The World of Quantum Mechanics
Quantization of Light
The fact that light can be experimentally treated as both a wave pattern and a particle under particular circumstances was known by the time of development of Bohr's atom.
The particle characteristics are described by E=hn and E2= m2c4 +p2c2 , which comes from the Special Theory of Relativity.
For particles at rest, E=mc2
For waves, E=p2c2
Thus, waves have momentum, p. This more clearly points to the wave/particle duality of light.
Thus final name for a 'particle' of light - photon.
Particles as Waves
De Broglie was the first to speculate that 'if light can have both wave and particle-like characteristics, why can't true particles?'
He developed the mathematics to test the idea and experiments which finally verified it.
Electrons, protons, and neutrons all have some wavelike characteristics. Amount inversely proportional to mass. Thus electrons are most wavelike. This led to amore truly quantum-mechanical view of the atom.
Now we replace orbits with wave functions surrounding the nucleus. The placement of electrons depends on four quantum numbers - three known to Bohr plus a fourth, spin.
Spin really describes the symmetry of particles, all particles with mass have spins of n*1/2 where n is an integer. The meaning of a 1/2 spin is that they can be rotated by 180 degrees and look the same. Light has integer spin need to rotate 360 degress or higher to look the same.
For electrons in an atom, the final piece was Pauli's Exclusion Principle - no two electrons can have the same four quantum numbers and reside in the same atom.
Mid 20th century World of Particles
The idea of spin permitted all 'particles' to be broken into two groups, those with integer spin and those with 1/2 integer spin.
All particles with real rest mass are Fermions and have 1/2 integer spin. These include electron, proton, neutron, etc.
All 'particles' with no rest mass are Bosons and have integer spin. These include photons (light), neutrinos, etc.
Matrix mechanics and wave mechanics are two equivalent methods of explaining wave/particle duality and placement of electrons in atoms.
Both calculate momentum (p) and position (q) for particles. Both turn out to develop the following equation:
PQ - qp = hi where p and q are matrices defining position and momentum, h is planck's constant and I is the square-root of negative 1. It turns out that matrix multiplication depends on order. The term 'i' points to wavelike characteristics.
One interpretation of this equation is Heisenberg's uncertainty principle. One cannot exactly define position and momentum of a particle at the same time. Only know one or both with a certain level of probability. This destroys an older Newtonian notion that all physics is predictable because it is all deterministic (simple equations for motion, just know starting positions of everything!).
Wave mechanics is actually used today but both are equally correct.
Problem is that we are left with a mathematical description of position of electrons in atom or overall position/momentum of anything with a wave/particle duality, but no easy vision to go with it. That makes the quantum world very difficult for non-specialists to comprehend.
Inside the Nucleus
Because of wave/particle duality, we can also consider protons and neutrons (the particles of the nucleus) in the same way. It turns out that the nucleus must have another force, called the strong force which operates only over very short distances, to keep all positive protons together in nucleus. Outside the nucleus, other protons are repelled from the nucleus by electrostatic forces.
Neutrons add stability to nucleus. Most stable nuclei have 2, 8, 18,... protons+neutrons. Same way electrons fill orbitals in all atoms.
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