Our target is the Schrödinger equation which is the key tool that can generate the wavefunctions that are at the heart of quantum mechanics.
In order to get there, we will follow a long and twisty path which will allow us to discover and enjoy several other remarkable and important things along the way:
We'll start with the strange constancy of the speed of light, irrespective of whatever movement there may be between the source and observer of the light. Exploring the consequences of this will lead us to the Special Theory of Relativity.
We'll find that in order to preserve the principle of conservation of momentum, once special relativity is taken into account, we'll need to redefine momentum and kinetic energy.
Considering the physical interpretation of the kinetic energy expression we'll develop leads to Einsteins' mass-energy relation (E = mc2) and also to the important relativistic energy-momentum relation.
One strange consequence of special relativity which we'll uncover is that photons have momentum, despite having no mass.
Next we'll learn to write equations to describe travelling and standing waves. We'll derive the wave equation which is a very important partial differential equation that must be satisfied by any wave.
Then we'll be ready to start bringing things together, to lead us into Quantum Mechanics. From photon momentum we'll derive the de Broglie equation and, with experimental support suggest that this applies to particles with mass, as well as photons: thus we introduce wave-particle duality.
We consider the Born interpretation of a matter wave, which makes a connection between the amplitude of a wave at any point in space and the probability that the particle will be located at this point if we take steps to detect it.
We explore how superposition of waves with different wavelengths can lead to a wave packet which, in the case of a matter wave, corresponds to a spatially localised particle. We develop the uncertainty principle, which quantifies the idea that increasing localisation comes at the expense of increased uncertainty in the momentum of a particle.
In consequence of the uncertainty principle we discover that localisation comes at a kinetic energy cost. This will enable us to explain why atoms are stable and do not collapse and to estimate the size of an atom.
We'll finally get to the Schrödinger equation by proposing that a particle's wavefunction is that of a standing wave. This allows us to bring together the wave equation, the de Broglie equation and the principle of conservation of energy to construct a second order differential equation which must be satisfied by a matter wave. If you like, You can get a sneak preview of a schematic summary of your projected journey to the Schrödinger equation here.
Applying the equation to some model systems, particularly the particle-in-a-box, we'll see how boundary conditions on the wavefunction lead to quantisation. We'll see where physically important things like quantum numbers and zero point energy come from and we'll see how the weird properties of systems in states with low quantum numbers become indistinguishable from classical physics when the quantum numbers are high.
We'll have a quick look at the problem of superposition states and the wavefunction collapse that is hypothesised to happen when an observation of the system is made.