The founders of quantum mechanics largely ignored relativity when they first applied quantum theory to chemistry:

"The general theory of quantum mechanics is now almost complete... they give rise to difficulties only when high-speed particles are involved, and are therefore of no importance in the consideration of atomic and molecular structure and ordinary chemical reactions."- Paul Dirac, 1929

Treat an atom as a two-body problem with one electron and a central nucleus with a charge \(Z\) and a mass much greater than the electron.

Was Dirac correct? At what element in the periodic table would relativistic effects become important for an electron in a ground state?

**Note:** Significant errors start to appear between classical and relativistic dynamics when the Lorentz factor \(\gamma = \frac{1}{\sqrt{1-v^2/c^2}} \geq 1.05\). It may be useful to perform this calculation in atomic units, where \(\hbar, e, m_e,\) and \(4 \pi \epsilon_0\) are defined as \(1.\) In these units, the speed of light is \(c = 137\).

Youâ€™re lost in the woods, and your only hope of reaching civilization is to find your way back to the only road.

- You know you are exactly 1000 meters from the infinitely long, straight road.
- You have perfect relative bearings, are able to track your position perfectly, and can travel along any line or curve.
- You don't know the direction of the road and cannot see it until you are upon it.

What is the **shortest possible walking distance**, to the nearest meter, that will **ensure** you reach the road?

**Be warned**, your initial thoughts may need to be refined to truly minimize the length of your path.

Within a \( 100 \times 100 \) square, draw in a shape that maximizes the ratio \( \frac{ \text{ Area } } { \text{ Perimeter } } \).

What is this ratio (to 2 decimal places)?

For example, if we drew in a \( 100 \times 100 \) square, we would get \( \frac{ 100 ^ 2 } { 4 \times 100 } = 25 \).

The caustic appearing on the circular surface with radius \(3\text{ cm}\) of this cup of coffee, with a spotlight source on the perimeter, turns out to be a cardioid.

What is the area \(\big(\)in \(\text{cm}^2\big)\) of the region that the cardioid encloses?

\(x\) can be expressed as \(\frac{a}{b}\), where \(a\) and \(b\) are coprime positive integers.

What is \(a+b?\)

**Note:** Synthetic proofs welcome.

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