Metals are crystalline solids. They typically have a fairly straightforward crystal structure that is characterized by dense atom packing and a high level of symmetry. Metal atoms often have fewer than half of their total number of electrons in their outermost shell. This property makes metals less likely to combine to form compounds. However, non – metals (such as oxygen and sulfur), which typically contain more than half the maximum amount of valence electrons, interact with them more easily. The chemical reactivity of different metals varies greatly. Aluminum is strong and lightweight, and is used in many applications ranging from construction to cookware. Aluminum is often alloyed with other metals to improve its strength and mechanical properties. One particularly useful alloy is duralumin, an alloy of aluminum with about 4% copper, and small amounts of magnesium, manganese, iron, and silicon. Aluminum and its alloys are used in aircraft and automobile frames, rockets, window frames, doors, siding, kitchenware, packaging (aluminum cans, aluminum foil), electrical wiring (although aluminum has a higher resistance than copper, it is cheaper), as a silvering agent in paints (usually in powdered form), in heat sinks, and many other uses [1].
Gallium is a soft, silvery-white metal with a low melting point (29.8ºC); unlike most metals, it expands when cooled. It is found in the Earth's crust at a concentration of 18 ppm, making it the 34th most abundant element. It is found in trace amounts in bauxite, coal, diaspore, germanite, and sphalerite[2]. The existence of gallium was predicted by Dimitri Mendeleev in 1869 from a blank space in his periodic table beneath aluminum.
The theory of quantum mechanics is arguably one of the greatest discoveries of physics; it revolutionized our understanding of molecules, atoms, radiation, and the world of the sub-atomic particles[3]. The study of matter and its interactions with energy on the scale of atomic and subatomic particles is known as quantum mechanics. Classical physics, on the other hand, only explains matter and energy on a scale familiar to human experience, including the behavior of astronomical bodies such as the moon. Much of modern science and technology is still based on classical physics. Toward the end of the nineteenth century, however, scientists discovered phenomena in both the large (macro) and small (micro) worlds that classical physics could not explain. The basic equation that was develop to suit the purpose of understanding the atomic scale of particles wave mechanics was the Schrödinger wave equation[4].
Schrodinger wave equation is a mathematical expression describing the energy and position of the electron in space and time, taking into account the matter wave nature of the electron inside an atom. It is based on three considerations. They are; Classical plane wave equation, Broglie’s Hypothesis of matter-wave, and Conservation of Energy.
The Schrödinger equation, or Schrödinger's wave equation, is a partial differential equation that utilizes the wave function to describe the behavior of quantum mechanical systems[5]. The Schrödinger equation can be used to determine the trajectory, location, and energy of these systems. The time-dependent Schrödinger equation in three dimensions (for a non-relativistic particle) is:
1
the basic form of Schrödinger's wave equation is as follows:
2
Ψ is called a wave function; H, with, hat, on top is known as the Hamiltonian operator; and E is the binding energy of the electron. Solving Schrödinger's equation yields multiple wave functions as solutions, each with an allowed value for E. The Schrodinger equation serves the same function in classical mechanics as Newton's laws and the conservation of energy, i.e., it forecasts how a dynamic system will behave in the future. In terms of the wave function, it is a wave equation that precisely and analytically forecasts the likelihood of events or outcomes. The precise result is not predetermined, but the Schrodinger equation may forecast the distribution of outcomes given a large number of events. The Hamiltonian, which works on the wave function to cause the wave function to evolve in time and space, is created by transforming the kinetic and potential energies. The Schrödinger equation provides the system's quantized energies as well as the wave function’s shape, allowing other parameters to be determined.
The Schrödinger wave equation and its wave mechanics that encompassed the subject of the first principle application of quantum mechanics can be applied to both single particles and many particle systems[6]. The study of systems of particles at the atomic scale is called the many body system.