This is only the briefest of introductions to QM (quantum mechanics.) No high school physics curriculum has a comprehensive discussion of QM. As such, I can’t stress enough how important it is to read more, if one wants a meaningful understanding of this topic.
I do warn readers away from many popular works (even Stephan Hawking’s Brief History of Time) because despite how readable they are, they do not welldescribe what is going on. Instead, I suggest these books. They require more careful reading and thought, but if one puts in the effort, one comes out with a much better understanding of the subject.
The Cosmic Code: Quantum Physics as the Language of Nature, Heinz R. Pagels
Quantum Reality: Beyond the New Physics, Nick Herbert
In Search of Schrödinger’s Cat: Quantum Physics and Reality, John Gribbon
_________________
Outline and some text adapted from Giancoli Physics. Chapter 28: Quantum Mechanics of Atoms
A New Theory
Quantum mechanics unifies the wave–particle duality into a single consistent theory
This new theory, when it is applied to macroscopic phenomena, must be able to produce the old classical laws. This, the correspondence principle is satisfied fully by quantum mechanics. This doesn’t mean we should throw away classical theories such as Newton’s laws. In the everyday world, classical laws are far easier to apply and they give sufficiently accurate descriptions.
The Wave Function
For material particles such as electrons, quantum mechanics relates the wavelength to momentum according to de Broglie’s formula.
But what corresponds to the amplitude or displacement of a matter wave?
The amplitude of an EM wave is represented by the electric and magnetic fields, E and B.
In quantum mechanics, this role is played by the wave function, which is given the symbol (Greek capital letter ψ, psi, pronounced “sigh”).
This represents the wave displacement, as a function of time and position, of a new kind of field: a matter wave.
 Giancoli Physics
______________
The following is adapted from Wikipedia
The physical reality underlying de Broglie waves is a subject of ongoing debate.

Some theories treat either the particle or the wave aspect as its fundamental nature, seeking to explain the other as an emergent property.

Some theories, such as the hidden variable theory, treat the wave and the particle as distinct entities.

Some theories propose some intermediate entity that is neither quite wave nor quite particle, but only appears as such when we measure one or the other property.

The Copenhagen interpretation of QM states that the nature of the underlying reality is unknowable and beyond the bounds of scientific inquiry.
– Wikipedia, Matter Wave
from Giancoli Physics:
In QM we consider the wavefunction, ψ, as the fundamental mathematical rule describing the particle. The wavefunction interacts with its environment – similarly to how a light or water wave interacts with its environment. The waveparticle duality manifests itself when you measure it:
the square of the wavefunction ψ2ψ2 gives you the probability of finding your particle at a given position.
But remember that matterwave functions are nothing like water waves, or even EM waves:
Water waves are waves of water
EM waves are varying degrees of an Elec or Mag field.
But matter waves are not waves of matter! ψ is a purely mathematical constructs that tell us the probability of finding a particle at a particular point in space and time.
—————————————–
DoubleSlit Interference Experiment for Electrons
from Giancoli:
Consider two slits whose size and separation are on the order of the wavelength of whatever we direct at them, either light or electrons, Fig. 28–3.
We know very well what would happen in this case for light, since this is just Young’s doubleslit experiment: an interference pattern would be seen on the screen behind.
If light were replaced by electrons – with wavelength comparable to the slit size – they too would produce an interference pattern.
If we reduced the flow of e (or photons) so they passed through the slits one at a time, we would see a flash each time one struck the screen. At first, the flashes would seem random. Indeed, there is no way to predict just where any e would hit the screen.
But if we let the experiment run for a long time, and kept track of where each e hit the screen, then we’d see a pattern emerging—the interference pattern predicted by the wave theory.
Although we could not predict where a given e would strike, we could predict probabilities. (The same can be said for photons.)
The probability, as we saw, is proportional to Psi^2
Where Psi^2 is 0, we get a minimum in the interference pattern.
Where Psi^2 is a maximum, we would get a peak in the interference pattern.
The interference pattern would thus occur even when electrons (or photons) passed through the slits one at a time. So the interference pattern could not arise from the interaction of one electron with another.
It is as if an electron passed through both slits at the same time, interfering with itself.
This is possible because an e is not really a particle. It is as much a wave as it is a particle, and a wave could travel through both slits at once.
What would happen if we covered one of the slits so we knew that the e passed through the other slit, and a little later we covered the second slit so the e had to have passed through the first slit?
Result: no interference pattern would be seen. We would see, instead, two bright areas (or diffraction patterns) on the screen behind the slits.
If both slits are open, the screen shows an interference pattern as if each e passed through both slits, like a wave. Yet each e would make a tiny spot on the screen as if it were a particle.

if we treat e (and other particles) as if they were waves, then represents the wave amplitude.

If we treat e as particles, then we must treat them on a probabilistic basis.
The square of the wave function, gives the probability of finding a given e at a given point. We cannot predict—or even follow—the path of a single e precisely through space and time.
28.3: The Heisenberg Uncertainty Principle
Whenever a measurement is made, some uncertainty is always involved…. We expect that by using more precise instruments, the uncertainty in a measurement can be made indefinitely small.
But according to quantum mechanics, there is actually a limit to the precision of measurements. This limit is not a restriction on how well instruments can be made; rather, it is inherent in nature.
It is the result of two factors: the wave–particle duality, and the unavoidable interaction between the thing observed and the observing instrument.
Another useful form of the uncertainty principle relates energy and time.
An object has an uncertainty in position Δx ≈ λ
The photon that detects it travels with speed c
It takes time Δt ≈ Δx/c = λ/c to pass through the distance of uncertainty.
Hence, the measured time when our object is at a given position is uncertain by about Δt ≈ λ/c
Since the photon can transfer some or all of its energy ( = hf = hc/ λ ) to our object, then the uncertainty in energy of our object as a result is
This tells us that the energy of an object can be uncertain (or can be interpreted as briefly nonconserved) by an amount ΔE for a time Δt (as defined above)
28.4: Philosophic Implications; Probability versus Determination
RK writes:
Matter waves are not “just a theory” – they make specific predictions which can be tested. QM is the most fantastically tested description of reality ever made. Every experiment ever done has produced results fully in accord with predictions made by QM.
But what does QM mean? We picture subatomic particles as classical particles, like billiard balls. They have to have a definite position in space, and a definite velocity, at any point in time, right?
Yet if we make such an assumption, the meaning of the matterwave is mysterious, and QM produce results which violently contradict commonsense.
QM theory – and all experiment – shows that these particles do not follow the laws of classical physics, or indeed, any form of common sense.
(A) This has resulted in decades of philosophical theorizing about how this matterwave controls realworld objects. What are “objects”? What physical reality lies behind this mathematical “matterwave”?
(B) At the same time, we obviously live in a world where causeandeffect, and Newton’s laws of motion, do apply. So whatever truth lies behind QM, we already know that when many particles interact with each other, their resulting emergent behavior becomes classical:
In other words, the rules of the quantum world usually create a situation which corresponds to the normal world that we experience. This is called the correspondence principle.
(*) There are examples of seemingly impossible phenomenon, in which the weirdness of QM seeps through to our everyday world, which an astute observer may notice. Examples include superconductivity and the Meissner effect
One modern approach is to let go of the idea that subatomic particles are particles at all: The matterwave is the reality, and the “particle” is an emergent property that results from it.
If all matter shows wavelike behaviour, why is this not observable in everyday life?
Comic from SMBC (Saturday Morning Breakfast Cereal) by Zachary Alexander Weinersmith
text
Okay, so why don’t we notice objects in the real world havingwavelike behavior and diffraction?
Interpretations of Quantum Mechanics
Schrödinger’s cat thought experiment
Parallel universes in quantum mechanics
QuantumMechanical View of Atoms
e (electrons) do not exist in circular orbits
e act as if they are spread out in space, as a “cloud.”
The size/shape of the cloud can be calculated for a given state of an atom.
For the ground state in the hydrogen atom, the electron cloud is spherically symmetric.
An e cloud at its higher densities roughly indicates the “size” of an atom.
Just as a cloud may not have a distinct border, atoms do not have a precise boundary or a welldefined size.
Not all e clouds have a spherical shape (the same is true for any subatomic particle)
Check out these wild shapes
In the 2nd row those yellow barbells aren’t 2 different e nearby each other – that is the probability cloud of 1 e !
In the 3rd row, those blue shapes aren’t 3 or 4 different e ,
rather that is the probability of cloud of just 1 e.
And so on for the 4th row (pictured here)
and for other rows of e exicted to even higher energy levels (not pictured)
An e cloud, spread out in space, is a result of the wave nature of e.
e clouds can be interpreted as probability distributions for a particle.
We cannot predict the path an e will follow (thinking of it as a particle).

a path would assume that the e is an “object” with classical properties of position, momentum, etc

But we now know that e aren’t such classical objects at all.

Rather, e only sometimes behave as if they are, in certain welldefined circumstances
After a measurement of its position we cannot predict where it will be at a later time.We can only calculate the probability that it will be found at different points.
If you make 500 different measurements of an e position, in an atom, the majority of the results would show it where the probability is high. Only occasionally would it be found where the probability is low.
The e probability cloud becomes very small at places far away, but never becomes zero.
Erwin Rudolf Josef Alexander Schrödinger (1887 – 1961) was a Nobel Prizewinning Austrian physicist who developed a number of fundamental results in the field of quantum theory, which formed the basis of wave mechanics: he formulated the wave equation, and discovered that his mathematical theory was equivalent to the other mathematical model of Quantum mechanics, matrix mechanics. Schrödinger proposed an original interpretation of the physical meaning of the wave function.
 Wikipedia
External links
James Schombert Cosmology course: Quantum mechanics. Univ. Oregon
https://www.preposterousuniverse.com/eternitytohere/quantum/
Deeper Than Quantum Mechanics—David Deutsch’s New Theory of Reality: Constructor Theory
Constructor Theory of Information David Deutsch, Chiara Marletto
Learning Standards
SAT Subject Test: Physics
Quantum phenomena, such as photons and photoelectric effect
Atomic, such as the Rutherford and Bohr models, atomic energy levels, and atomic spectra
Nuclear and particle physics, such as radioactivity, nuclear reactions, and fundamental particles
Relativity, such as time dilation, length contraction, and massenergy equivalence
A Framework for K12 Science Education: Practices, Crosscutting Concepts, and Core Ideas (2012)
Electromagnetic radiation can be modeled as a wave of changing electric and magnetic fields or as particles called photons. The wave model is useful for explaining many features of electromagnetic radiation, and the particle model explains other features. Quantum theory relates the two models…. Knowledge of quantum physics enabled the development of semiconductors, computer chips, and lasers, all of which are now essential components of modern imaging, communications, and information technologies.
Massachusetts Science and Technology/Engineering Curriculum Framework 2006
Chemistry: Atomic Structure and Nuclear Chemistry
Atomic models are used to explain atoms and help us understand the interaction of elements and compounds observed on a macroscopic scale. Nuclear chemistry deals with radioactivity, nuclear processes, and nuclear properties. Nuclear reactions produce tremendous amounts of energy and lead to the formation of elements.
2.1 Recognize discoveries from Dalton (atomic theory), Thomson (the electron), Rutherford (the nucleus), and Bohr (planetary model of atom), and understand how each discovery leads to modern theory.
2.2 Describe Rutherford’s “gold foil” experiment that led to the discovery of the nuclear atom. Identify the major components (protons, neutrons, and electrons) of the nuclear atom and explain how they interact.
2.3 Interpret and apply the laws of conservation of mass, constant composition (definite proportions), and multiple proportions.
2.4 Write the electron configurations for the first twenty elements of the periodic table.
2.5 Identify the three main types of radioactive decay (alpha, beta, and gamma) and compare their properties (composition, mass, charge, and penetrating power).
2.6 Describe the process of radioactive decay by using nuclear equations, and explain the concept of halflife for an isotope (for example, C14 is a powerful tool in determining the age of objects).
2.7 Compare and contrast nuclear fission and nuclear fusion.
AP Physics Curriculum Framework
Essential Knowledge 1.D.1: Objects classically thought of as particles can exhibit properties of waves.
a. This wavelike behavior of particles has been observed, e.g., in a doubleslit experiment using elementary particles.
b. The classical models of objects do not describe their wave nature. These models break down when observing objects in small dimensions.
Learning Objective 1.D.1.1:
The student is able to explain why classical mechanics cannot describe all properties of objects by articulating the reasons that classical mechanics must be refined and an alternative explanation developed when classical particles display wave properties.
Essential Knowledge 1.D.2: Certain phenomena classically thought of as waves can exhibit properties of particles.
a. The classical models of waves do not describe the nature of a photon.
b. Momentum and energy of a photon can be related to its frequency and wavelength.
Content Connection: This essential knowledge does not produce a specific learning objective but serves as a foundation for other learning objectives in the course.