Physical significance of wave function: In 1923 Louis de Broglie, then a graduate student, introduced the conjecture that particlelike objects (say electrons) should display wave properties. “It would seem” he wrote, ... that the basic idea of the quantum theory is the impossibility of imagining an
'isolated quantity of energy 'without associating with it 'a certain frequency'.
For now, not only was light to be treated sometimes as a wave and sometimes as a particle,but matter itself- the ultimate, the final repository of atomic, corpuscular properties- the atoms of Democritus, Gassendi, and Newton- now had associated with them in some mysterious way a wave.
The physicist of the 1920s had become accustomed to treating light or matter as a wave in diffraction or interference and as a particle in emission, absorption, or transfer of energy. Shortly after de Broglie introduced the idea of the associated wave for an electron, Erwin Schrödinger (in 1926) proposed an answer to the question of what happens to the associated wave if a force acts on it. He introduced now famous equation to study many of the basic problems of the quantum theory.
A wave is a disturbance that propagates through space and time, usually with transfer of energy. Wave is distributed in space and the distribution increases with time. Thus the wave energy per unit volume decreases with time about a space point. A particle (micro-particle) is endowed with a mass. It may also have charge, like electron or proton. The mass is concentrated about a space point. Again the energy associated with the particle is concentrated about the particle. Thus wave and particle form two distinctly different physical phenomenon.

Schrödinger's equation gives the de Broglie wave associated with an electron, any other particle, or finally any quantum system. Given the mass of the particle, and given the forces to which it is subjected, let us say gravitational, or electromagnetic, then Schrödinger's equation gives the possible waves associated with this particle: the waves (functions of position and time) give a number associated with any position in space at an arbitrary time. And they are designated by the hardest working symbol in twentieth-century physics: the wave function ψ(x,y,z;t)
The essence of the Schrödinger equation is that, given a particle, and given the force system that acts, it yields the wave function solutions for all possible energies. The wave function satisfies the most fundamental property of waves-the property of superposition. Just as in classical theory, this means that a trough and a crest can be added to cancel one another. Thus, one can have interference, that most characteristic wave phenomenon. And this now is associated with what were thought of previously as particles: electrons or protons and finally even with entire systems.
An idea of Einstein's gave Max Born to propose an interpretation of the wave function whose consequence are possibly as revolutionary as those of any other idea of the twentieth century. Einstein had tried to make the duality of particles-'light quanta or photons'-and waves comprehensible by interpreting the square of the optical wave amplitudes as probability density for the occurrence of photons. This concept could at once be carried over to the ψ-function: |ψ|2 ought
to represent the probability density for micro-particles.
The probability interpretation, as developed by Born, now is the standard interpretation of the associated wave with any moving micro-particle. The square of the wave function, ψ2 , represents a possibility (probability) density. Thus the particle is likely to be found at the place where the wave function is large and not at the place where the wave function is small. Again, ψ2 is chosen rather than ψ, because ψ itself can be negative and an interpretation for a negative probability is hard to find. With this interpretation, one arrives at the requirement that the total area under the curve ψ2 as a function of x (in one dimensional case) must be equal to 1, because that represents the probability that the electron be in some place-which is to be certain.
As a general rule, for bound systems, not all energy levels are possible. The reason is related closely to that proposed initially by de Broglie: an integral number of waves must fit into a closed orbit in order that the orbit be stable. Thus, from the point of view of the Schrödinger equation for a particle that is in a bound
state (confined to a finite region of space), not all energies, not all momenta, and not all wavelengths are allowed. Only a discrete set, a small portion of the energies that would have been allowed in Newtonian mechanics, occurs. One gets analogous results, although differing in quantitative detail, for particles contained in three dimensions, for particles contained by walls that are not rigid, and for particles contained by a potential such as that which produces the hydrogen atom. This, then, from Schrödinger's point of view, is the origin of discrete levels of the Bohr atom, and the discrete levels characteristic of all bound quantum mechanical system.

Metals can be thought of as three dimensional boxes with the surfaces as boundaries and the valence electrons as relatively free particles. As the dimesion of the box increases the energy levels are very close to each other forming some sort of quasi-continuous states.