In the early 1900s physicists performed experiments that probed reality at the nanometer scale.
They found that matter was made of particles they named atoms.
Inside each atom they found the subatomicparticles: electrons, protons, and neutrons.
They also found that light was made of particles that we now call photons.
Strangely they found that all particles produced wave-like interference patterns, not just light.
wave and particle models of reality
In physics a model is a mathematical simplification of reality.
Particles model reality as a point-like object with no internal structure, just properties like position, velocity, mass, and momentum.
When particles collide they bounce off each other. Imagine a sports ball.
Wave model reality in a less intuitive way.
They don't exist at a point. They spread out in every direction at a constant speed.
They can overlap with each other in superposition to form interference patterns.
Imagine sound waves.
waves or particles as fundamental
When I studied physics I was originally taught the mystery of wave particle duality.
Things like electrons were sometimes wave-like and sometimes particle-like according to rules that seems to bend reality.
Wave-particle duality is nice because you mostly imagine simple particles and you only think about waves in special cases.
These days there is an growing consensus among physicists that fundamentally stuff is just waves, not particles.
The various particle-like properties have been found to naturally emerge out of the math of waves.
We don't need special rules anymore to explain particle-like behavior.
How particle-like behavior emerges from quantum waves is explained in detail in advanced physics topics like quantum field theory, decoherence, and the many-worlds interpretation.
Those are some big ideas, and they take time to fully digest.
So the big picture is that everything is just waves, but particles are a helpful simplification.
(in my opinion)
The overlap between the wave and particle models at the nanometer scale led to some exciting experiments.
Based on these results, 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 an energy well.
The work function represents the minimum energy the electrons need 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 electron volts 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.
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, but they aren't otherwise dangerous.
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 lead to cell death and an increased 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.
CO₂ takes roughly 50-200 years to be removed from the atmosphere, so the effects of our actions will be felt for a long time.
Exactly how increased CO₂ levels will shape global climate change is complex.
Limiting CO₂ emissions is probably the cheapest way to reduce the impact of climate change.
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.
What if two photons hit at the same time?
It can happen, but the probability is very low, and the energy of two photons is still below the ionization threshold for just 2 microwave photons.
If there was a high enough intensity of microwave photons to ionize atoms, the microwave oven would turn produce so much heat that it would produce a plasma.
Plasma is the state of matter, where electrons are stripped from atoms, and yes this would be very dangerous.
Plasma is generally limited to extreme situations like lighting or stars.
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 contribute to a vitamin D deficiency.
Also, not being in the sun can just make you feel sad.
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 act like a wave.
As the electrons 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.
principal (n)
1,2,3, ...
energy or number of standing wave nodes
azimuthal (ℓ)
0,1,2, ...
shape or number of angular nodes (s,p,d)
magnetic (mℓ)
-ℓ to ℓ
spatial orientation of the standing wave
spin (ms)
+½ or −½
electron’s possible internal spin states
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.
Particles outside an atom don't normally form standing waves so they aren't quantized.
Although, photons can inherit quantization when they are emitted from electrons in atoms.
Electron and photon quantization is similar to guitar strings and musical notes.
Sound waves in air can have any frequency.
Guitar strings only have quantized frequencies because the strings form standing waves.
Sound waves leaving a plucked guitar string will inherit this quantization, so a guitar only makes specific notes.
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 solar 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.
The double slitexperiment is a good example of wave interference.
A wave passing through a slit spreads out. It diffracts.
This pattern had been seen in water waves, but in 1801, Thomas Young produced a double slit interference pattern with beams of light.
At the time this was interpreted as evidence that light was a wave...
single slit wave diffractiondouble slit wave interference
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.
separation =
Question: How many constructive interference peaks are on the far right edge of this diagram?
answer
There are 5 constructive interference peaks.
In the constructive interference section both waves are high and low at the same time.
In the destructive regions the waves are out of phase by half a wavelength so that one wave is up while the other wave is down.
In 1909, the experiment was repeated with a single photon at a time.
After passing through the double slit, each photon was measured at a single location,
but the sum of repeated experiments produced a wave interference pattern.
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: Even when one particle at a time is fired, the distribution of hits produces an wave-like interference pattern.
What could the single particle be interfering with? What's going on?
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.
Even a single particle has wave-like properties.
Explaining why is still unsolved, but there are a few options.
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 double slit experiment isn't limited to photons.
In 1927, the experiment was repeated with one electron at a time.
In 2019, an interference pattern was produced by a 2000 atom molecule.
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.
Also, you might notice if you play with the simulation, the size of the slit needs to be similar to the size of the wave, and larger objects have smaller wavelength.
Wavelength and Momentum
We can calculate the quantum wavelength of a particle in terms of it's momentum with the equation below.
derivation of particle wavelengths
Special relativity explains the relationship between energy and momentum.
We can use that relationship to find how wavelength and momentum relate.
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.
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 waves 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 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 the odds will be very close to 50:50 heads vs. tails.
This leaves each experimental 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 classical macroscopic world.
Decoherence acts like a filter on quantum states.
It isolates classical states while removing nonclassical superposition states.
It's pretty confusing how our classical reality emerges from quantum rules.
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 questions are potentially answered by the different interpretations of quantum mechanics.
foundations 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 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 quantum mechanics describing 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.
Quantum mechanics is describing only a wave, with particles that emerge as excited wave states.
Quantum effects are transmitted locally.
Measurement causes decoherence, which isolates different parts of 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.
Quantum mechanics is describing 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.
Quantum mechanics is describing 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.
In case you wanted more practice I used AI to make some more problems. The rest of the site I made by hand, but generating endless problems seemed safe. I did find mistakes in the AI generated problems, and there are probably some I didn't find. Let me know if something could be fixed. I also added a practice problem on each page with no solution. That's intentional. Have fun!
Question: A double slit experiment sends one photon at a time toward the slits. Each photon is measured at one spot on the screen, but after many photons the hits form an interference pattern. What does that tell us about using only the particle model for light?
answer
The particle model is not enough by itself. Each photon is detected like a particle at one location, but the overall pattern is wave-like.
This is why the page treats photons as quantum objects. They are measured as packets, but their probability pattern can behave like a wave.
Example: A green photon from a laser pointer has frequency 5.60 × 1014 Hz. The beam looks continuous to your eye, but each photon carries a separate packet of energy. What is the energy of one photon in joules and electron-volts?
solution
Use Planck's equation for the energy of one photon.
Example: A detector records photons with energy 4.80 × 10-19 J. What frequency do those photons have, and what wavelength would they have in vacuum?
solution
Example: Neutral hydrogen in space emits a famous 21 cm photon that radio telescopes can detect from huge clouds of gas. What are the frequency and energy of one 21 cm photon?
solution
The frequency is 1.43 × 109 Hz, and the energy is 9.48 × 10-25 J.
Question: Why can dim ultraviolet light release electrons from some metals, while very bright red light usually cannot?
answer
The photoelectric effect depends on energy per photon, not just total brightness.
Ultraviolet photons have higher frequency and enough energy per photon to dislodge electrons. Red photons have lower energy per photon. Adding more red photons usually means more low-energy packets, not one packet with enough energy to release an electron.
Example: A small red laser pointer outputs 5.0 mW of optical power at wavelength 635 nm. The pointer uses two AAA batteries, but that does not matter for this calculation. Assume all the optical power is carried by 635 nm photons. About how many photons leave the laser each second?
solution
The battery information is extra context. Use the optical power and the energy of one photon.
Example: A blue LED flash releases 0.020 J of light at wavelength 450 nm. The plastic lens spreads the light into a wide cone, but assume every photon has the same wavelength. About how many photons are in the flash?
solution
The spread of the beam does not change the energy per photon for this question.
Example: An unknown metal surface is hit by 422 nm light from a 3.0 mW laser for 2.0 s. What is the maximum kinetic energy of the released electrons?
solution
This cannot be determined from the information given.
The maximum kinetic energy in the photoelectric effect depends on the work function of the metal.
$$K = hf - \Phi$$
The wavelength can tell us the photon energy, and the power and time can tell us total energy delivered. But without the metal's work function, the maximum kinetic energy of the released electrons cannot be solved.
Example: A clean metal surface has work function 4.2 eV. It is hit by ultraviolet photons with energy 7.0 eV each. What is the maximum kinetic energy of the released electrons in eV and joules?
solution
No electrons are released because the photon energy is less than the work function.
Example: A photoelectric experiment measures electrons leaving a metal with maximum kinetic energy 1.8 eV when each incoming photon has energy 6.5 eV. What is the work function of the metal?
solution
Question: A greenhouse gas molecule absorbs infrared photons strongly, but it lets most visible photons pass through. Why does the photon energy matter for this kind of absorption?
answer
Molecules only absorb photons well when the photon energy matches an allowed change in the molecule.
Infrared photons can match changes in molecular vibration and rotation. Visible photons usually have different energies, so many greenhouse gases do not absorb them strongly.
Example: A metal has work function 2.0 eV. What is the threshold frequency for releasing electrons from this metal?
solution
At the threshold, the photon energy equals the work function.
The longest wavelength is 4.14 × 10-7 m, or about 414 nm.
Example: An electron leaves a metal with maximum kinetic energy 2.4 eV. Using electron mass 9.11 × 10-31 kg, estimate the electron's speed. Assume the electron is slow enough that the usual kinetic energy equation is reasonable.
solution
Question: Why is ionizing radiation described using energy per particle instead of only total power?
answer
Ionization depends on whether a single particle has enough energy to break a chemical bond or remove an electron.
A large number of low-energy photons can heat a material, but each photon may still be non-ionizing. A single high-energy ultraviolet, x-ray, or gamma photon can be dangerous in a different way because it can disrupt atoms or molecules directly.
Example: In hydrogen, the n = 3 energy level is -1.51 eV and the n = 2 energy level is -3.40 eV. If an electron drops from n = 3 to n = 2, what photon energy is emitted, and what wavelength does that photon have?
level
energy
n = 2
-3.40 eV
n = 3
-1.51 eV
solution
The photon gets the energy lost by the electron.
$$\Delta E = -3.40\,\mathrm{eV} - (-1.51\,\mathrm{eV})$$
$$\Delta E = -1.89\,\mathrm{eV}$$
The photon energy is the magnitude of that change.
The emitted photon has energy 1.89 eV, or 3.02 × 10-19 J. Its wavelength is about 658 nm.
Example: A hydrogen atom absorbs ultraviolet light with wavelength 121 nm. Find the photon energy in eV, then decide whether it is above the 10 eV ionizing-radiation guideline on the page.
solution
This is slightly above 10 eV, so it is in the ionizing range by the page's guideline.
Question: If an electron in an atom drops from a higher energy level to a lower energy level, is a photon emitted or absorbed?
answer
A photon is emitted.
The electron loses energy when it moves to a lower energy level. That lost energy leaves the atom as a photon with a frequency and wavelength set by the size of the energy difference.
Example: A green photon has wavelength 532 nm. What is its momentum?
solution
Photons have no mass, but they still have momentum.
Example: An electron in an electron microscope moves at 2.0 × 106 m/s. Using electron mass 9.11 × 10-31 kg, what is its de Broglie wavelength?
solution
For a particle with mass, momentum is mass times velocity.
Why can sunlight push a solar sail even though light has no material medium like air or water?
answer
Light is an electromagnetic wave with electric and magnetic fields that interact with charged particles in matter. Those interactions transfer momentum and create a small forward force.
The force on a solar sail is tiny. Why can it still change a spacecraft's path by a meaningful amount?
answer
Sunlight can keep applying the force for a very long time. In space there is little resistance, so the continuing small acceleration can build a large change in velocity.
How is a reflective solar sail different from an ordinary photovoltaic panel on a spacecraft?
answer
A photovoltaic panel converts light into electrical energy. A solar sail is a large reflective surface designed to receive radiation pressure and provide thrust.
