# Blake Test

Quantum POTW sequence.

#### Contents

## Tests

Great question!

The simple answer is that all particles (including neutrons) are always undergoing random thermal motion at any temperature above absolute zero (\(-273 ^ \circ C\) or \(0 ^ \circ K\)). The velocity you'd expect to observe for a particle at a given temperature \(T\) depends on the Maxwell-Boltzmann distribution.

According to this distribution, for a neutron at room temperature (\(20 ^ \circ C\) or \(293 ^ \circ K\)), the average speed would be around \(\SI{2}{\kilo \meter \per \second}\). For a neutron emerging from a hypothetical thermonuclear oven, the temperature might be close to that of a nuclear explosion, yielding neutrons with an average speed greater than \(\SI{10 000}{\kilo \meter \per \second}\). Since the distribution has a wide spread, not all the particles would be moving at that speed; but every particle would be expected to have a non-zero speed. In the oven, these neutrons would be rapidly colliding with their container and one another, very quickly yielding a random distribution of velocity vectors and orientations.

This is just a discussion of translational motion, but at any reasonable temperature we'd expect the neutron's rotational motion to behave similarly, yielding random orientations of neutrons.

## Basic

The wavelength of green light is about \(\SI{500}{\nano \meter}\). As the wavelength of light gets smaller, its energy gets larger. Green light is about \(200 \times\) smaller than the width of a hair, and about \(2000 \times\) larger than most atoms.

We use visible light in microscopes to observe forensic samples of hair, and even to visualize the inside of cells, which are only about \(2 \times\) larger than the wavelength of green light. But we can't observe individual atoms with a light microscope: In fact as a rule, you can't really observe objects much smaller than the wavelength you're using to look at them. It would be like trying to track a speck of dust in the sky with a radar station.

What does this imply about subatomic particle physics?

- Subatomic particles can only be studied with very long wavelength light.
- As we probe more deeply into subatomic particles, we'll need to build higher energy accelerators.
- To observe individual subatomic particles, the experiment must be extremely cold.

## Intermediate

Hydrogen is the simplest atom, with only one proton and one electron, interacting with a Coulomb potential:

\[U = \frac{e^2}{4 \pi \epsilon_0 r}\]

where \(e\) is the elementary charge and \(\epsilon_0\) is the permitivity of free space.

If the electron were a classical charged particle, its orbit around the nucleus would decay until it came to rest at the nucleus. Since the electron is a quantum object, this doesn't happen. The spread in position and momentum of the electron is limited by the uncertainty principle:

\[\Delta x \Delta p \geq \frac{\hbar}{2}\]

where \(\hbar\) is the reduced Planck's constant.

The radius of the electron orbit **must** be larger than the spread in position, likewise the momentum of the electron **must** be larger than this spread in momentum. Otherwise, the Hydrogen atom would be violating this fundamental limit.

Treat the Hydrogen atom as a two body problem where the mass of the proton is much greater than the electron. Estimate the minimum radius of the electron orbit.

## Advanced

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

"The general theory of quantum mechanics is now almost complete ... these 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

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. Find the characteristic length scale of a ground state electron orbit. 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 \(\frac{v}{c} \geq 0.1\).*