§ 00 · Visual overview

Matter is a wave.
Until you look.

Quantum mechanics is not spooky hand-waving — it is one equation moving one complex wave, and every strange headline follows from it. Every idea here — interference, uncertainty, quantized energy, tunneling — lives as a simulation you can push, drag, and replay. The math is honest. The intuition is buildable.

Double slit → Uncertainty → Quantized energy → Tunneling →
|ψ|² — where the particle would be found Re ψ — the wave underneath A wave packet bouncing inside a box, evolving live under the Schrödinger equation. It spreads, shatters against the walls, and — wait for it — periodically reassembles itself.
§ 01 · Wave–particle duality

One particle, two paths, at once

Photons arrive at the screen below one at a time — each lands at a single point, like a particle. But let thousands accumulate and stripes emerge: each photon interfered with itself, passing through both slits as a wave. Now switch on the detector that watches which slit each photon uses. The stripes vanish. Knowing the path destroys the wave.

P = |ψ₁ + ψ₂|² = |ψ₁|² + |ψ₂|² + 2 Re(ψ₁*ψ₂)  —  the crossed term is the stripes
detection histogram quantum prediction |ψ₁+ψ₂|² Top: the photographic plate — every dot is one photon. Bottom: the same data binned, with theory overlaid.
Photons detected
0
Fringe spacing λL/d
–mm
Which-path info
–
Pattern
–

Notice what the detector does not do: it doesn't nudge the photons or block a slit. Merely making the path knowable kills the crossed term, and the stripes dissolve into one smooth, featureless hump. Turn it back off, clear the plate, and the universe resumes adding amplitudes instead of probabilities. Also try widening the slits apart — finer stripes — or reddening the light — coarser ones. The geometry obeys Δy = λL/d exactly.

§ 02 · The uncertainty principle

Sharper here, blurrier everywhere else

A quantum particle doesn't have a position and a momentum — it has one wave, and position and momentum are two readings of the same shape. Squeeze the wave narrow in space and it must contain many wavelengths: momentum blurs. Uncertainty isn't ignorance. It's geometry.

Δx · Δp ≥ ħ/2   —  a Gaussian packet starts exactly at the limit
|ψ(x)|² — position Re ψ — the wave itself |Φ(p)|² — momentum Left: the packet, released and evolving freely. Right: its momentum content — which never changes for a free particle.
Δx now
–
Δp (frozen)
–
Δx·Δp
–ħ
Heisenberg floor
0.500ħ

Drag Δx₀ down to 0.4 and watch the momentum hump on the right balloon — then watch the released packet fly apart in a blur, because it now contains fast and slow pieces travelling together. A wide packet barely spreads. This trade-off is why atoms don't collapse: the electron can't sit on the nucleus without acquiring enormous momentum. Uncertainty is the pressure that holds up matter.

§ 03 · Quantization

Energy comes in steps, not slopes

Trap a wave between two walls and only certain shapes survive — the ones that fit a whole number of half-wavelengths, exactly like a guitar string. Each allowed shape has one allowed energy. Everything in between is forbidden. That is the whole origin of the word “quantum”.

En = n²·h²/8mL²    n = 1, 2, 3, …  —  and E₁ > 0: a trapped particle can never stop
energy ladder Re ψ — standing wave, rotating in phase |ψ|² — probability A pure state's |ψ|² is frozen — nothing observable moves. Mix two states and probability starts sloshing.
Energy
–E₁
Nodes
–
de Broglie λ
–·L
|ψ|² motion
–

Two things to catch. First, the spacing: E grows as n², so the rungs of the ladder spread apart — jumps between them emit photons of specific colors, which is why every element has a barcode. Second, press superpose: a mixture of two energies is the only way this box gets moving probability — the lump sloshes at frequency (E₂−E₁)/ħ. Motion in quantum mechanics is interference between energy levels.

§ 04 · Quantum tunneling

Through the wall, with odds

This wave packet does not have enough energy to climb the barrier — a classical ball would bounce off every single time. But a wave doesn't stop dead at a wall: it leaks in, decaying exponentially, and if the far side arrives before the amplitude dies, part of the wave simply continues. Measured afterwards, the particle is sometimes on the other side.

T ≈ e−2κw,   κ = √(2m(V₀−E))/ħ  —  every extra bit of width w costs exponentially
|ψ|² barrier V(x) packet energy E Watch the moment of impact: most of the packet reflects, a ghost crosses inside the barrier, and a transmitted packet escapes right.
Transmitted
–%
Reflected
–%
WKB estimate e⁻²ᵏʷ
–%
Classical ball
–

Play with the exponential: double the width and transmission doesn't halve — it roughly squares, collapsing by orders of magnitude. That savage sensitivity is a feature: a scanning tunneling microscope hangs a needle one atom's-width above a surface and reads the tunneling current — which changes by ~7× per ångström — to photograph individual atoms. The sun runs on the same trick: protons tunnel through their electric repulsion to fuse. No tunneling, no sunlight.

§ 05 · Practice

Now you do the math

Each problem needs one idea from above and arithmetic you can do on paper. Hints reveal one step at a time — try before you peek, and use the simulators to check yourself.

1 · The electron that thinks it's a wave

An electron microscope accelerates electrons to v = 3.0 × 10⁶ m/s (1% of light speed). What is their de Broglie wavelength — and why does that number explain the machine's existence?

Hint 1 — Everything with momentum has a wavelength: λ = h/p = h/(mv), with h = 6.63×10⁻³⁴ J·s and mₑ = 9.11×10⁻³¹ kg.
Hint 2 — p = 9.11×10⁻³¹ · 3.0×10⁶ ≈ 2.7×10⁻²⁴ kg·m/s. Divide h by that.
Answer — λ ≈ 2.4×10⁻¹⁰ m ≈ 0.24 nm — the size of an atom, and ~2000× shorter than visible light. A microscope can never resolve detail smaller than its wavelength, so light microscopes stop at ~200 nm while electron beams resolve individual atomic columns. You are also a wave: at walking speed your λ ≈ 10⁻³⁵ m, which is why you have never diffracted through a doorway.

2 · An electron in a nano-box

An electron is trapped in a 1.0 nm box (roughly a small molecule). Find the ground-state energy E₁, and the wavelength of the photon emitted when it drops from n = 2 to n = 1.

Hint 1 — Use § 03: E₁ = h²/(8mL²). Keep everything in SI and convert to eV at the end (1 eV = 1.6×10⁻¹⁹ J).
Hint 2 — E₁ = (6.63×10⁻³⁴)² / (8 · 9.11×10⁻³¹ · (10⁻⁹)²) ≈ 6.0×10⁻²⁰ J. For the photon: ΔE = E₂ − E₁ = 3E₁, then λ = hc/ΔE (shortcut: λ[nm] = 1240/ΔE[eV]).
Answer — E₁ ≈ 0.38 eV, so ΔE ≈ 1.13 eV and λ ≈ 1100 nm — near-infrared. Shrink the box and colors shift blue as 1/L²: this is literally how quantum dots are tuned, and why the 2023 Nobel Prize in Chemistry went to people who got very good at making tiny boxes.

3 · Why atoms don't collapse

Classically, the electron should spiral into the proton and hydrogen should be ruined. Use Δx ≈ 0.05 nm (atomic confinement) to estimate the minimum kinetic energy uncertainty forces on the electron.

Hint 1 — From § 02: Δp ≥ ħ/(2Δx), with ħ = 1.05×10⁻³⁴ J·s. The electron's momentum is at least of order Δp.
Hint 2 — Δp ≈ 1.05×10⁻³⁴/(2 · 5×10⁻¹¹) ≈ 1.1×10⁻²⁴ kg·m/s. Kinetic energy: KE ≈ Δp²/(2m).
Answer — KE ≈ 6×10⁻¹⁹ J ≈ 4 eV — the right scale (hydrogen's true value is 13.6 eV). Squeezing the electron tighter raises this cost as 1/Δx², while the electric attraction only pays back as 1/Δx. The balance point is the atom's size. Matter has volume because localization is expensive.

4 · The microscope that feels atoms

An electron tunnels through a vacuum gap with barrier V₀ − E = 1.0 eV. Compare the transmission for gaps of w = 0.5 nm and w = 1.0 nm. What does this do for a scanning tunneling microscope?

Hint 1 — From § 04: T ≈ e^(−2κw) with κ = √(2m(V₀−E))/ħ. Compute κ first — convert 1.0 eV to 1.6×10⁻¹⁹ J.
Hint 2 — κ = √(2 · 9.11×10⁻³¹ · 1.6×10⁻¹⁹)/1.05×10⁻³⁴ ≈ 5.1×10⁹ m⁻¹. Then 2κw = 5.1 for the small gap and 10.2 for the large one.
Answer — T(0.5 nm) ≈ e^(−5.1) ≈ 0.6%, but T(1.0 nm) ≈ e^(−10.2) ≈ 0.004% — doubling the gap didn't halve the current, it cut it ~150×. The exponential converts sub-ångström height changes into large current changes, which is how an STM traces the bumps of single atoms. Verify the trend in § 04: double w and watch the transmitted percentage crater.