The foundation of quantum mechanics is the principle of particle-wave duality, according to which everything has a dual nature: particle and wave, at the same time. This principle is expressed through the following two equations:
$$ E = \hbar \omega,\quad p = \hbar k $$
where $E$ is the energy of the particle, $\omega = 2\pi f$ is the angular frequency of the wave, $p$ is the momentum of the particle and $k = 2\pi / \lambda$ is the wave-number of the wave.
These two equations relate the particle-nature to the wave-nature of a particle. If we know one, then we know the other. Specifically, the wave’s frequency gives us its energy, and its wavelength gives us its speed.
The equations are also a window as to why quantum mechanics is a theory about the microcosm. To detect wave-like properties from, say, a basketball, we’d have to come up with a way to measure its wavelength $\lambda$. According to the particle-wave equations shown above, this can be done like $\lambda = 2\pi\hbar/mu$. But $m$ is very large for objects above the molecular or even atomic scale, driving $\lambda$ to be infinitesimal and therefore undetectable. Quantum mechanics is therefore a theory of particles in the atomic and subatomic scale.
[TODO] One would be right to ask, what motivated physicists to theorize such a duality between particles and waves? How did the equations above come to be?
Why did we start with particle-wave duality? What does this have to do with quantization? From a very wide perspective, if we accept that particles in nature have wave-like properties, then we would have to ask about the nature of those waves.
What are waves in the first place? Understanding some of the fundamentals from classical physics is very useful before tackling quantum mechanics because it will help you build intuition and useful mechanical analogs.
[TODO] Fundamentals of waves in classical physics.
To fully describe the wave nature of a particle as it moves in space, time and under the influence of various forces in nature, we use Schrodinger’s equation:
$$ i\hbar\frac{\partial \psi}{\partial t} = \hat{H}\psi $$
This equation allows us to solve for the unknown function $\psi = \psi(x,t)$, the wavefunction describing the material wave. We will discuss extensively about the meaning of $\psi$ in physics, but from a mathematical perspective, we only need to solve Schrodinger’s equation for a particle in a certain potential in order to determine its quantum-mechanical properties.
Schrodinger’s equation includes the Hamiltonian operator $\hat{H}$. This operator is a linear operator acting on the space of wave-functions, a space we will shortly define. It is defined through the expression for the total energy of the particle in the potential, is a similar way to how classical physics defines energy:
$$ \hat{H} := \frac{\hat{p}^2}{2m}+V(\hat{x}) = -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} + V(x) $$
Note that in the expression above we substituted: