1. Bohr’s model failed
For atoms other than Hydrogen Bohr’s theory failed and Bohr new this himself as in physics:
A charged body (ie. an electron) following a circular orbit will emit energy and as energy decreases so does the distance to the nucleus. Thus as the electron orbits it will get closer and closer to the nucleus and the atom will eventually collapse.

2. Next comes Schrodinger
Electrons moving around the nucleus can be thought of as forming standing waves which can be described by the de Broglie equation.
Standing waves = do not travel because they have fixed start and end points. Thus, they must contain a fixed number of half wavelengths.

3. Schrodinger
Thus only certain circularly shaped standing waves with a whole number of half wavelengths would be possible without destructive interference at the ends.
These standing wave orbits (of sorts) would then only be possible at fixed distances from the nucleus thus supporting the energy quantization observation.

4. At the same time…Heisenberg
Heisenberg uncertainty principle:
There is a fundamental limitation to just how precisely we can know both the position and momentum of a particle at a given time. In layman’s terms: The more precisely we know a particles position the less we know about its momentum and vice versa.

5. Schrodinger + Heisenberg = Orbitals
Regardless of the method used to arrive at the theory of orbitals (either schrodinger’s or Heisenberg’s) the result is the same:
A precise location for an electron cannot be determined however regions which describe the probable location for electrons can be and these regions are known as orbitals.

6. Particle in a Box We are looking to find a way to describe an
electron’s position.
This of a 1D box.
V=0 in the box and V=∞ outside the box.
This defines the particles a trapped in the box and
it would take an infinite amount of energy to
remove the particle from the box.

7. Particle in a Box
x=0
x=l

8. Schrodinger’s Method Based on the wave properties of the atom
 = wave function
H = mathematical operator
E = total energy of the atom
A specific wave function is often called an orbital.

9. Schrodinger Solutions = orbitals
The Schrodinger equation has many possible solutions for a particular atom.
A solution to the Schrodinger equation is a function which satisfies it.
These wave functions represent orbitals and this explains why atoms have several orbitals.
The functions squared represent the probability density of the electrons. We then limit the area we describe at the orbital to 90% of the probability area.
These probability functions are used to picture in 3D space the probable location of an electron and are what is most commonly referred to as an orbital.