Look at a sodium street lamp and you see a colour that is not a blend of anything: a narrow, almost pure yellow-orange. Heat a pinch of table salt in a gas flame and the flame turns the same yellow. Nothing in Newton's mechanics says why a hot atom should give out only certain colours, or why every sodium atom in the universe gives out the same ones.
Quantum mechanics says why, and the answer is stranger than the question. A small thing such as an electron is not a tiny ball on a path. It is described by a wave, and the wave is not a ripple in anything. It is a list of how likely each outcome would be if you looked. Confine that wave and, like a guitar string, it can only vibrate in certain patterns. Each pattern has its own energy. Those energies are the colours.
A physics graduate carries a handful of pictures from this course for life. This piece builds six of them with sliders you can play with. If you want the gentlest possible start, read Quantum Mechanics: The Weirdness at the Bottom (13/06) first. This one goes further: it computes.
- Who takes it
- Physics majors, usually in year 2 or 3 (at MIT it is 8.04, then 8.05). Chemists, materials scientists and electrical engineers meet a shorter version. Programmes in the style of MIT Course 8 and the Cambridge Natural Sciences physics route teach it before statistical physics and particle physics.
- What it is for
- Predicting what a small system can do (its energies, its spins, its odds) from a wavefunction and the equation that moves it.
- Comes after
- Linear Algebra (34/04), Waves & Optics (34/80) and Partial Differential Equations (34/79).
- Leads to
- Statistical Mechanics (34/84), Particle & Nuclear Physics (34/86) and Condensed Matter Physics (34/87).
- What this piece gives you
- You will not be able to solve the hydrogen atom. You will see why it has the colours it has, why uncertainty is a property of waves, and what a Bell test actually tested. Easier first reads: Quantum Mechanics (13/06) and The Standard Model (13/07).
The state is a wave, and its square is a probability
In classical mechanics the state of a ball is two lists: where it is and how fast it moves. In quantum mechanics the state of an electron is a function ψ(x), the wavefunction, a number (in general a complex one) for every place. You cannot see ψ. What you can get from it is the odds of finding the electron near x. Max Born proposed this reading in 1926, and it is called the Born rule.
The wave is fixed; each look gives one random place. Collect many looks and the pile follows the square of the wave. Here is the simplest wave worth knowing: a particle trapped on a line of length L, with walls it cannot pass. The allowed waves are standing waves, ψn = √(2/L) sin(nπx/L), with n = 1, 2, 3 …
The curve is |ψn|² computed from the formula. The bars are positions drawn at random from that curve, in 20 bins. The middle third of the box is where n = 1 is most likely to be found and where n = 2 is least likely to be.
With n = 1 you find the particle in the middle third of the box 61% of the time. With n = 2 only 20%. A classical bead rattling between walls would be found there 33% of the time, wherever you look. The wave has nodes, places where it is exactly zero, and the particle is never found at a node, even though it passes from one side to the other.
Squeeze a wave and its energy comes in steps
A standing wave that fits the box has a whole number of half-wavelengths: L = nλ/2. A short wavelength means a large momentum (de Broglie, 1924: p = h/λ), and momentum means energy. So the allowed energies are not a continuum. They are a ladder.
Worked number: an electron in a box 1 nm wide (a few atoms across) has E1 = 0.376 eV. The step from n = 2 to n = 1 is 3 × 0.376 = 1.13 eV, which is a photon of wavelength 1,099 nm, in the near infrared. Shrink the box to 0.1 nm, the size of an atom, and the same step is 113 eV, in the ultraviolet. Squeezing a wave raises its energy, which is why atoms do not collapse: the electron's lowest wave is never at zero energy.
En = n²h²/(8mL²) with CODATA h and masses. The slider runs from 0.01 nm to 1 cm. For the grain the ladder is drawn, but its real steps are far below anything measurable.
The grain row is the point. Quantization is not absent for large things; it is unmeasurably fine. The same formula that gives an electron a 1 eV step gives a milligram grain in a 1 cm box a step of about 10−38 eV. Hydrogen is the real-world version. Its levels are En = −13.6 eV ÷ n², and a jump from n = 3 to n = 2 releases a photon at 656.5 nm in vacuum (656.3 nm in air), the red line astronomers call H-alpha. Sodium's yellow doublet near 589 nm comes from the same logic with a more complicated atom.
Superposition: a wave can be a mix, and the mix sloshes
Waves add. If ψ1 and ψ2 are both allowed, so is any blend a·ψ1 + b·ψ2. That is a superposition, and it is the same fact as adding two guitar notes, not a mystical one. What is odd is the reading: the particle is not "secretly in one of the two states". The blend is the state.
The square of a sum contains a cross term, and the cross term is where interference lives. Each pattern's energy sets how fast its phase turns, so the two parts drift in and out of step and the probability sloshes back and forth at the beat frequency (E2 − E1) ÷ h. For the 1 nm electron box that is one slosh every 3.67 femtoseconds. This is how an excited atom radiates: its charge cloud oscillates, and an oscillating charge makes light.
|ψ|² = (1−a)ψ₁² + aψ₂² + 2√(a(1−a)) ψ₁ψ₂ cos(2πt). The average position is integrated numerically. At a = 0 or 1 nothing sloshes: a single energy has a frozen density.
At 50:50 the average position swings by about 18% of the box width either way, and then back. At 0% or 100% nothing moves at all: a state with one energy is a stationary state. That is why atoms are stable in their lowest level.
Uncertainty: not clumsiness, but what waves are
A short wave packet, a pulse, is a sum of many wavelengths. A pure single wavelength is spread over all space. You cannot have a wave that is both narrow in place and sharp in wavelength, and this is true of sound, of water and of a radio signal long before quantum mechanics. Quantum mechanics adds one thing: for a particle, wavelength is momentum (p = h/λ). So the same trade-off becomes Heisenberg's relation (1927).
It is not about the disturbance of measuring. It says the quantity "exact position and exact momentum at once" does not exist for the wave. Worked number: pin an electron into an atom-sized region, Δx = 0.1 nm, and the smallest possible momentum spread gives a velocity spread of 580 km/s. That is a quarter of the electron's speed in hydrogen's lowest orbit (2.19 × 10⁶ m/s). This is why an atom has its size: squeeze the electron closer and its motion grows faster than the squeeze can pay back.
Δp = ħ/(2Δx), the minimum for a Gaussian packet; Δv = Δp/m. The two curves are schematic: their widths are drawn so that the product stays fixed.
Now slide to a baseball and 1 mm. The velocity spread is about 10−30 m/s, which is why nobody ever noticed uncertainty in daily life.
Spin and measurement: asking changes the answer
An electron has an inner two-valued property called spin. Send a beam through a magnet whose field changes with position and each electron comes out on one of exactly two paths, "up" or "down" relative to the magnet's axis. That is the Stern–Gerlach experiment (Frankfurt, 1922, with silver atoms). Rotate the magnet and it is still two paths, never a smear.
The state of a spin is a two-component arrow, and measuring along another axis gives "up" with probability cos²(θ/2), where θ is the angle between the two axes. Measuring also changes the state: after you filter for up along z, the electron really is z-up, and an x measurement then gives 50:50 and erases the z information. Put three filters in a row and the odd result follows: z-up, then x-up, then z again lets some z-down electrons through, even though the first filter removed all of them. A pair of crossed polarising sunglasses with a third lens slid between them does exactly this to light.
Start with a beam that is already z-up. Middle filter passes "up along angle θ" with probability cos²(θ/2). The last filter asks z again. Final z-down fraction = cos²(θ/2) · sin²(θ/2) = sin²(θ)/4. Runs use the browser's random numbers, so they differ each time.
At 90 degrees, a quarter of the beam ends up z-down. With no middle filter, none do. At 0 or 180 degrees the middle filter either does nothing or flips the beam cleanly, and the final count is 0 or 100%.
Entanglement: correlations no pre-agreed script can fake
Prepare two spins in the combined state with total spin zero (the "singlet"), and fly them apart. Each one alone looks random. But when both are measured along the same axis they always disagree, and for axes at angle Δ the correlation is E = −cos Δ (+1 means always agree, −1 always disagree).
Is this like two sealed envelopes, one with a red card and one with a blue? Einstein, Podolsky and Rosen argued in 1935 that quantum mechanics must then be incomplete: hidden "instruction sets" travelling with the particles. In 1964 John Bell showed that any such local scheme must obey a bound. Using the CHSH form (Clauser, Horne, Shimony and Holt, 1969), pick two settings a, a′ on one side and b, b′ on the other:
E(α, β) = −cos(α − β) for the singlet. The run draws each pair's two ±1 results with the matching probabilities; it estimates S with a statistical error of about ±0.1.
Experiments sided with quantum mechanics: Clauser and Freedman (1972), then Aspect (1982), then "loophole-free" tests that closed the remaining gaps in 2015 (Delft, NIST and Vienna). The 2022 Nobel Prize in Physics went to Alain Aspect, John Clauser and Anton Zeilinger for this line of work. Bell's result does not say you can send a message faster than light: each side's results alone are random. It says the world cannot be explained by local pre-agreed instructions alone.
Four stories about what it means
Everyone who uses the maths gets the same numbers. What the maths is about is still argued. The main readings, in their holders' own terms:
- Copenhagen (Bohr, Heisenberg and descendants): the wavefunction is a tool for predicting results of measurements; asking what the electron "really did" between measurements has no answer.
- Many-worlds (Everett, 1957; later DeWitt, Deutsch, Wallace): the wave never collapses. Every outcome happens, in branches that no longer interact.
- Pilot-wave (de Broglie 1927; Bohm 1952): particles always have positions, guided by the wave. It reproduces the same predictions, at the price of non-local guidance, which Bell's theorem says any such theory needs.
- QBism (Fuchs, Mermin and others): the wavefunction is an agent's personal degrees of belief about what they will experience.
No experiment so far separates these as they are usually stated, and physicists are split. Treat any claim that physics has settled the question with caution.
What a graduate just sees
- The state is a wave; the square of the wave is the odds. Nodes are places the particle is never found.
- Confine a wave and its energy comes in steps; the smaller the box, the larger the steps and the higher the lowest energy.
- Spectra of atoms are the fingerprints of their energy ladders.
- Superposition is addition of waves, and interference lives in the cross term.
- Uncertainty is the narrow-versus-sharp trade-off of any wave, with ħ setting the exchange rate.
- Measuring along one axis resets the state; a third filter can restore what a second one hid.
- Entangled outcomes are correlated more strongly than any pre-agreed script allows (|S| up to 2.83 against a bound of 2), yet no message travels.
- The maths is agreed; the meaning is not.
Where it shows up next
- Statistical Mechanics (34/84): counting quantum levels is how temperature and entropy are defined for real matter.
- Particle & Nuclear Physics (34/86): fields are quantum waves; spin and superposition run the Standard Model.
- Condensed Matter Physics (34/87): billions of boxes coupled together make the energy bands of a chip.
- In the world: LEDs and lasers (levels and photons), MRI (spin), atomic clocks (the hydrogen-like levels again), flash memory (tunnelling), quantum cryptography (Bell tests), and the first small quantum computers.
If you want to go deeper
- 📘 Textbook. David Griffiths and Darrell Schroeter, Introduction to Quantum Mechanics (Cambridge, 3rd edition, 2018). The most widely assigned first course, with worked problems and a clear style.
- 📘 Textbook. J. J. Sakurai and Jim Napolitano, Modern Quantum Mechanics (Cambridge, 3rd edition). Starts from spin and states, closer to the linear-algebra view; a standard for the next course.
- 📘 Gentler. Leonard Susskind and Art Friedman, Quantum Mechanics: The Theoretical Minimum (Basic Books, 2014). Real maths, written for self-study, with no university needed.
- 🌐 Free course. MIT OpenCourseWare 8.04 Quantum Physics I (Spring 2016), followed by 8.05 Quantum Physics II.
A quantum system is a wave whose square gives the odds. Confine it and its energies become a ladder, which is why atoms have fixed colours and sizes. Mix waves and they interfere; ask a question and the answer rewrites the state; entangle two systems and their answers correlate beyond any script. The equations are agreed; what they mean is still open.
Try this: this week, take three polarising lenses (old polarised sunglasses can be cut or popped out, or use a pair of passive 3D cinema glasses) and cross two of them until the light goes dark, then slip the third between them at 45 degrees. Light comes back. That is the three-filter result, in photons instead of electrons.
Sources. The box, hydrogen, uncertainty and spin results are standard textbook quantum mechanics (Griffiths & Schroeter; Sakurai). Constants are CODATA 2018 values: h = 6.62607015 × 10⁻³⁴ J·s, electron mass 9.1093837015 × 10⁻³¹ kg, proton mass 1.67262192369 × 10⁻²⁷ kg, 1 eV = 1.602176634 × 10⁻¹⁹ J. Hydrogen H-alpha from the Rydberg formula with the reduced-mass correction (656.47 nm vacuum). Born 1926, Heisenberg 1927, Stern–Gerlach 1922, Bell 1964, CHSH 1969, Clauser–Freedman 1972, Aspect 1982, and the 2022 Nobel Prize in Physics (Aspect, Clauser, Zeilinger): Nobel Prize press release. The loophole-free tests of 2015 (Hensen et al., Delft; Shalm et al., NIST; Giustina et al., Vienna) are cited from the literature, not re-checked for this piece. Interpretation summaries follow the Stanford Encyclopedia of Philosophy entries on Copenhagen, many-worlds and Bohmian mechanics. Every number an interactive shows was computed in node: for example E1 = 0.376 eV at 1 nm, beat period 3.67 fs, 61% and 20% in the middle third for n = 1 and 2, Δv = 579 km/s at 0.1 nm, final z-down fraction 25% at 90 degrees, and S = 2.828 at the best settings.
- 🌐 Nobel Prize in Physics 2022: press release
- 🌐 Stanford Encyclopedia of Philosophy: Copenhagen Interpretation
- 🌐 MIT OCW 8.04 Quantum Physics I
- 📘 Griffiths & Schroeter, Introduction to Quantum Mechanics