with respect to the magnetic field. In 1925, Werner Heisenberg published the first consistent mathematical formulation of quantum mechanics (matrix mechanics). One year earlier, Louis de Broglie had proposed the de Broglie hypothesis: that all particles behave like waves to some extent, and in 1926 Erwin Schrödinger used this idea to develop the Schrödinger equation, a mathematical model of the atom (wave mechanics) that described the electrons as three-dimensional waveforms rather than point particles. A consequence of using waveforms to describe particles is that it is mathematically impossible to obtain precise values for both the position and momentum of a particle at a given point in time. This became known as the uncertainty principle, formulated by Werner Heisenberg in 1927. In this concept, for a given accuracy in measuring a position one could only obtain a range of probable values for momentum, and vice versa. This model was able to explain observations of atomic behavior that previous models could not, such as certain structural and spectral patterns of atoms larger than hydrogen. Thus, the planetary model of the atom was discarded in favor of one that described atomic orbital zones around the nucleus where a given electron is most likely to be observed. Discovery of the neutron The development of the mass spectrometer allowed the mass of atoms to be measured with increased accuracy. The device uses a magnet to bend the trajectory of a beam of ions, and the amount of deflection is determined by the ratio of an atom's mass to its charge. The chemist Francis William Aston used this instrument to show that isotopes had different masses. The atomic mass of these isotopes varied by integer amounts, called the whole number rule. The explanation for these different isotopes awaited the discovery of the neutron, an uncharged particle with a mass similar to the proton, by the physicist James Chadwick in 1932. Isotopes were then explained as elements with the same number of protons, but different numbers of neutrons within the nucleus. Fission, high-energy physics and condensed matter In 1938, the German chemist Otto Hahn, a student of Rutherford, directed neutrons onto uranium atoms expecting to get transuranium elements. Instead, his chemical experiments showed barium as a product. A year later, Lise Meitner and her nephew Otto Frisch verified that Hahn's result were the first experimental nuclear fission. In 1944, Hahn received the Nobel Prize in Chemistry. Despite