### Section 1.5 - Quantum Mechanics

#### The Four Fundamental Forces of Nature

We've just discussed gravity, but there are in fact four fundamental forces in nature:

Gravity is by far the weakest force. To get a sense of the difference in strength between gravity and the strong force, consider this. It takes the entire mass of Earth to exert 175 lbs of force on me (my weight), while just 14 lbs of plutonium obliterated Nagasaki when the atom bomb was dropped. The only reason gravity has meaningful effect is because it has infinite range and because there are some really massive objects in the Universe

Gravity (and by extension general relativity) defines the realm of the macroscopic. The other three exist in the infinitesimally small world of quantum mechanics, so we need to understand that first before we can move on

#### Quantum Mechanics

Classical physics had accurately described natural phenomena for centuries, but by the early 1900s, scientists were uncovering observations that could no longer be reconciled with existing theories

The world of quantum theory can be spooky, counter-intuitive, and downright bizarre, but it has come to supersede classical theory because it can describe subatomic particles with unbelievable precision. For our purposes, there are a few simple postulates that need to be known and are explained below.

#### Quantization

The term "quantized" simply means the observed value of some physical phenomenon can only exist at discrete steps. For our purposes, we're generally referring to electron states. In an atom, a given electron can occupy certain energy states (n=1, n=2, n=3...) but can never be in between energy states (so no n=1.5). The analogy is that you can stand on any rung on a ladder that you wish, but you can never stand in between the rungs.

The above diagram shows the energy states of a simple hydrogen electron. The electron will usually be at its base state, but if given enough energy, it'll jump to a higher energy state. When the electron drops back to its base state, it will radiate the excess energy off as light with a very distinct wavelength that can be used to identify the element at hand.

#### Wave-Particle Duality

Early in the 20th century, physicists realized that light behaved as both a wave and a particle

• Light as a wave: interference pattern formed from the famous double slit experiment can only occur if light is a wave

• Light as a particle: the photoelectric effect occurs when light of a sufficiently high frequency discharges electrons from the surface of a metal. This makes sense if we define light as a stream of particles ("photons") striking the surface of the metal
 Red light (low frequency) lacks the energy needed to display electrons off the metal plate, but green and blue light does. Learn more here

In 1924, Louis de Broglie demonstrated that all matter, not just light, has wave-particle duality by experimentally proving it for the electron. As a result, all matter can be described as having a wavelength. It’s just that for macroscopic objects, the wavelength is negligible (in high school freshman physics, I once calculated the de Broglie wavelength of a baseball)

In summary, this is a great video explaining wave-particle duality

#### The Heisenberg Uncertainty Principle

First introduced by Werner Heisenberg in 1927, this principle states that it is impossible to perfectly know both the position and the momentum of an object. He defined it with this inequality:

$\sigma_x\sigma_p \geq \frac{\hbar}{2}$

where $\sigma_x$: standard deviation of position, $\sigma_p$: standard deviation of momentum, and $\hbar$: reduced Planck's constant

This uncertainty has nothing to do with experimental error or faulty instrumentation; it is a fundamental truth of the Universe that cannot be avoided. We cannot know both of these quantities because any sort of observation will disturb the subject. For example, if we want to shine a light on a particle so we can know its position, the photon of light will impact the particle and affect its momentum

The deeper truth lies more in the wave-particle duality of matter. Particles are understood to occupy an exact location. But waves are disturbances; we can identify key features like wavelength (which is tied to momentum), but the wave as a whole has a probability distribution of where it could be

This is a good summary video of a confusing topic that is often hard to describe.

#### Schrodinger's Cat (Quantum Superposition)

The previous topics relied on the concept of quantum superposition, the idea that different quantum states can be "added together" to produce another valid state. For example, the double-slit experiment implies that a given electron passes through not one slit or the other, but both slits at the same time (what on Earth?!) And only when it hits the detector do we know for sure where it ended up

Schrodinger's thought experiment was this: say we took a cat and put it in a box with a radioactive device that had a 50/50 chance of killing the cat in the next hour. After one hour, it's time to check. Common sense would tell us that at this time, the cat is either alive or dead. But quantum superposition tells us that the cat is both alive and dead at the same time! Only once we check does the cat's true state get revealed.

#### Pauli Exclusion Principle

Wolfgang Pauli demonstrated in 1925 that two electrons cannot occupy the same quantum state (the details of what an electron's quantum state are not important for our purposes, but in case it rings a bell from AP Chemistry, it's defined by four quantum numbers that determine an electron's shell, subshell, orbital, and spin). For example, the electron in a regular hydrogen atom occupies the simplest, lowest energy orbital. To add another proton-electron pair and up it to helium, the next electron needs to occupy a new (and possibly higher energy) quantum state
The point to take a way is that two electrons cannot hold the same place in an atom at the same time, just like how you and I can't both occupy the same 3D macroscopic space at the same time. This will be important for the astrophysics section on white dwarves.

#### Book Recommendation

Final word on quantum mechanics, if you've gone through this primer and want something more technical, I suggest the book Thirty Years That Shook Physics by George Gamow. It was my AP Chemistry summer reading assignment, several years ago...