Quantum mechanics is the extension of wave-particle duality to all matter! Everything becomes probabilistic. Some fundamental ideas are:
- We have probabilities, not determined outcomes
- We often find quantized phenomena: energy levels, flux, etc
- Only operates at the small scale. More rigor on this in just a second.
Definition: de-Broglie Relations
The de-Broglie wavelength gives a length scale for when quantum mechanical effects matter. It is defined as
$$\lambda = \frac hp \iff p = \hbar k.$$
There is also a de-Broglie frequency,
$$f = \frac Eh \iff E = \hbar \omega.$$
Theorem: Heisenberg Uncertainty Principle
There are several pairs of quantities which have an uncertainty principle, but the two most famous are $x \leftrightarrow p$ and $E \leftrightarrow t$. Mathematically, it states
$$\Delta x \Delta p \ge \frac{\hbar} 2, \Delta E \Delta t \ge \frac \hbar 2.$$
In most problems, the numerical factors will not matter.
Idea: WKB Approximatino
In the regime where the debroglie wavelength is much smaller than the length scale of the problem, we can treat particles semi-classically! Using this we have that for a quantum "standing wave",
$$\oint k dx = 2\pi n \implies \oint \frac{p}{\hbar} dx = 2\pi n.$$
This reduces to
$$\oint p dx = nh.$$
Most other "modern physics" olympiad questions are just questions that apply a lot of the old concepts that we’ve learned so far. The most popular topics to use are Relativistic Dynamics, Electrodynamics, and Gravity.