Matrix mechanics
In 1925 W. Heisenberg introduced matrix mechanics. His work was based on the correspondence principle of Bohr, which can be formulated as follows: In the limit of the quantum numbers approaching infinity the result of quantum theory should agree with that of the classical theory.
Heisenberg considered the Bohr atom at very large orbits. There the orbital frequency would be the radiation frequency and the atom would be like a simple linear oscillator. From the largest orbit, where he could get answers from classical theory, he then tried to extrapolate to the inner orbits. Heisenberg found that in his atomic theory pq does not equal qp, where p represents the position of a particle and q the momentum. He postulated that pq - qp = h / 2pi. M. Born pointed out that Heisenberg's strange multiplication law could be understood in terms of matrix multiplication. Together with P. Jordan he transposed Heisenberg's theory into a systematic matrix language. In 1926 W. Pauli applied the Heisenberg theory to the hydrogen atom problem and did not only deduce the spectrum of hydrogen but also the additional lines produced by electric and magnetic fields.
Wave mechanics
In 1926 E. Schrödinger introduced the equation obeyed by the de Broglie waves, and he demonstrated that the quantization conditions emerge from the solution of the eigenvalue problem for his wave equation. He applied his equation to the hydrogen atom and he found that both the quantization of angular momentum and the quantization of energy emerge from his equation. The Schrödinger equation describes the behavior of quantum particles by means of waves and thereby reconciles, in a consistent manner, the wave / particle duality.
