In the early 1900s physicists performed experiments that probed reality at its fundamental level.
They found that matter was made of particles they named atoms.
Inside each atom they found the subatomic particles:
electrons, protons, and neutrons.
They also found that light was made of particles that we now call photons.
The strangest result was that all these particles produced wave-like interference patterns.

The double slitexperiment is a good example of wave interference.
A wave passing through a slit spreads out. It diffracts.

single slit wave diffraction

2 slits produce 2 separate wave diffractions.
Once both waves get to their target, they have traveled different distances so they have different phases.
When both waves are in the same phase they add constructively.
When both waves are half a cycle out of phase they add destructively.
The phase differences lead to an alternating constructive and destructive interference pattern.

double slit wave interference

gap separation =

In 1801, Thomas Young produced a double slit interference pattern with beams of light.
In 1909, the experiment was repeated with a single photon at a time.
After passing through the double slit, each photon was measured at single location, but the distribution of
measurements still produced an interference pattern.

wavelength =
slit separation =
slit width =

The double slit experiment
isn't limited to photons.
In 1927, the experiment was repeated with electrons.
In 2019, an interference pattern was produced by a 2000 atom molecule.

In order to produce a wave interference pattern, particles must be perfectly isolated from their environment.
That's why we still can't produce a double slit interference pattern for everyday sized objects, like cats.
It's just too hard to isolate large objects long enough to run the experiment.

Simulation: What happens when one slit is blocked? Why?
answer

The interference pattern goes away, because the waves are all originating from the same point, so they have
the same phase.

Although, there is still a diffraction pattern related to the width of the slit.

Simulation: When one particle at a time is fired, the distribution of hits still produces an
interference pattern.
What could the single particle be interfering with?
answer

A single particle goes through both slits.
It interferes with itself. Probably...

This experiment shows that we can't think of particles as simple balls.
There isn't agreement on what particles actually are.
At the bottom of this web page I go into more detail on the various interpretations of quantum mechanics.
Each interpretation has a wildly different explanation of the double slit experiment.

The overlap between the wave and particle models at the nanometer scale led to many more exciting experiments.
Over time, physicists developed and refined a new model, called quantum mechanics.
Predictions based on quantum mechanics can only give
probabilities, but when applied to repeated experiments they are very accurate.

Photon Energy

In 1899, after investigating the thermal radiation spectrum,
Maxwell Planck reluctantly hypothesized that the energy of light is only released in small quantities determined
by its frequency.
This idea was the beginning of quantum mechanics.

These packets of light were eventually called photons.
Photons have a frequency and a wavelength, but no mass.
They are produced anytime a charged particle loses energy.
Photons spread out propagating through space at the speed of light.
They can transfer their energy to a charged particle in a process called absorption.

$$E=hf$$

\(E\) = energy of a photon [J, joules, kg m²/s²]
\(h\) = 6.626 × 10^{-34} = Planck's constant [J s]
\(f\) = frequency [Hz, 1/s]

The color is not visible, because it is in the ultraviolet.

Example: Find the energy for a photon that has a wavelength of 0.3 m. What part of the E-M
spectrum is the photon in?
solution
$$c = \lambda f $$
$$f = \frac{c}{\lambda} $$
$$f = \frac{ 3 \times 10^{8}}{ 0.3 } $$
$$f = 10^{9} \, \mathrm{Hz}$$
$$ E = hf $$
$$ E = (6.626 \times 10^{-34})(10^{9}) $$
$$ E = 6.626 \times 10^{-25} \, \mathrm{J}$$

The photon is a microwave.

Example: Find the wavelength of a photon that has 2.65 × 10^{-19} J of energy.
solution
$$ c = f \lambda $$
$$ f = \frac{c}{ \lambda }$$
$$E=hf$$
$$E=\frac{hc}{\lambda}$$
$$\lambda=\frac{hc}{E}$$
$$\lambda=\frac{(6.626 \times 10^{-34})(3 \times 10^8)}{2.65 \times 10^{-19}}$$
$$ \lambda = 7.5 \times 10 ^{-7} \, \mathrm{m} $$
$$ \lambda = 750 \, \mathrm{nm} $$

Example: A helium
neon laser outputs 5 mW of optical power. Lookup the
wavelength of the light and then calculate the number of photons produced every second.
strategy

Wikipedia says the wavelength of each photon is 632.8 nm. Convert the wavelength into energy.

The number of photons is the energy output divided by the energy for each photon. The energy output is 0.005
J every second.

The photoelectric effect occurs when metals dislodge
electrons after being hit by light. Light with a frequency above the visible spectrum is required to produce the
effect. Bright red light can't produce the effect, but even dim UV light can.

In 1905, Albert Einstein published an explanation of the
photoelectric effect that
supported Max Planck's concept of quantized light. Einstein suggested that light is made up of many small packets
of energy, and each packet interacts with a single electron.
Only high frequency light releases electrons because it has enough energy per packet.

Einstein published an equation that describes the photoelectric effect using conservation of energy. The kinetic energy of a released
electron can't exceed the difference between the energy of an incoming photon and the energy needed to dislodge
the electron.
If the maximum kinetic energy is below zero, an electron is not released.

$$K_{max}=hf - \Phi$$

\(K_{max}\) = maximum kinetic energy of released electron [J, joules]
\(h\) = 6.626 × 10^{-34} = Planck's constant [J s]
\(f\) = frequency of incoming light [Hz, 1/s]
\(\Phi\) =
Work function, the
minimum energy to dislodge an electron [J]

Electrons can be thought of as being stuck in a energy
well. The work function represents the minimum energy the electrons needs to escape.
The work function depends on the material. Metals have a low work function, so it is easier to dislodge an
electron from a metal.

Question: Why does dim ultraviolet light produce the photoelectric effect, yet very bright red
light doesn't?
answer

Most of the time just one photon is hitting one electron.
Low energy photons don't have enough energy to dislodge an electron.

A two photon photoelectric effect can happen, but it's probability is very low at normal light intensities.
A high intensity red laser produces enough photons for a two photon effect, but it would also turn the metal
into a hot plasma.

Energy at the atomic level is often calculated in electron volts (eV). We can convert between
eV and J by multiplying or dividing by the charge of an electron.

Example: After being hit by light with 7.0 eV per photon, the rare earth metal terbium releases
electrons. The electrons have a maximum kinetic energy of 4.0 eV. What is the work function of terbium in
electron-volts?
solution
$$K_{\text{max}} = hf - \Phi$$
$$\Phi = hf - K_{\text{max}}$$
$$\Phi = 7.0 \, \mathrm{eV} - 4.0 \, \mathrm{eV}$$ $$\Phi = 3.0 \, \mathrm{eV}$$

Example: Find the max kinetic energy of a magnesium electron after being hit by a photon with a
frequency of 600 THz. Look up the work function for
magnesium and convert it into joules.
metric prefixes

A negative kinetic energy means the electron will not escape the atom.

Example: A 100 nm photon strikes a lump of magnesium. How much kinetic energy could the released
electron have?
solution
$$ c = f \lambda $$
$$ f = \frac{c}{\lambda} $$
$$ f = \frac{(3.00 \times 10^{8})}{100 \times 10^{-9}} $$
$$ f = 3.00 \times 10^{15} \, \mathrm{Hz} $$
$$ \Phi = 3.66 \, \mathrm{eV} \left( \frac{1.6 \times 10 ^{-19} \, \mathrm{J}}{1 \, \mathrm{eV}} \right) =
5.856 \times 10^{-19} \, \mathrm{J}$$
$$K_{\text{max}} = hf - \Phi$$
$$K_{\text{max}} = (6.626 \times 10^{-34})(3.00 \times 10^{15}) - 5.856 \times 10^{-19}$$
$$K_{\text{max}} = 19.878 \times 10^{-19} - 5.856 \times 10^{-19}$$
$$K_{\text{max}} = 14.022 \times 10^{-19} \, \mathrm{J}$$
How fast could the electron be moving as it escapes the
magnesium atom?
solution
$$m_e = 9.10 \times 10^{-31} \, \mathrm{kg} $$
$$ K = \tfrac{1}{2}m_ev^2 $$
$$ v = \sqrt{\frac{2K}{m_e}}$$
$$ v = \sqrt{\frac{2(14.022 \times 10^{-19})}{9.10 \times 10^{-31}}}$$
$$v = \sqrt{3.08 \times 10^{12}}$$ $$ v = 1.76 \times 10 ^6 \, \mathrm{\tfrac{m}{s}}$$

Example: Use the work function table to
decide which elements would release electrons from 427 nm wavelength light.
strategy

The lowest energy photon that can dislodge electrons will have the same energy as the work function. Convert
the photon's wavelength into electronvolts and compare it to the work functions in the table.

Any element with a work function below 2.91 eV. Like: Ce, Gd, Li

Radiation

Radiation is a wave or particle that transmits energy through space.
Typical radiation takes the form of a protons, electrons, nuclei, or photons.

A large amount of radiation can hurt living things in a direct way, by increasing their temperature.
A toaster and a microwave oven both use radiation to cook food.

Radiation from particles with very high energy per particle can harm living things in a more subtle way.
High energy particles can ionize atoms, breaking chemical bonds.
This can led to cell death and an increased the risk of cancer.

Non-ionizing radiation is generally below 1.60 × 10^{-18} J (10 eV).
This safe radiation doesn't have enough energy to break
chemical bonds.

Ionizing radiation is generally above 1.60 × 10^{-18} J (10 eV).
This unsafe radiation can potentially break chemical bonds.

Try this PhET simulation to see how molecules interact with photons.

Question: Greenhouse gasses need to interact with infrared light. Which molecules from the
simulation could be greenhouse gasses?
answer

CO | CO₂ | CH₄ | H₂O | NO₂ | O₃ all could deflect infrared light as it leaves the Earth,
preventing the Earth from cooling.

In the last 100 years humans have greatly increased the amount of CO₂. The extra CO₂ has led to a 1° Celsius
increase in average
global temperature.

Question: Can microwave ovens ionize atoms? Are microwaves dangerous?
answer

Microwave ovens make microwaves. Each microwave photon has an energy around 10^{-23} J. This is far
below the ionization threshold: 1.60 × 10^{-18} J.

Many studies have been done on the risks of microwaves, but no adverse health effects have been established. Microwaves can heat
up
things, but they will not directly cause cell damage or cancer.
Cell Phones and Wi-Fi also use microwaves to transmit data.

Question: Is sunlight dangerous?
answer

Too much sun can cause sunburns and skin cancer.
Sunlight has some ionizing radiation in the UV part of the electromagnetic spectrum.
Most of the UV is blocked by the ozone layer, but some still gets through.

Not enough sunlight is also unhealthy. A lack of Sun exposure can lead a vitamin D deficiency.
It is recommended that we get at least 15 minutes of direct Sun every day.

Question: Which would damage a person the most: an ultraviolet photon or a gamma-ray photon?
answer

Gamma Rays

UV is less likely to ionize cells. Gamma-rays have much more energy per photon, but gamma-rays also have
higher penetration. This means they can pass deeper into the body or not interact at all.

Emission and Absorption

Each electron in an atom can only have an energy that
exactly matches an atomic orbital.

quantum mechanics and atomic orbitals

Two bodies can orbit each other if they experience an
attractive force.
Gravitational attraction keeps the Earth orbiting the Sun every year.

The nucleus of an atom attracts electrons because it has an opposite electric charge.
This attraction can also lead to an orbit, but with some quantum mechanical restrictions.

Quantum mechanics tells us that electrons are waves.
As the waves try to orbit the nucleus, they destructively interfere with themselves.
This limits the wavelengths to values that allow constructive orbital standing waves to form.
We say that the allowed wavelengths are quantized.
Quantizing means to restrict a variable to specific values.

An electron in an atom can be described by 4 quantum numbers.
Quantizing these numbers adds another restriction.
Two electrons can't form the exact same wave due to a type of destructive interference called the Pauli exclusion principle.
This means each electron in an atom must have a different combination of the 4 numbers.

This image is a 2-D slice of the allowed 3-D states for
electrons in a Hydrogen atom.
Each electron in an atom or molecule must occupy a different wave state. This state determines its wavelength,
frequency and energy.

All quantum mechanical particles are not required to be quantized.
Particles outside an atom are free to have a wide range of values.
The quantization of particles comes from electrons in atoms forming standing waves only at specific values.
This quantization can spread to photons as they are emitted from electrons in atoms.

If an electron gains the energy difference between two states it can jump to a higher energy level.
For example, an electron will transition to a higher energy level if it collides with a photon that has an energy
equal to the energy difference between levels.

This process can also run in reverse.
An electron can drop to a lower energy level if an atom has an unoccupied energy level.
When the electron drops down it emits a photon equal to the difference in energy between each level.
This follows the law of conservation of energy.

Photon emission occurs when electrons transition to lower energy levels within an
atom.
Each electron transition emits a photon with an energy equal to the energy difference between levels.

Hydrogen has a very simple emission spectrum because it doesn't have very many possible energy states and therefore
few energy state transitions.

Iron's nucleus has more protons so it has more possible energy transitions.

Photon emission is the working principle behind fluorescent lights.
To make light, a tube is filled with various gases.
The gases are electrically charged up which brings the electrons to a higher energy level.
The electrons are unstable in the higher energy levels.
They eventually fall back down to their ground state and emit light.
Try looking at fluorescent light reflected off an old CD to see the separate bands of color.

Example: Find the frequency of the photon produced when an electron drops from an energy of -3.4
eV to -13.6 eV in a hydrogen atom?
solution
$$\Delta E = -13.6 \, \mathrm{eV} + 3.4 \, \mathrm{eV}$$
$$\Delta E = -10.2 \, \mathrm{eV} \left( \frac{1.6 \times 10^{-19} \, \mathrm{J}}{1 \, \mathrm{eV}} \right) =
1.632 \times 10^{-18} \, \mathrm{J}$$
$$E = hf$$
$$f = \frac{E}{h}$$
$$f = \frac{ 1.632 \times 10^{-18} \, \mathrm{J}}{6.626 \times 10^{-34} \, \mathrm{J \, s}}$$
$$f = 2.46 \times 10^{15} \, \mathrm{Hz}$$

The photon is in the ultraviolet spectrum.

Absorption is the reverse of emission. A single photon is absorbed by a single electron in a
single atom.
This causes the electron to transition to higher energy levels.

If there isn't an energy difference that matches the energy of the colliding photon, the material is transparent to that frequency. A substance may be clear
in one range of the spectrum but not in others. For example: glass is mostly transparent to visible light, but
it has many absorption frequencies in the infrared and UV.

Exposing an atom to a full range of light will produce an absorption spectrum that matches the energy difference
between electron energy levels.

Light absorption occurs at the same energies as emission. Hydrogen's absorption frequencies are the same as its
emission frequencies.

Analyzing the spectrum of emission or absorption can
actually be used like a fingerprint to
identify the elements or molecules being observed. This technique is used in fields like forensics and
astronomy.

Sunlight is mostly thermal radiation with large sections
of absorption from the molecules in Earth's atmosphere.

Question: What molecules are absorbing most of the Sun's infrared rays?
answer

H₂O, CO₂ and O₂ absorb infrared light and then re-emit the light in a random direction.

High H₂O and CO₂ concentrations lead to a warmer Earth.
This happens because incoming sunlight is mostly visible and is not absorbed by these gases, but light leaving
the Earth is mostly infrared and is absorbed.

Question: What molecule is responsible for absorbing the Sun's UV rays?
answer

O₃ (Ozone) absorbs some of the UV spectrum.
This reduces ionizing radiation.

This diagram shows the wavelengths of photons emitted or absorbed when an electron transitions between energy levels for Hydrogen.
Each wavelength, listed in nanometers, is for a photon with an energy equal to the difference between energy levels.

Question: What wavelength of light could make an electron jump from energy level n = 1 to n = 5?
answer
$$\lambda = 95 \, \mathrm{nm}$$

Example: Find the energy of a photon produced as an electron drops from energy level n = 4 to n =
2?
solution
$$\lambda = 486 \, \mathrm{nm} = 4.86 \times 10^{-7} \, \mathrm{m}$$
$$E = \frac{hc}{\lambda}$$
$$E = \frac{(6.626 \times 10^{-34})(3 \times 10^{8})}{4.86 \times 10^{-7}}$$
$$E = 4.09 \times 10^{-19} \, \mathrm{J} $$
$$E = 4.09 \times 10^{-19} \, \mathrm{J} \left(\frac{1 \, \mathrm{eV}}{1.6 \times 10^{-19} \, \mathrm{J}}
\right) = 2.56 \, \mathrm{eV}$$

Example: Find three possible light frequencies that could be absorbed by an electron at energy
level n = 3.
solution

We can see three wavelengths in the diagram that transition to n = 3:

1875 nm, 1282 nm, 1094 nm.

There are more transitions, but the diagram doesn't go past n = 6.

Photons have no mass, but they they still have momentum.
You can actually push a space ship like a sail boat, but with light instead of wind.
When photons collide with the light
sail, they bounce off pushing the sail forward.

Example: What is the momentum of a photon of green light? (540 nm)
solution
$$\lambda = \frac{h}{p}$$
$$p = \frac{h}{\lambda}$$
$$p = \frac{6.626 \times 10^{-34}}{540 \times 10^{-9}}$$
$$p = 1.2 \times 10^{-27} \, \mathrm{kg \tfrac{m}{s}}$$

If we fired photons at a massive object the photons should transfer their momentum because of conservation of
momentum.

Extra Credit: How many green photons would it take to accelerate a 1 kg body from rest to 1
m/s?
(assume the green photons reflect off the 1 kg object)
strategy

In 1924 Louis de Broglie proposed that particles with mass might have a wavelength.
This turned out to be accurate. Electrons, atoms, even large molecules all have a measurable quantum wavelength
based on their momentum.

Oxygen is made of smaller particles, so it's odd that it has a single wavelength.
Yet, particles made of smaller particles do behave as a single wave.
Wave patterns have been verified experimentally for molecules up to around 25,000 protons and neutrons in 2019.

A cat particle has an extremely small wavelength, but a cat isn't a single quantum particle in a practical sense.
The equation begins to lose meaning at the macroscopic scale as particles can no longer be isolated from their
environment.
This agrees with our everyday observations for cats. They don't have a measurable wave-nature, although they are
very sneaky.

Example: Find the wavelength of a proton moving at 30 m/s.
Subatomic Particles Data Table

Quantum mechanics seems strange.
It feels like our everyday experiences are normal, and quantum mechanics is a set of special rules for small
things.
It's actually the other way around.
Quantum mechanics models the fundamental rules of reality, and our everyday experience emerges from those rules.

Learning quantum mechanics
is typically done by studying the underlying math.
Building an intuition without the math can be confusing.
The grey boxes below have a nonmathematical and inevitably flawed description of quantum mechanics.
You may want to skip them.

how quantum mechanics makes predictions

The state of a quantum mechanical system is described mathematically by it's wave function.
The wave function contains all the measurable information about the system.
For simple situations the evolution of the wave function over time can be calculated
with the Schrödinger equation.
The Schrödinger equation can be applied to one particle, atom, molecule, or possibly the entire universe.

$$\Psi(x,t) = \text{wave function} $$

As a wave function
evolves with time it often looks like a field of positive and negative amplitudes oscillating back and forth like a sine wave.
It kinda looks like waves and ripples on surface of a pool of water.

These waves are modeled as complex numbers. This means they have a part made of real numbers and a part made of imaginary numbers.
This means the wave wiggle back and forth in the 3 spacial dimensions, but also in some other direction perpendicular to all the spacial dimensions.

measurement and decoherence

In quantum mechanics, the likelihood of finding a system in a particular state comes from the system's wave function.
This probability is calculated by squaring the wave function's amplitude.
For instance, the probability of locating a particle in a specific position increases if the square of its wave function's amplitude is greater at that position.

The wave function doesn't have a single location. It is spread out.
Yet, when an observer measures the results of quantum experiments they get results at precise locations.
The wave function only seems to predict the probability of where a measurement might be.
Why does a spread out wave give measurements at random locations?

Both the observer and the experiment are quantum objects with their own wave functions.
In order for the observer of an experiment to take a measurement they have to interact with the wave function of the experiment.
The observer and experiment together form a superposition of all possible experimental outcomes.

So why do we experience a single outcome if we are in a superposition of all possible outcomes?
This happens as the particles in an experiment interact with external particles outside the experiment.
These external particles might be stray air molecules, a random photon, or part of a detector that measures the results of the experiment.
Each interaction make the superposition messier.
Differences between the parts of the wave function increase and the the total wave function becomes less coherent.
Like how splashing water on a lake can blur your reflection.

Eventually each outcome of the wave function's superposition sees the total of the other outcomes as almost random noise.
The total superposition of all other random outcomes almost perfectly cancels out destructively.
Like how if you flipped 100 coins that odds are almost also 50:50 heads vs. tails.
This leaves each outcome increasingly more isolated after each external interaction.
This process is called decoherence.

When you measure the results of an experiment you become part of the experiment's superposition.
You become a superposition of different versions of yourself that have each seen separate outcomes of the experiment.
After decoherence occurs each version of you that saw each experimental outcome no longer agree.
The superposition of all the wave functions from your other versions cancel out destructively.
Each of yourselves has an nearly zero probability of interacting with other selves.
From the point of view of each self they measure only one coherent outcome.

Decoherence helps explain how we experience distinct, singular events out of calculations that imply numerous possibilities.
It bridges the gap between the probabilistic nature of quantum systems and the definite outcomes we observe in the macroscopic world.

It's pretty confusing how multiple outcomes produce a single outcome.
Not all physicists would explain measurement in the same way I did.
There are some philosophical disagreements.
Do outcomes you don't measure actually occur?
Are measurements truly random or just really complicated?
Is reality made of particles or waves?
These question are potentially answered by the different interpretations of quantum mechanics.

interpretations of quantum mechanics

Different interpretations
of quantum mechanics suggest suggest strange and exciting things about the fundamental nature of our reality.
Yet, each interpretation is just a different way to put math into words.
All valid interpretations of quantum mechanics should predict the same experimental outcomes, so it is hard to
test which is correct.

An important feature of quantum mechanical interpretations is how they handle the measurement problem.
Measurement of a particle's position can only be done by having a second particle interact with it.
For example, you only know the position of this text because photons of light are interacting with the electrons
in the text, and in your eyes.

Before measurement, an electron matches the wave model.
It is spread out, oscillating up and down, and forms wave interference patterns in superposition.
When measured by a photon, an electron is found at a random position.
This position is more likely to be found where it's wave function has a large amplitude.

This leads to some questions about measurement that different interpretations of quantum mechanics attempt to
answer.

Is nature made of waves, particles, both, or neither?

Are quantum effects transmitted faster than the speed of light?

What happens to the wave function during measurement

Is measurement random or deterministic?

The Many-Worlds Interpretations

If the mathematics of the Schrödinger equation is interpreted literally, it means that nature is a wave and all
possible outcomes of an interaction occur.
These different outcomes do not interact with each other, like how sound waves in superposition don't affect
each other.
The many-worlds interpretation suggests that a version of you exists that observes each possible outcome of a
quantum event.
No one has devised an experiment to test for these branching worlds. For now their existence is unknowable.

Having many worlds seems needlessly complicated, but other possible outcomes are already a part of the
Schrödinger equation.
Many-worlds just doesn't add an extra step to remove unseen outcomes.

Nature is modeled as only a wave, with particles that emerge as excited wave states.

Quantum effects are transmitted locally.

Measurement branches the wave function into many worlds.

Measurement isn't fundamentally random.
It seems random because a version of you experiences every outcome, and you don't know ahead of time which you you are.

The Copenhagen Interpretation

The Copenhagen interpretation was the first popular explanation of the math behind quantum mechanics.
Like the many-worlds interpretation, the evolution of the wave function is still calculated with the Schrödinger
equation.
It differs from many-worlds during interactions.
Instead of suggesting that all possible outcomes of an interaction occur, one random outcome is chosen.

The unchosen outcomes are said to be collapsed to a single value.
In this way, the Copenhagen interpretation imagines a particle during measurement,
but treats the particle as a wave of possible outcomes when it is not measured.

Nature is modeled as waves that become particles when measured.

Quantum effects are transmitted locally.

Measurement collapses the wave function to one outcome.

Measurement is fundamentally random.

Pilot Wave Theory

The pilot wave theory proposes a deterministic
universe made of particles.
It also proposes a hidden level of reality for the wave function, which is still calculated with the Schrödinger
equation.

The wave function guides particles down a path based on the particle's position.
When the position of the particle gets measured, the wave function doesn't change.
Measurement disrupts interference patterns because in order to measure something you have to hit it with another
particle, which changes it's velocity.

The wave function is determined by the instantaneous positions of particles, including particles that are very far away.
Knowing these positions instantly can't be used for faster than light communication, because the usable information in the wave function is hidden.

Nature is modeled as a wave that guides particles.

Quantum effects travel faster than the speed of light.

Measurement doesn't change the wave function.

Measurement isn't fundamentally random.
Measurement appears random because the initial position of a particle isn't known.

wave-particle duality

Is nature made of waves or particles?
It depends on how detailed we want to be and how we interpret quantum mechanics.
The wave model and the particle model are approximations of nature that are each accurate for different
conditions, but if you have to choose, the wave model is probably more accurate at the fundamental level.

Quantum field theory describes reality with wave
functions.
In this theory there are several field types spread across all of space and time.
Regions of these fields can evolve into localized excited states with consistently high amplitudes.
What we think of as particles are a simplification of these localized excited wave states.

For example, an electron is a region where the electron field has more amplitude, bigger ripples, higher probability.
The electron field exists everywhere, but electrons are more likely to be measured where the field amplitude is
larger.

Fields interact with other fields, and with themselves.
These interactions can be approximated with Feynmandiagrams, which treat everything as particles.

In Feynman diagrams electron field interactions are generally modeled well as particles.
This is because the most complex electron interactions are weak enough to be mathematically canceled out.
But Feynman diagrams don't work for all interactions.
Quarks, the particles inside protons and neutrons, interact through the "strong force".
This force has frequent complex interactions that don't cancel out, making the particle model less useful for calculations.

how a quantum particle moves

Quantum mechanics answers questions I never thought to ask.
Like, why do objects with no forces acting on them move in a straight line?

Calculating how something moves involves adding the amplitudes of every possible path of getting from one place to another.
The amplitudes of the possible paths cycle from positive to negative as time moves forward.

Some paths might take a very long time. They might zigzag or even go backwards.
These slow paths tend to have very different times and have dissimilar amplitudes.
The randomly dissimilar amplitudes mostly average to zero.

Paths that are more direct, have similar times and therefore similar amplitudes.
These similar amplitudes are constructive, not destructive.
The square of the superposition of all the amplitudes predicts the probability of observing a possible outcome,
so we are more likely to observe a direct straight path.

For very short periods of time, this bias towards straight line paths becomes reduced.
The less direct paths have less time to develop random phases.
This leads to particles moving in a less predictable way.

There is an uncertainty in the motion of a particle that increases as the total time decreases.
Even when a force should act as a barrier under the classical rules of physics, a particle might just quantumtunnel
through that barrier if the time scales involved are very short.

observing quantum physics

Quantum mechanics explains almost all observable phenomena.
It explains the color of materials, rigidity of solids, conductivity, superconductivity, magnetism, all of chemistry, and just about everything else in reality.
You are observing quantum mechanical phenomena in every moment.

Quantum mechanics says that small particles, like atoms, have a measurable wave nature.
Cats are made of atoms.
Why don't we notice cats acting like waves?

It's because atoms are hard to isolate.
They are frequently interacting with their environment.
This causes decoherence between the relative phases of atoms.
When coherence is lost, destructive and constructive interference patterns between waves become messy and begin to look like a classical system.
For large groups of atoms, decoherence happens so frequently that measurable wave interference patterns don't have time to develop.

A Bose-Einstein condensate
is one way to produce particles in superposition at the macroscopic scale.
They are made by cooling a class of particles to near absolute zero.
Cold particles move slower, and according to quantum mechanics slower particles have longer wavelengths.
As the temperature lowers, the wave functions of the particles begin to overlap.
This causes them to enter a shared quantum state.

Quantum mechanics is currently the most accurate theory for how our universe works, but it is still limited.
Quantum mechanics doesn't explain large scale phenomena, like gravitation, dark matter, or dark energy.
New theories that unite gravity with quantum mechanics are being developed, but so far there aren't any testable
predictions.