Quanta

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Accelerating voltage600 V Slit separation d272 nm Cover the lower slit Which-way detector at the slits Show the |ψ|² prediction on the histogram

Who saw it

1. c. 1801Thomas Young — fringes of light behind two slits: light is a wave.
2. 1924Louis de Broglie — every particle has a wave of wavelength λ = h/p.
3. 1927Davisson and Germer (and independently G. P. Thomson) — electron diffraction by crystals. Nobel Prize 1937.
4. 1974–89Merli, Missiroli, Pozzi (Bologna), then Tonomura et al. (Hitachi, 1989) — fringes built up from single electrons; instead of slits they used a so-called electron biprism.
5. 1999Arndt, Zeilinger et al. (Vienna) — interference of C₆₀ molecules on a diffraction grating.
6. 2013Bach, Batelaan et al. — electrons through two nanoslits that could be blocked.

Like Millikan (1916): for each color we look for the voltage that stops the fastest electrons. eU₀ = hf − W, so the points lie on a straight line with slope h/e — the same for every metal.

From Planck to Einstein

1. 1887Heinrich Hertz notices that ultraviolet light makes a spark jump more easily.
2. 1900Max Planck: radiation energy is exchanged in packets E = hf — “an act of desperation” in his calculation.
3. 1902Philipp Lenard: the electrons' energy does not depend on the brightness of the light.
4. 1905Albert Einstein: light itself is made of quanta (photons). For this — not for relativity — he won the Nobel Prize for 1921.
5. 1916Robert Millikan, skeptical of photons, carefully measures U₀(f) — and confirms Einstein's formula.

In everyday life

• Solar cells — a photon in silicon knocks an electron out of its bond (the internal photoelectric effect).
• Camera sensor — each pixel counts the electrons freed by photons.
• Night-vision goggles — a photocathode turns a handful of photons into electrons, which are then multiplied.

Excite the electron to level:

An energy staircase

Niels Bohr (1913) assumed that the electron can have only certain energies. When it jumps down, it gives off the difference as a single photon. Jumps down to level n = 2 make up the Balmer series — four lines of visible light: 656, 486, 434 and 410 nm. It is hydrogen's fingerprint in the spectra of stars (see Spectroscopy).

The energies from Bohr's model are correct (for hydrogen!), but the picture of a tiny orbiting planet is not. According to electrodynamics, an orbiting electron should radiate and fall into the nucleus in a fraction of a second. The true description comes from the Schrödinger equation: instead of a path there is an orbital — a cloud of probability. The orbit radii in the drawing are not to scale (in reality they grow as n²).

Stimulated emission on the same transitions powers lasers.

An electron is not a tiny planet

Each point of the cloud is one possible place where a measurement could find the electron. We sample the points from the |ψ|² distribution of the real hydrogen wave functions — dense where the probability is high.

Nodes are places where ψ = 0 and the electron is never found: spheres (radial nodes, e.g. in 2s and 3s) and planes or cones (angular nodes, e.g. in 2p). Turn on the cross-section to see them. Colors show the sign of the wave function — important when orbitals combine into bonds (Molecules).

Orbitals p_x, d_xy and so on are real combinations of states with different m — the ones drawn in chemistry. The cloud does not show motion along a path: an electron in a stationary state does not “orbit,” and its probability distribution does not change.

Click an orbital box: empty → ↑ → ↓ → ↑↓ → empty. The number of electrons determines the element (neutral atom).

Three rules

• Pauli exclusion principle (1925): one orbital holds at most two electrons, and they must have opposite spins (↑↓).
• Aufbau principle: lower-energy subshells fill first: 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p…
• Hund's rule: within a subshell, electrons first occupy separate orbitals with parallel spins; only then do they pair up.

That is why the periodic table has its shape: the s-block has 2 columns (one orbital), the p-block 6 (three orbitals), the d-block 10 (five orbitals). Elements with a similarly filled outermost subshell have similar chemistry.

The filling order is only an approximate rule: for example, chromium (3d⁵ 4s¹) and copper (3d¹⁰ 4s¹) are exceptions.

A wave that counts odds

The wave function ψ is a complex number at every point. Its squared modulus |ψ|² tells us where we are most likely to find the particle (Max Born, 1926). The simulation solves the equation with the Crank–Nicolson method on a grid of more than a thousand points, in units where ħ = m = 1.

Box: quantized energies

Tunneling

• Alpha decay — an α particle tunnels through the barrier of the nucleus (Gamow, Gurney and Condon, 1928); see Radioactivity.
• The Sun — protons in the Sun's core have too little energy to overcome their repulsion classically; fusion happens thanks to tunneling.
• Scanning tunneling microscope (Binnig and Rohrer, Nobel 1986) — the tunneling current changes steeply with distance, so you can “see” individual atoms.
• Flash memory and the tunnel diode (Esaki, 1957) — electrons pass through thin insulating layers. Without quantum physics there would be no transistors (Computer).

Width in position1.00

Δx·Δp = —

units: ħ = 1; the Heisenberg limit is 0.50

Not a clumsy measurement

Werner Heisenberg (1927) realized that position and momentum cannot both be pinned down arbitrarily precisely at the same time. It is not the instruments' fault but a property of waves: a short packet must be made of many wavelengths (see Fourier), and wavelength is momentum (λ = h/p). A Gaussian packet reaches the limit exactly; every other shape has a larger product.

Why don't atoms collapse? Squeezing an electron closer to the nucleus increases the spread of its momentum, and therefore its kinetic energy. The balance between this “uncertainty pressure” and the pull of the nucleus sets the size of the atom (≈ 0.05 nm for hydrogen).

Zero-point energy: a trapped particle never sits still — not even at absolute zero. That's why helium at ordinary pressure doesn't freeze even at 0 K.

The slit measures position

Passing through a slit of width a, the photon “learns” where it is to within Δy ≈ a. The price is a sideways spread of momentum: the first dark spot lies at an angle sin θ = λ/a, so Δp_y ≈ p λ/a = h/a.

The same formula works for electrons: that's why an electron microscope, with its very short wavelength, sees finer detail than an optical one.

Measurement changes the state

Otto Stern and Walther Gerlach (1922, Frankfurt) sent silver atoms through a nonuniform magnetic field. Instead of a smeared streak they got two spots: the projection of spin onto a chosen axis has only two values, “up” or “down.”

Set up z → x → z: after the first magnet only “z up” atoms remain. Measuring along x gives 50 / 50. Then z again… and it's back to 50 / 50, even though we had filtered out “z down”! Measuring x destroyed the information about z.

settings	pairs	E quantum	E local model

Correlations stronger than classical

For a pair of photons in an entangled state, the probability of matching results is cos²(a − b). No model in which each photon “carries” predetermined answers (local hidden variables) can give S > 2 — that is Bell's inequality (1964) in the CHSH form (1969). Experiments (Clauser 1972, Aspect 1982, loophole-free tests in 2015) confirmed the quantum predictions. Nobel Prize 2022: Aspect, Clauser, Zeilinger.

Can this send a message faster than light? No. Each of them sees random 50 / 50 results on their side, whatever the other one does. The correlation only shows up after comparing the lists of results — over an ordinary channel, slower than light.

The local model in the simulation: each pair carries a hidden angle λ, and the result is the sign of cos 2(angle − λ). It gives exactly S = 2 at the optimal angles — it can't do any better.

|ψ⟩ = 1·|0⟩ + 0·|1⟩

Superposition is not “both at once”

A qubit's state is a direction on the sphere: the north pole is |0⟩, the south pole |1⟩. A state on the equator is not “zero and one at the same time” in the everyday sense — it is a perfectly definite state; it's just that a measurement along z gives a random result for it (50 / 50). Gates are rotations of the sphere: X — by 180° about the x axis, H — about the diagonal axis between x and z.

What are quantum computers for?

For certain tasks: Shor's algorithm (1994) factors numbers faster than any known classical method (a threat to today's RSA encryption), and simulations of molecules and materials could help chemistry. They are not “faster at everything.” Today they have from a few dozen to over a thousand noisy qubits; error correction needs many physical qubits per logical one, and that is still the main challenge.

Polish connections

Artur Ekert (born 1961 in Wrocław, works in Oxford and Singapore) proposed quantum cryptography based on entanglement and Bell's inequality in 1991 (the E91 protocol). Wojciech Żurek (born 1951 in Mysłowice, Los Alamos) is one of the founders of the theory of decoherence and co-author of the no-cloning theorem for quantum states (1982). Important contributions to the theory of entanglement also came from the Horodecki family of Gdańsk, among others.

Top: the potential V(x) — click to release the ball from a different spot. Bottom: phase space (x, p) — drag to pick any initial state.

The Hamiltonian — a recipe for motion

The state of the system is two numbers: position x and momentum p. Every point of the (x, p) plane — phase space — is one possible state. The function H, the total energy, tells you which way and how fast that point moves. Energy is conserved, so the point never leaves its contour H = const: a drawing of the contours is a map of every possible motion at once.

William Rowan Hamilton (Dublin, work from 1833–1835) rewrote Newton's mechanics this way. Almost a hundred years later, the same function H became the operator Ĥ in quantum mechanics — see the next section.

We compute the motion with the Verlet (“leapfrog”) method. It is a symplectic method: it preserves areas in phase space, so the energy wobbles only slightly but doesn't drift away. The simple Euler method with the same step adds energy on every lap — the orbit unwinds into a spiral. Arbitrary units, m = 1.

Parameter—

Energy levels

n	En	En / E1	En − En−1

Units: ħ = m = 1, region width L = 1 (infinite walls at the edges). Grayed out: states above the rim of the well.

In quantum mechanics the momentum p becomes the operator −iħ d/dx, and the Hamilton function becomes a recipe that acts on the wave function. States that Ĥ merely multiplies by a number are stationary states: their |ψ|² does not change in time. The numbers E are the only energies a measurement can give — that is why the energy in a well is quantized.

• Infinite well: E_n ∝ n², so E₂ = 4E₁, E₃ = 9E₁.
• Oscillator: levels evenly spaced by ħω, and the lowest has energy ½ħω, not zero — that is the zero-point energy (“Uncertainty” mode, key T).
• Finite well: only a few bound states; above them the particle is not trapped.
• Double well: levels come in close pairs — tunnel splitting. This happens in the ammonia molecule NH₃ (the nitrogen above or below the plane of the hydrogens); the transition within such a pair, about 24 GHz, was used in the first maser (Townes et al., 1954).
• “Crystal”: the levels cluster into bands separated by gaps. How the bands are filled decides whether a solid is a metal, a semiconductor or an insulator (Computer).

—

Finding energies = diagonalizing the Hamiltonian

On a grid of points, the wave function is simply a list of numbers, and the Hamiltonian is a table — a matrix. On the diagonal sits the “on-site energy”: the potential V plus the cost of squeezing the particle into one point. Right next to the diagonal are the hopping elements — they let the particle move to a neighboring point, and that is where kinetic energy comes from. Everything farther out is zero.

Diagonalization is a change of viewpoint: we rotate the coordinate system until all the off-diagonal elements vanish. Then the diagonal holds the energy eigenvalues, and the combined rotations give the eigenstates. Jacobi's method (1846) does this by rotating two rows and two columns at a time, each time zeroing the largest element.

That is how quantum chemistry programs compute the energies of molecules — only with enormous matrices and cleverer methods. For many electrons, the size of the matrix grows exponentially with their number; that is one reason simulating molecules is considered one of the most promising uses of quantum computers (mode Y).

The potential comes from the “Quantum” section. A 24-point grid is coarse, so E₁ from the matrix differs slightly from the 240-point result.

Two states, one Hamiltonian

Without coupling (Δ = 0) the energies of states |1⟩ and |2⟩ simply cross. Coupling pushes the levels apart: an avoided crossing forms with a gap of 2Δ, and the stationary states become mixtures of |1⟩ and |2⟩. A system prepared in |1⟩ is therefore not an eigenstate — the probability sloshes back and forth between |1⟩ and |2⟩. A complete transfer happens only at resonance (ε = 0). On the Bloch sphere (as in mode Y), the state simply rotates about the axis set by Ĥ.

• Ammonia NH₃: |1⟩ — nitrogen above the plane of the hydrogens, |2⟩ — below it. Δ comes from tunneling, and the gap corresponds to microwaves with a frequency of about 24 GHz.
• A qubit in a microwave field: switching to a frame that rotates along with the field, we get the same Hamiltonian — Δ sets the pulse strength, ε the detuning from resonance. That is how an X gate is performed.
• Neutrino oscillations (simplified to two flavors): a neutrino of a given flavor is a mixture of two states with different masses, so in flight it partly changes flavor — following a formula of the same sin² form. Nobel Prize 2015: Kajita and McDonald.