Bohrs model of the atom was essentially a planetary one, with the electrons orbiting around the nuclear "sun". However, the uncertainty principle states that an electron cannot simultaneously have an exact location and velocity in the way that a planet does. Instead of classical orbits, electrons are said to inhabit atomic orbitals'. An orbital is the "cloud" of possible locations in which an electron might be found, a distribution of probabilities rather than a precise location. Each orbital is three dimensional, rather than the two-dimensional orbit, and is often depicted as a three-dimensional region within which there is a 95 percent probability of finding the electron.
Schrödinger was able to calculate the energy levels of hydrogen by treating a hydrogen atoms electron as a wave, represented by the "wave function" , in an electric potential well, , created by the proton. The solutions to Schrödingers equation  are distributions of probabilities for electron positions and locations. Orbitals have a range of different shapes in three dimensions. The energies of the different orbitals can be calculated, and they accurately match the energy levels of the Bohr model.
Within Schrödinger's picture, each electron has four properties:
# An "orbital" designation, indicating whether the particle-wave is one that is closer to the nucleus with less energy or one that is farther from the nucleus with more energy;
# The "shape" of the orbital, spherical or otherwise;
# The "inclination" of the orbital, determining the magnetic moment of the orbital around the -axis.
# The "spin" of the electron.
The collective name for these properties is the quantum state of the electron. The quantum state can be described by giving a number to each of these properties; these are known as the electron's quantum numbers. The quantum state of the electron is described by its wave function. The Pauli exclusion principle demands that no two electrons within an atom may have the same values of all four numbers. 
The first property describing the orbital is the principal quantum number, , which is the same as in Bohr's model.  denotes the energy level of each orbital. The possible values for  are integers:
The next quantum number, the azimuthal quantum number, denoted , describes the shape of the orbital. The shape is a consequence of the angular momentum of the orbital. The angular momentum represents the resistance of a spinning object to speeding up or slowing down under the influence of external force. The azimuthal quantum number represents the orbital angular momentum of an electron around its nucleus. The possible values for  are integers from 0 to  (where  is the principal quantum number of the electron):
The shape of each orbital is usually referred to by a letter, rather than by its azimuthal quantum number. The first shape (=0) is denoted by the letter  (a mnemonic being "sphere"). The next shape is denoted by the letter  and has the form of a dumbbell. The other orbitals have more complicated shapes (see atomic orbital), and are denoted by the letters , , , etc.
The third quantum number, the magnetic quantum number, describes the magnetic moment of the electron, and is denoted by  (or simply m). The possible values for  are integers from  to  (where  is the azimuthal quantum number of the electron):
The magnetic quantum number measures the component of the angular momentum in a particular direction. The choice of direction is arbitrary; conventionally the z-direction is chosen.
The fourth quantum number, the spin quantum number (pertaining to the "orientation" of the electron's spin) is denoted , with values + or −.
The chemist Linus Pauling wrote, by way of example:
It is the underlying structure and symmetry of atomic orbitals, and the way that electrons fill them, that leads to the organization of the periodic table. The way the atomic orbitals on different atoms combine to form molecular orbitals determines the structure and strength of chemical bonds between atoms.
The field of quantum chemistry was pioneered by physicists Walter Heitler and Fritz London, who published a study of the covalent bond of the hydrogen molecule in 1927. Quantum chemistry was subsequently developed by a large number of workers, including the American theoretical chemist Linus Pauling at Caltech, and John C. Slater into various theories such as Molecular Orbital Theory or Valence Theory.