Shortly after de Broglie published his ideas that the electron in a hydrogen atom could be better thought of as being a circular standing wave instead of a particle moving in quantized circular orbits, Erwin Schrödinger extended de Broglie’s work by deriving what is now known as the Schrödinger equation. When Schrödinger applied his equation to hydrogen-like atoms, he was able to reproduce Bohr’s expression for the energy and, thus, the Rydberg formula governing hydrogen spectra.

In a beam of charged particles created by a heated cathode, the particles move at different speeds. However, many applications need a beam with uniform particle speeds. An arrangement known as a velocity selector uses electric and magnetic fields to pick particles with a particular speed from the beam.
A particle with charge q, speed v, and mass m enters an area from the top, where the magnetic and electric fields are perpendicular both to the particle's motion and to one another. The magnetic...

Overview
Electrons are negatively charged subatomic particles that are attracted to an orbit around the positively-charged nucleus of an atom. They reside in locations that are associated with energy levels called shells and are further organized into sub-shells and orbitals within each shell.
Electrons Orbit the Nucleus
Electrons are found in specific locations outside of the nucleus. The shell in which an electron resides indicates the general energy level of the electron: those closer to the...

Electrons are negatively charged subatomic particles attracted to and orbit around the positively-charged nucleus of an atom. They reside in spaces associated with energy levels called shells and are further organized into subshells and orbitals within each shell.
Electrons Orbit the Nucleus
Electrons are found in specific locations outside of the nucleus. The shell in which an electron resides indicates the general energy level of the electron: those closer to the nucleus have less energy,...

Werner Heisenberg considered the limits of how accurately one can measure properties of an electron or other microscopic particles. He determined that there is a fundamental limit to how accurately one can measure both a particle’s position and its momentum simultaneously. The more accurate the measurement of the momentum of a particle is known, the less accurate the position at that time is known and vice versa. This is what is now called the Heisenberg uncertainty principle. He...

An applied magnetic field causes loosely bound π-electrons in organic molecules to circulate, producing a local or induced diamagnetic field over a large spatial volume. As the molecules tumble in solution, the field generated by π-electrons in spherical substituents results in a zero net field. However, the net field generated by π-electrons in non-spherical substituents is not zero. The effect of this induced field depends on the orientation of the molecule with respect to B0,...