de Broglie Wave Equation

Einstein’s idea that light can exhibit both wave properties and particle properties suggested to Louis de Broglie (1892–1987) that very small particles, such as electrons, might also display wave properties under the proper circumstances. Other researchers found that wavelength associated with electrons of known energy is exactly that predicted by de Broglie.

The wave behavior of electrons is exploited in the electron microscope. This instrument allows magnification of objects far too small to be seen with an ordinary light microscope.

Heisenberg Uncertainty Principle

The Heisenberg Uncertainty Principle, stated in 1927 by Werner Heisenberg (1901–1976), is a theoretical assertion that is consistent with all experimental observations and states: 

 It is impossible to determine accurately both the momentum and the position of an electron (or any other very small particle) simultaneously. 

Because electrons are so small and move so rapidly, their motion is usually detected by electromagnetic radiation. Photons that interact with electrons have about the same energies as the electrons. Consequently, the interaction of a photon with an electron severely disturbs the motion of the electron. It is not possible to determine simultaneously both the position and the velocity of an electron, so we resort to a statistical approach and speak of the probability of finding an electron within specified regions in space.

Schrödinger Wave Equation

In 1926, an Austrian physicist, Erwin Schrödinger, used mathematics and statistics to combine: 
(i)	de Broglie’s idea of matter waves
(ii)	Einstein’s idea of quantized energy particles (photons). 
(iii)	with another idea called Heisenberg’s uncertainty principle 
resulted in the birth of the field of quantum mechanics. This is a branch of physics that uses mathematical equations to describe the wave properties of sub-microscopic particles such as electrons, atoms, and molecules. Schrödinger used concepts from quantum mechanics to propose a new atomic model: the quantum mechanical model of the atom. This model describes atoms as having certain allowed quantities of energy because of the wave-like properties of their electrons.

Schrödinger used a type of equation called a wave equation to define the probability of finding an atom’s electrons at a particular point within the atom. There are many solutions to this wave equation, and each solution represents a particular wave function. Each wave function gives information about an electron’s energy and location within an atom. Chemists call these wave functions orbitals.

Each orbital has its own associated energy, and each represents information about where, inside the atom, the electrons would spend most of their time. Scientists cannot determine the actual paths of the moving electrons. However, orbitals indicate where there is a high probability of finding electrons.

Comparison of Bohr’s Orbits with Schrödinger Orbitals
Orbits	Orbitals	
2-D path	3-D region of space	
fixed distance from the nucleus	variable distance from the nucleus	
circular or elliptical path	no path; varied shape in region	
2n2 electrons per orbit	2 electrons per orbital