Fire electrons one at a time. No detector: interference pattern — the electron goes through both slits. Detector on: two bands — the electron chooses one slit. The universe enforces a strict rule: path knowledge and wave behavior cannot coexist.
The wavefunction ψ encodes all possible positions as probability amplitudes. The probability of finding the particle at position x is |ψ(x)|² — the squared magnitude. Before measurement: spread out like a wave. After: collapsed to a point.
A quantum system exists in a genuine combination of all possible states until measured. Bell's theorem proves this isn't hidden ignorance. The Schrödinger equation governs the deterministic evolution of this state — randomness only enters at measurement.
It is not "switching rapidly between states." It is not "we don't know which state." Bell's theorem (confirmed by Aspect 1982 and loophole-free tests 2015) proves no pre-existing hidden value exists. The indeterminacy is ontological — baked into reality itself.
Position and momentum are Fourier conjugates. Sharpen the wavepacket in position (narrow Δx) and it must broaden in momentum space (large Δp). This is pure mathematics applied to physical reality — not about disturbing particles with measurements.
Short-lived states have uncertain energy. Virtual particles borrow energy from the vacuum for Δt ≈ ℏ/ΔE. The Casimir effect — measurable attraction between uncharged plates — is direct experimental proof.
Even at absolute zero, quantum systems retain ground-state motion. A harmonic oscillator's minimum energy is ℏω/2 — not zero. Confirmed by liquid helium, which refuses to freeze at 0K at atmospheric pressure.
A qubit is a point on the Bloch sphere — any superposition of |0⟩ and |1⟩. Gates are rotations of this sphere. Quantum algorithms engineer interference so correct-answer paths amplify and wrong-answer paths cancel.
Shor's Algorithm factors integers in polynomial time — breaking RSA encryption. Grover's Algorithm searches N items in √N steps. Quantum simulation models molecules exactly — impossible classically beyond ~50 atoms. These aren't incremental speedups — they're exponential separations.
Two particles entangled in a Bell state share a single quantum state spanning any distance. Measure one — the other's state is instantly correlated. Einstein called it spooky. Bell made it testable. Aspect proved Einstein wrong.
Hidden variables predict correlations ≤ 2 (CHSH). Quantum mechanics predicts up to 2√2 ≈ 2.83. Aspect 1982 measured ≈ 2.70. Loophole-free tests 2015 confirmed it definitively. Local realism is experimentally refuted.
The correlation is instant but you can't use it to send information. Your measurement gives a random result — B's observer also sees random results. The correlation only appears when you compare notes through a classical (slower-than-light) channel.
The wavefunction doesn't stop at a barrier — it decays exponentially through it. If the barrier is thin enough, nonzero amplitude survives on the other side. Transmission probability: T ∝ e^(−2κL).
Proton core temp is 10× too low for classical barrier crossing. Tunneling makes stars shine.
Alpha particles tunnel out of nuclei. Half-life changes 10²³× for small changes in nuclear radius.
Electrons tunnel through insulating layers billions of times per second in your storage devices.
When a quantum system entangles with its environment, quantum information leaks into billions of environmental degrees of freedom. Superposition doesn't collapse — it becomes observationally indistinguishable from a classical mixture.
Everett (1957): there is no collapse. The Schrödinger equation applies universally. Every measurement entangles the observer with all outcomes — all branches of the universal wavefunction are equally real.
| Interpretation | Collapse? | What's Real | Main Problem |
|---|---|---|---|
| Copenhagen | Yes | Only measurement outcomes | What collapses? How? |
| Many-Worlds | Never | All branches equally | Deriving Born rule |
| Pilot Wave | Apparent | Particles + guiding wave | Nonlocal, excess structure |
| QBism | Epistemic | Agent's beliefs only | Solipsism-adjacent |
| Relational QM | Relative | Observer-relative facts | No absolute reality |
Special relativity eliminates a universal "now." Two observers in relative motion have different simultaneity slices. Since no slice is privileged, the entire 4D spacetime block exists equally. You are a worldline, not a moving point.
Observer A moving toward a distant galaxy has a "now" that includes events 200 years in observer B's future. Both slices are equally valid — therefore those future events already exist. The block universe follows necessarily from special relativity.
If the block is static, why does time feel like it flows? The thermodynamic arrow — entropy always increasing — creates a direction. Memory only forms in the entropy-increasing direction. The experience of "flow" is neurological tracking of entropy gradients in a static 4D structure.
Electrons have intrinsic angular momentum — spin — with no classical analog. Spin-½ means a 720° rotation is needed to return to the original state. This is measurable and real, confirmed by interferometry experiments.
Spin-½ particles (electrons, quarks) are fermions — Pauli exclusion principle applies. No two can share the same quantum state. This is why matter is solid. Integer-spin particles (photons, gluons) are bosons — they pile into the same state. This is why lasers work.
A spin-½ state acquires a phase of −1 under 360° rotation and returns to itself only after 720°. This is not metaphor — neutron interferometry experiments confirmed it in 1975. Spinors are a deeper mathematical object than vectors.
Bell proved that hidden variable theories must satisfy |S| ≤ 2 (CHSH inequality). Quantum mechanics predicts |S| ≤ 2√2 ≈ 2.828. Every experiment confirms the quantum prediction. Local realism is experimentally dead.
Electrons don't orbit nuclei in classical paths — they inhabit probability density clouds. Each orbital is a solution to Schrödinger's equation. Energy quantization is why atoms emit discrete spectral lines — the fingerprint of every element.
Energy quantization means electrons occupy specific orbital energies. The Pauli exclusion principle means each orbital holds at most 2 electrons. These two facts — quantization + exclusion — determine the shell structure of every element and produce the entire periodic table.
Particles are not fundamental objects — they are quantized excitations of fields that permeate all space. The electron field, the photon field, the Higgs field. Everything is ripples in these underlying quantum fields.
Quantum electrodynamics predicts the electron g-factor to 12 decimal places. Most accurate theory in science.
12 fermions + 4 force bosons + Higgs. Describes all known particles and three of four fundamental forces.
The missing piece. QFT + general relativity remain incompatible. The frontier of fundamental physics.
Near the event horizon, the gravitational field interacts with quantum vacuum fluctuations. Virtual particle pairs form — one falls in, one escapes. The black hole appears to emit thermal radiation and slowly evaporates.
If a black hole evaporates completely as thermal radiation, what happens to the information that fell in? Thermal radiation carries no information. But quantum mechanics requires information conservation (unitarity). This remains unresolved — likely requiring a full theory of quantum gravity.