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Build a foundations-first understanding of quantum states, their evolution, and predictions for measurement.
For: Learners at different starting points, from curious beginners to students with physics and mathematics preparation.
After this page you can: Choose a suitable learning path and explain a basic quantum prediction using a model or equation.
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Quantum mechanics is a mathematical framework for describing physical systems at scales where classical physics does not account for their observed behavior. In wave mechanics, the wave function is used to calculate probability distributions for measurement outcomes rather than a classical trajectory.
Students study the framework by connecting experiments to mathematical models: they represent a state, use an equation of motion or an operator, and compare its predictions with possible measurements. Introductory courses develop this through superposition, wave functions, uncertainty, Schrödinger’s equation, and examples such as potential wells and the harmonic oscillator. This hub offers routes through those ideas, with exercises to help you check your understanding.
Start with OpenStax’s chapter on quantum mechanics, which introduces wave functions, uncertainty, Schrödinger’s equation, and basic model systems.
Start with: University Physics Volume 3, Chapter 7: Quantum Mechanics
Use MIT’s introductory course if you have a good grasp of Newtonian mechanics, electricity and magnetism, and waves.
Start with: Quantum Physics I
Read OpenStax’s focused section on position and momentum uncertainty, then try its examples.
Start with: The Heisenberg Uncertainty Principle
A wave function is used to calculate the probability density for where a particle may be found.
MIT’s lecture notes include an introduction to superposition.
Wave function
A mathematical function used to calculate probabilities for measurement outcomes.
If a position wave function has squared magnitude 0.25 at a point, what does that value represent?
It is the probability density at that point; it is not by itself the probability of finding the particle in a finite interval.
Schrödinger’s equation relates a system’s wave function to its energy and potential, and can be used to predict how the state changes.
The available notes cover operators, Schrödinger’s equation, and time evolution.
Schrödinger’s equation
A differential equation used to describe the evolution of a quantum wave function.
Potential energy
An energy function that appears in Schrödinger’s equation and helps specify the system being modeled.
In the time-dependent Schrödinger equation, identify what mathematical object the equation is used to solve for.
It is used to solve for the wave function.
Introductory models such as a particle in a box and a harmonic oscillator show how solving Schrödinger’s equation can yield energy states.
This section works through the quantum harmonic oscillator as a model system.
Energy eigenstate
A state with a definite energy value under the system’s energy operator.
Name one model system used in introductory quantum mechanics and identify the equation commonly solved for it.
The harmonic oscillator is one such model; its states can be found by solving Schrödinger’s equation.
The position-momentum uncertainty relation places a lower limit on the product of the uncertainties in position and momentum.
The syllabus includes the uncertainty principle among the course’s introductory wave-mechanics topics.
Position-momentum uncertainty relation
The product of position and momentum uncertainties is at least one-half of the reduced Planck constant.
Measurement
A position measurement yields a particle position whose likelihood can be determined from the wave function.
If position uncertainty becomes smaller, what must happen to momentum uncertainty to satisfy the position-momentum uncertainty relation?
Momentum uncertainty must increase enough that their product remains at least one-half of the reduced Planck constant.
No recent updates are available.
David J. Griffiths and Darrell F. Schroeter · 2018
A clear starting point for building a systematic understanding of the subject.
Ramamurti Shankar · 1994
A broad course text whose mathematical introduction and path-integral treatment make it useful beyond a first pass.
J. J. Sakurai and Jim Napolitano · 2020
A strong next step for readers ready for a more advanced and mathematically developed treatment.
Claude Cohen-Tannoudji, Bernard Diu, and Franck Laloë · 2006
A useful reference to consult alongside a more linear course text.
P. A. M. Dirac · 1958
A historically significant complement for readers interested in the theory’s formal foundations.
Richard P. Feynman, Robert B. Leighton, and Matthew Sands · 1970
An engaging conceptual companion to more systematic textbook treatments.
Of the courses with “quantum mechanics” or “quantum physics” in their title or in a topic tag, these are listed as Quantum Mechanics courses: