Spin a coin on a table. While it whirls, "heads or tails?" has no answer yet — it is somehow leaning toward both, and only settles when it topples and you look. A quantum object is stranger still: before you look it is genuinely a blend of its choices, that blend can interfere with itself, and the act of looking forces a single answer at random. That blend-then-pick is the whole story of quantum mechanics in miniature.
Naming the blend
The simplest quantum object is a two-choice system — a "qubit." Call its two choices $|0\rangle$ and $|1\rangle$. A blended state is written $|\psi\rangle=\alpha|0\rangle+\beta|1\rangle$. The numbers $\alpha$ and $\beta$ are amplitudes, and the rule that turns them into something you can bet on is the Born rule: the chance of measuring $0$ is $|\alpha|^2$ and the chance of measuring $1$ is $|\beta|^2$. They must cover all possibilities, so $|\alpha|^2+|\beta|^2=1$.
Work one number. Set $\alpha=\beta=\tfrac{1}{\sqrt2}\approx0.707$. Then $|\alpha|^2=0.5$ and $|\beta|^2=0.5$: a perfect 50/50 coin. Slide $\alpha$ up to $1$ and the chance of $0$ becomes certain — the blend has collapsed into a plain answer even before you measure.
The precise picture
Amplitudes are complex numbers, so each carries a phase as well as a size, and phase is physically real: $(|0\rangle+i|1\rangle)/\sqrt2$ and $(|0\rangle-i|1\rangle)/\sqrt2$ give the same 50/50 odds yet are different states. A tidy way to hold both size and phase is the Bloch sphere, $|\psi\rangle=\cos(\tfrac{\theta}{2})|0\rangle+e^{i\varphi}\sin(\tfrac{\theta}{2})|1\rangle$, where $\theta$ and $\varphi$ point to a spot on a globe. For a spin-½ particle every measurement of $S_z$ returns just $+\hbar/2$ or $-\hbar/2$. And two particles can share a state, like $|\Psi^-\rangle=(|\!\uparrow\downarrow\rangle-|\!\downarrow\uparrow\rangle)/\sqrt2$, that cannot be split into "Alice's part times Bob's part" — measuring one instantly fixes the other. The CHSH test sharpens this: any local-hidden-variable world obeys $|S|\le2$, but quantum mechanics reaches $|S|=2\sqrt2\approx2.83$.
Try this in the sim above
Drag the amplitude slider to $\alpha=1.000$ and watch $|\alpha|^2$ lock to 1 — a pure $|0\rangle$ that always measures the same. Pull it back to $0.707$ and the two basis circles grow equal. Then switch to Entanglement mode and sweep the Bell angles $a$ and $b$: the correlation traces $E(a,b)=-\cos(b-a)$, dipping below anything a classical coin pair could manage.
Superposition. A quantum state can be a linear combination of basis states: $|\psi\rangle=\alpha|0\rangle+\beta|1\rangle$. Before measurement: system is genuinely in both states simultaneously (not just unknown). After measurement: collapses to one eigenstate with probability $|\alpha|^2$ or $|\beta|^2$. The interference between $\alpha$ and $\beta$ (via phase) is what makes quantum computing powerful.
2
Spin-½ particles. Electron spin has only two measurement outcomes: $+\hbar/2$ (up) or $-\hbar/2$ (down) along any axis. Measuring $S_z$ on a spin-$x$ eigenstate gives 50/50 result. Sequential measurements along different axes yield QM statistics that violate classical hidden-variable predictions (Bell inequality). Confirmed experimentally to high precision.
3
Entanglement. A two-particle state $|\Psi^-\rangle=(|\uparrow\downarrow\rangle-|\downarrow\uparrow\rangle)/\sqrt{2}$ cannot be written as $|\psi_A\rangle\otimes|\psi_B\rangle$. Measuring particle A instantly determines particle B's state, regardless of distance. This violates Bell's inequality — nature is non-local (but cannot be used for FTL signalling, as measurement outcomes are random).
4
CHSH Bell inequality. For any local hidden variable theory: $|S|\leq2$. Quantum mechanics predicts $|S|=2\sqrt{2}\approx2.83$ at optimal angles. Aspect et al. (1982) confirmed QM prediction. Recent loophole-free experiments (Delft 2015, Vienna 2015) confirm the violation — Hensen et al. measured $|S|\approx2.42\pm0.20$ — decisively ruling out local realism.
Ref: Griffiths — Introduction to Quantum Mechanics (3rd Ed.), Chs. 1–4; Feynman Lectures on Physics Vol. III; Aspect et al. (1982) Phys. Rev. Lett. 49, 1804.
Section 04
Frequently Asked Questions
A particle in superposition $|\psi\rangle=\alpha|0\rangle+\beta|1\rangle$ does not have a definite value of the observable — the question "is it 0 or 1?" has no answer before measurement. This is not just ignorance: the interference term in $|\alpha+\beta|^2\neq|\alpha|^2+|\beta|^2$ shows the two possibilities interact. After measurement, the superposition collapses to a definite state.
Key takeaway: Superposition: no definite value before measurement. Phase interference is physically real.
MRI (nuclear spin alignment), laser operation (stimulated emission, quantum levels), transistors (quantum tunnelling in MOSFET gates), LED/laser diodes (band structure), atomic clocks (hyperfine transition), electron microscopes, photovoltaics, and emerging: quantum computers (IBM, Google), quantum cryptography (BB84 protocol), and quantum sensors.
Key takeaway: QM underlies MRI, lasers, transistors, LEDs, atomic clocks, and all of modern electronics.
No. When you measure a particle and get a result, you cannot control what result you get — it's random. So you cannot encode a message. Your partner measures their particle, also gets a random result. The correlation only becomes apparent after comparing notes through a classical channel (no faster than light). The Holevo bound formally proves QM cannot transmit information FTL via entanglement.
Key takeaway: Entanglement: instantaneous correlations but no FTL signalling — Holevo bound and no-communication theorem.
$|\alpha|^2=|1/\sqrt{2}|^2=1/2$. $|\beta|^2=|i/\sqrt{2}|^2=1/2$. Both outcomes $+\hbar/2$ and $-\hbar/2$ have probability 50%. But the phase $e^{i\pi/2}=i$ matters: this state is an eigenstate of $\hat{S}_y$ (not $\hat{S}_z$). Measuring $S_y$ gives $+\hbar/2$ with probability 1. Phase contains real physical information.
Key takeaway: Phase matters: $(|0\rangle+i|1\rangle)/\sqrt{2}$ is $|+y\rangle$. Measuring $S_y$ gives +ℏ/2 with certainty.
Resources: Khan Academy; HyperPhysics (hyperphysics.phy-astr.gsu.edu); MIT OCW 8.962.
Section 05
Common Misconceptions
❌ Quantum superposition means the particle is in multiple places at once.
✅ Superposition in position space means the wavefunction is spread over multiple locations — the particle has no definite position. But this is not the same as "being in multiple places" in a classical sense. The particle is described by a probability amplitude, not a collection of classical copies. Position is undefined, not multiply defined.
📖 Griffiths — Introduction to Quantum Mechanics, §1.1; Feynman Lectures Vol. III, §1.
❌ Entanglement allows particles to communicate with each other instantaneously.
✅ Entangled particles have correlated states, but no information is transmitted. Each measurement result is random and independent. Only after comparing results classically (at light speed or slower) can the correlation be detected. The no-communication theorem (provable from QM) forbids using entanglement for superluminal information transfer.
📖 Peres — Quantum Theory: Concepts and Methods, Ch. 6.
❌ The wavefunction collapse is a physical process that takes time.
✅ The timing of "collapse" is a matter of interpretation, not empirical prediction. In Copenhagen, collapse is instantaneous upon measurement (not physical). In Many-Worlds, no collapse occurs — the universe branches. In GRW spontaneous collapse models, it takes a finite time. All interpretations agree on measurement statistics (Born rule) — the "collapse" timing is not observable.
📖 Griffiths — Introduction to Quantum Mechanics, §1.2; Penrose — The Road to Reality, Ch. 29.
Misconception research: Muller & Weissman (2002) Am. J. Phys.; Johnston et al. (2002) Phys. Educ.