atom Schrodinger's wave equationmatter

In 1926 the Schrödinger equation, essentially a mathematical wave equation, established quantum mechanics in widely applicable form. In order to understand how a wave equation is used, it is helpful to think of an analogy with the vibrations of a bell, violin string, or drumhead. These vibrations are governed by a wave equation, since the motion can propagate as a wave from one side of the object to the other. Certain vibrations in these objects are simple modes that are easily excited and have definite frequencies. For example, the motion of the lowest vibrational mode in a drumhead is in phase all over the drumhead with a pattern that is uniform around it; the highest amplitude of the vibratory motion occurs in the middle of the drumhead. In more-complicated, higher-frequency modes, the motion on different parts of the vibrating drumhead are out of phase, with inward motion on one part at the same time that there is outward motion on another.

Schrödinger postulated that the electrons in an atom should be treated like the waves on the drumhead. The different energy levels of atoms are identified with the simple vibrational modes of the wave equation. The equation is solved to find these modes, and then the energy of an electron is obtained from the frequency of the mode and from Einstein’s quantum formula, E = hν. Schrödinger’s wave equation gives the same energies as Bohr’s original formula but with a much more-precise description of an electron in an atom (see figure). The lowest energy level of the hydrogen atom, called the ground state, is analogous to the motion in the lowest vibrational mode of the drumhead. In the atom the electron wave is uniform in all directions from the nucleus, is peaked at the centre of the atom, and has the same phase everywhere. Higher energy levels in the atom have waves that are peaked at greater distances from the nucleus. Like the vibrations in the drumhead, the waves have peaks and nodes that may form a complex shape. The different shapes of the wave pattern are related to the quantum numbers of the energy levels, including the quantum numbers for angular momentum and its orientation.

The year before Schrödinger produced his wave theory, the German physicist Werner Heisenberg published a mathematically equivalent system to describe energy levels and their transitions. In Heisenberg’s method, properties of atoms are described by arrays of numbers called matrices, which are combined with special rules of multiplication. Today physicists use both wave functions and matrices, depending on the application. Schrödinger’s picture is more useful for describing continuous electron distributions because the wave function can be more easily visualized. Matrix methods are more useful for numerical analysis calculations with computers and for systems that can be described in terms of a finite number of states, such as the spin states of the electron.

In 1929 the Norwegian physicist Egil Hylleraas applied the Schrödinger equation to the helium atom with its two electrons. He obtained only an approximate solution, but his energy calculation was quite accurate. With Hylleraas’s explanation of the two-electron atom, physicists realized that the Schrödinger equation could be a powerful mathematical tool for describing nature on the atomic level, even if exact solutions could not be obtained.