Resources
Quantum Playground is a public-engagement piece, not a research artifact — but it was built to hold up under a scientist's scrutiny, not just a general audience's. This page states the method plainly, derives the equations behind each topic, and lists every primary source. For a hands-on two-qubit simulator, see Build Your Own Experiment.
What the demos are, and aren't
Each topic pairs a small interactive visualization with the physics behind it. Don't read too much into the visuals, though — the glowing orbs and canvases are stand-ins, not the real thing. None of them are numerically solving the actual wave equation under the hood. They're built to make an idea click, not to model it precisely, and I'd rather be upfront about that than let the demo oversell itself.
Same goes for where a demo simplifies, or where physicists themselves still argue about what's really going on — I call it out directly, right in that topic's explanation and in its "A bit of the math" panel, instead of smoothing it over.
Standards this build holds to
- Checked against primary literature. Every claim was verified against the original papers or an authoritative secondary source, not restated from memory or general pop-science summaries.
- Popular shorthand is labeled as such. Where a common explanation is known to be an oversimplification — the "virtual particle pairs" picture of Hawking radiation, for instance — the page says so explicitly and explains the better-supported mechanism alongside it.
- Speculative ideas are attributed, not presented as settled. "Quantum foam" is credited to John Wheeler's 1955 conjecture and marked as still experimentally unconfirmed, rather than presented as established physics.
- Open interpretational questions are named as open. The measurement problem, for example, distinguishes the experimentally solid Born-rule statistics from the philosophically unresolved question of what "collapse" physically is.
Technical appendix — equations by topic
The formula shown in each topic's "A bit of the math" panel on the interactive build is deliberately the minimal version. This is the fuller derivation behind each one, with the primary source it traces back to.
Any pure qubit state can be written with just two real angles (θ∈[0,π], φ∈[0,2π)) instead of two free complex numbers — normalization and an unobservable global phase remove two of the four real degrees of freedom. θ sets the measurement odds; φ is invisible to a single-qubit measurement but becomes physical the moment the qubit interacts with anything else.
Standard Dirac / Bloch-sphere formalism — see Nielsen & Chuang, §1.2.
The Born rule is the one part of this story that's an experimentally confirmed law, not an interpretation. What's genuinely unsettled is the mechanism — whether |ψ⟩→|n⟩ is a real physical process, an artifact of decoherence with no true collapse (many-worlds, consistent histories), or something else.
Born, M. (1926). Zur Quantenmechanik der Stoßvorgänge. Zeitschrift für Physik.
The fringes come entirely from the cross term 2√(I₁I₂)cosδ — block either path (close a slit, or entangle a which-path marker) and it vanishes, leaving the flat classical sum I₁+I₂. This holds whether one particle or 10¹² go through; the single-electron result is the direct demonstration that the fringes build up one particle at a time.
Two-path interference — Feynman, Leighton & Sands, Lectures on Physics Vol. III, ch. 1.
Local hidden-variable theories are bound to |S|≤2; a Bell state (|00⟩+|11⟩)/√2 measured at the right angles saturates 2√2≈2.828. The 2022 Nobel-winning experiments closed the remaining loopholes and measured violations consistent with quantum mechanics, not local realism.
Clauser, Horne, Shimony & Holt (1969). Proposed Experiment to Test Local Hidden-Variable Theories. PRL 23, 880.
Position–momentum is the special case usually quoted, but the general Robertson–Schrödinger relation holds for any two observables with a nonzero commutator. It's a property of non-commuting operators on Hilbert space, not a statement about measurement disturbance — a common but imprecise popularization.
Robertson, H.P. (1929) — full citation in References.
The general form is the WKB approximation to the Schrödinger equation in the classically forbidden region V(x)>E; the site's demo uses the rectangular-barrier case, where the integral collapses to a closed form. A scanning tunneling microscope's exponential sensitivity to gap width follows this exact relation.
Standard WKB treatment — Griffiths, Introduction to Quantum Mechanics, §8.2.
Grover's algorithm is a geometric rotation in the 2D subspace spanned by the marked and unmarked states — each iteration rotates the amplitude vector by a fixed angle 2θ toward the marked state. Bennett, Bernstein, Brassard & Vazirani proved this Θ(√N) scaling is provably optimal for unstructured search.
Grover (1996); Bennett et al. (1997) — full citations in References.
Alice and Bob each pick a random basis per bit; they publicly compare only which basis was used (never the value) and discard mismatches. Because measuring in the wrong basis unavoidably disturbs a qubit (no-cloning theorem), an eavesdropper intercepting and resending leaves a detectable ~25% error rate in a sampled portion of the key.
Bennett & Brassard (1984) — full citation in References.
The rigorous derivation is a Bogoliubov transformation between the quantum field's vacuum state before and after spacetime curves around a collapsing star — not literally particle pairs popping into existence at the horizon. That picture is a widely used teaching shorthand this site flags explicitly, because it gets the qualitative result right without the correct mechanism.
Hawking (1975); Unruh (1976) — full citations in References.
The Planck length is where "quantum foam" becomes concrete: the scale at which quantum fluctuations of the gravitational field itself are conjectured to rival spacetime's own curvature. No experiment has probed anywhere near it — the shortest distance directly tested is still ~10¹⁸ times larger — so this remains a theoretical extrapolation, not a measured effect.
Wheeler, J.A. (1955) — full citation in References.
Citing this project
This is a public-engagement piece, not a peer-reviewed publication — but if you want to reference it (a talk, a syllabus, a blog post), here's a suggested informal citation:
References
The primary papers and results the physics on this site is drawn from, in the order they're first used across the ten topics.
- Feynman, R.P. (1948). Space-Time Approach to Non-Relativistic Quantum Mechanics. Reviews of Modern Physics, 20(2), 367–387. — path integral, Topic 1.
- Kennard, E.H. (1927). Zur Quantenmechanik einfacher Bewegungstypen. Zeitschrift für Physik, 44, 326–352; Robertson, H.P. (1929). The Uncertainty Principle. Physical Review, 34(1), 163–164. — rigorous form of Δx·Δp ≥ ħ/2, Topic 5.
- The Nobel Prize in Physics 2022, awarded to Alain Aspect, John F. Clauser & Anton Zeilinger "for experiments with entangled photons, establishing the violation of Bell inequalities." The Royal Swedish Academy of Sciences. — rules out local hidden-variable explanations, Topics 2 & 4.
- Merli, P.G., Missiroli, G.F. & Pozzi, G. On the statistical aspect of electron interference phenomena. American Journal of Physics, 44(3), 306–307 (experiment run 1974, published 1976); Tonomura, A., Endo, J., Matsuda, T., Kawasaki, T. & Ezawa, H. (1989). Demonstration of single-electron buildup of an interference pattern. American Journal of Physics, 57(2), 117–120. — single-particle double-slit interference, Topic 3.
- Bennett, C.H. & Brassard, G. (1984). Quantum Cryptography: Public Key Distribution and Coin Tossing. Proceedings of IEEE International Conference on Computers, Systems and Signal Processing, Bangalore, 175–179. — the BB84 protocol, Topic 8.
- Grover, L.K. (1996). A Fast Quantum Mechanical Algorithm for Database Search. Proceedings of the 28th Annual ACM Symposium on Theory of Computing (STOC '96), 212–219; Bennett, C.H., Bernstein, E., Brassard, G. & Vazirani, U. (1997). Strengths and Weaknesses of Quantum Computing. SIAM Journal on Computing, 26(5), 1510–1523. — Grover's algorithm and its proven optimality, Topic 7.
- Hawking, S.W. (1975). Particle Creation by Black Holes. Communications in Mathematical Physics, 43(3), 199–220; Unruh, W.G. (1976). Notes on Black-Hole Evaporation. Physical Review D, 14(4), 870–892. — black hole evaporation and its link to the Unruh effect, Topic 9.
- Wheeler, J.A. (1955). Geons. Physical Review, 97(2), 511–536. — the original, still-speculative "quantum foam" conjecture, Topic 10.
- Born, M. (1926). Zur Quantenmechanik der Stoßvorgänge. Zeitschrift für Physik, 37, 863–867. — the Born rule, Technical Appendix §2.
- Clauser, J.F., Horne, M.A., Shimony, A. & Holt, R.A. (1969). Proposed Experiment to Test Local Hidden-Variable Theories. Physical Review Letters, 23, 880–884. — the CHSH inequality, Technical Appendix §4.
Every "A bit of the math" panel throughout the interactive build carries its own caveats and citation notes closer to the specific claim they support — this list gathers the primary sources behind them in one place. Corrections are welcome — see Contact.