how to each function of the position- and momentum-co-ordinates there may be related a matrix in such a manner, that these matrices, in every case, satisfy the formal calculating rules of Born and Heisenberg (among which I also reckon the so-called "quantum condition'' or "interchange rule''; see below). This relation of matrices to functions is general; it takes no account of the special mechanical
system considered, but is the same for all mechanical systems. (In other words: the particular Hamilton function does not enter into the connecting law.) However, the relation is still indefinite of an arbitrary complete orthogonal system of functions having for domain entire configuration space (N.B.-not "pq-space'', but " q-space''). The provisional indefiniteness of the relation lies in the fact that we can assign the auxiliary rôle to an
arbitrary orthogonal system.
After matrices are thus constructed in a very general way, so as to satisfy the general rules, I will show the following in S 4. The special system of algebraic equations, which, a special case, connects the matrices of the position and impulse co-ordinates with the matrix of the Hamilton function, and which the authors call "equations of motion'', will be completely solved by assigning the auxiliary rôle to a definite
orthogonal system, namely, to the system of proper functions of that partial differential equation which forms the basis of my wave mechanics. The solution of the natural boundary-value problem of this differential equation is completely equivalent to the solution of Heisenberg's algebraic problem. All Heisenberg's matrix elements, which may interest us from the surmise that they define "transition probabilities'' or "line intensities'', can be actually evaluated by differentiation
and quadrature, as soon as the boundary-value problem is solved. Moreover, in wave mechanics, these matrix elements, or quantities that are closely related to them, have the perfectly clear significance of amplitudes of the partial oscillations of the atom's electric moment. The intensity and polarization of the emitted light is thus intelligible on the basis of the Maxwell-Lorentz theory. A short preliminary sketch of this relationship is given in S 5.
Related references
More (earlier) information appears in
E. Schrödinger, Annalen der Physik. Leipzig 79 (1926) 489;
E. Schrödinger, Annalen der Physik. Leipzig 79 (1926) 361;
See also
Courant-Gilbert, Methods of Mathematical Physics I. Berlin, Springer (1924) 36;
P. A. M. Dirac, Proc. Roy. Soc. A110 (1926) 561;
P. A. M. Dirac, Proc. Roy. Soc. A109 (1925) 642;
K. Lanczos, Z. Phys. 35 (1926) 812;
Analyse data from
A. Einstein, Sitzber. Pr. Akad. Wiss. 23 (1925) 9;
M. Born, W. Heisenberg, and P. Jordan, Z. Phys. 35 (1926) 557;
M. Born and P. Jordan, Z. Phys. 34 (1925) 858;
W. Heisenberg, Z. Phys. 33 (1925) 879;
L. de Broglie, Ann. de Physique 3 (1925) 22;
