The development of modern quantum mechanics had its beginning in the
early summer of 1925 when Werner Heisenberg, recuperating on the island
,f He1igoland from a heavy attack of hay fever, conceived the idea of
representing physical quantities by sets of time-dependent complex numbers.'
As Max Born soon recognized, the "sets" in terms of which Heisenberg
had solved the problem of the anharmonic oscillator were precisely
those mathematical entities whose algebraic properties had been studied by
ever since Cayley published his memoir on the theory of matrices (1858). Within a few months Heisenberg's new approach was
elaborated by Born, Jordan, and Heisenberg himself into what has become
known as matrix mechanics, the earliest consistent theory of quantum
phenomena.
At the end of January 1926 Erwin Schrodinger, at that time professor at the University of Ziirich, completed the first part of his historic paper "Quantization as an Eigenvalue Problem."~ He showed that the usual, although enigmatic, rule for quantization can be replaced by the natural requirement for the finiteness and single-valuedness of a certain space function. Six months later Schrodinger published the fourth communicatbn4 of this paper, which contained the time-dependent wave equation and time-dependent perturbation theory and various other applications of the new concepts and methods. By the end of February of that year, after having completed his second communication, Schrodinger5 discovered, to his surprise and delight, that his own formalism and Heisenberg's matrix calcu1us are mathematically equivalent in spite of the obvious disparities in their basic assumptions, mathematical apparatus, and general tenor.
Schrodinger's contention of the equivalence between the matrix and wave mechanical formalisms gained further clarification when John von Neumanq6 a few years later, showed that quantum mechanics can be formalized as a calculus of Hermitian operators in Hilbert space and that the theories of Heisenberg and Schrodinger are merely particular representations of this calculus. Heisenberg made useof the sequence space 12, the set of all infinite sequences of complex numbers whose squared absolute values yield a finite sum, whereas Schrodinger made use of the space C2 (- co, + co) of all complex-valued square-summable (Lebesgue) measurable functions; but since both spaces, I2 and C2 , are infinitedimensional realizations of the same abstract Hilbert space X, and hence isomorphic (and isometric) to each other, there exists a one-to-one correspondence, or mapping, between the "wave functions" of C2 and the "sequences" of complex numbers of 12 , between Hermitian differential operators and Hermitian matrices. Thus solving the eigenvalue problem of an operator in C2 is equivalent to diagonalizing the corresponding matrix in 12.
That a full comprehension of the situation as outlined was reached only after 1930 does not change the fact that in the summer of 1926 the mathematical formalism of quantum mechanics reached its essential completion. Its correctness, in all probability, seemed to have been assured by its spectacular successes in accounting for practically all known spectroscopic phen~mena,~ with the inclusion of the Stark and Zeeman effects, by its explanation, on the basis of Born's probability interpretation, of a multitude of scattering phenomena as well as the photoelectric effect. If we recall that by generalizing the work of Heisenberg and Schrodinger Dirac soon afterward, in his theory of the electron,* accounted for the spin whose existence had been discovered in 1925, and that the combination of these ideas with Pauli's exclusion principle gave a convincing account of the system of the elements, we will understand that the formalism established in 1926 was truly a major breakthrough in the development of modern physics.
To continue reading check the book at the page 21 in the following link:
https://philosophy-of-sciences.blogspot.com/2018/12/the-philosophy-of-quantum-mechanics.html
At the end of January 1926 Erwin Schrodinger, at that time professor at the University of Ziirich, completed the first part of his historic paper "Quantization as an Eigenvalue Problem."~ He showed that the usual, although enigmatic, rule for quantization can be replaced by the natural requirement for the finiteness and single-valuedness of a certain space function. Six months later Schrodinger published the fourth communicatbn4 of this paper, which contained the time-dependent wave equation and time-dependent perturbation theory and various other applications of the new concepts and methods. By the end of February of that year, after having completed his second communication, Schrodinger5 discovered, to his surprise and delight, that his own formalism and Heisenberg's matrix calcu1us are mathematically equivalent in spite of the obvious disparities in their basic assumptions, mathematical apparatus, and general tenor.
Schrodinger's contention of the equivalence between the matrix and wave mechanical formalisms gained further clarification when John von Neumanq6 a few years later, showed that quantum mechanics can be formalized as a calculus of Hermitian operators in Hilbert space and that the theories of Heisenberg and Schrodinger are merely particular representations of this calculus. Heisenberg made useof the sequence space 12, the set of all infinite sequences of complex numbers whose squared absolute values yield a finite sum, whereas Schrodinger made use of the space C2 (- co, + co) of all complex-valued square-summable (Lebesgue) measurable functions; but since both spaces, I2 and C2 , are infinitedimensional realizations of the same abstract Hilbert space X, and hence isomorphic (and isometric) to each other, there exists a one-to-one correspondence, or mapping, between the "wave functions" of C2 and the "sequences" of complex numbers of 12 , between Hermitian differential operators and Hermitian matrices. Thus solving the eigenvalue problem of an operator in C2 is equivalent to diagonalizing the corresponding matrix in 12.
That a full comprehension of the situation as outlined was reached only after 1930 does not change the fact that in the summer of 1926 the mathematical formalism of quantum mechanics reached its essential completion. Its correctness, in all probability, seemed to have been assured by its spectacular successes in accounting for practically all known spectroscopic phen~mena,~ with the inclusion of the Stark and Zeeman effects, by its explanation, on the basis of Born's probability interpretation, of a multitude of scattering phenomena as well as the photoelectric effect. If we recall that by generalizing the work of Heisenberg and Schrodinger Dirac soon afterward, in his theory of the electron,* accounted for the spin whose existence had been discovered in 1925, and that the combination of these ideas with Pauli's exclusion principle gave a convincing account of the system of the elements, we will understand that the formalism established in 1926 was truly a major breakthrough in the development of modern physics.
To continue reading check the book at the page 21 in the following link:
https://philosophy-of-sciences.blogspot.com/2018/12/the-philosophy-of-quantum-mechanics.html
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