Heisenberg sets out this view as follows (5, p. 685). In the classical theory, given an electron with one degree of freedom, in harmonic oscillation, the elongation at time can be represented by a Fourier series: where is a constant and is the number of the harmonic. The single terms of this series, namely: would contain the quantities which have been signalized as directly observable—namely, frequency, amplitude, and phase.[Pg 44] But in virtue of the fact that, in atoms, frequencies are found to be the differences of "terms" we shall have to replace the above by: and the collection (not the sum) of such terms represents what was formerly the elongation . The sum of all these terms has no longer any physical significance. Thus the atom comes to be represented by the numbers , arranged in an infinite rectangle or "matrix." It is possible to construct an algebra of matrices, which differs formally from ordinary algebra in only one respect, namely, that multiplication is not commutative. A new operation is defined which, when the quantum numbers become large, approximates to differentiation. By using this operation, Hamilton's equations of motion can be preserved in a form which is applicable equally to periodic and to unperiodic motions, so that it is no longer necessary to distinguish a certain sphere of quantum phenomena, to which different laws are applied from those applied to the phenomena amenable to classical dynamics: "A distinction between 'quantized' and 'unquantized' motions loses all meaning in this theory, since in it there is no question of a quantum condition which selects certain motions from a great number of possible ones; in place of this condition appears a quantum-mechanical fundamental equation ... which is valid for all possible motions, and is necessary in order to give a definite meaning to the problem of motion" (3, p. 558). The fundamental equation alluded to in the above is as follows: Let be a Hamiltonian co-ordinate, and the corresponding (generalized) momentum, both being matrices. It will be remembered that multiplication is not commutative for matrices; in fact, we have as the fundamental equation in question (2, p. 871): [Pg 45] where represents the matrix whose diagonal consists of 's, and whose other terms are all zero. The above is the sole fundamental equation containing (Planck's constant), and it is true for all motions. Heisenberg does not claim that the new theory solves all difficulties. On the contrary, he says (5, p. 705):