Particle in a Box

Last session we met the Schrödinger equation, the engine that moves wavefunctions through time. Today we bolt it to the simplest problem that still behaves in a genuinely quantum way: the infinite square well, nicknamed the particle in a box. Inside a region of width L the potential energy V(x) is zero, so the particle feels no force; outside, V is infinite. No real wall is infinitely high, but the idealization captures any tightly confined particle: an electron in a semiconductor nanocrystal, an electron roaming along a dye molecule, a proton rattling inside a nucleus.

What do the walls do to the wavefunction? No finite energy can pay an infinite potential, so the particle is never found outside and ψ = 0 there. In chapter 10 we demanded that acceptable wavefunctions be continuous, so ψ(0) = 0 and ψ(L) = 0. Requirements imposed at the edges of a region are called boundary conditions, and they are the entire secret of this chapter: they alone manufacture quantization.

Inside the box V = 0, so the time-independent Schrödinger equation keeps only its kinetic term: −ħ²/(2·m) times the curvature of ψ equals E times ψ. In words, the curvature of ψ is proportional to minus ψ itself, so the wavefunction always bends back toward the axis. Functions that behave like that are sines and cosines, with wavenumber k = √(2·m·E)/ħ, where k is 2π divided by the wavelength. A cosine equals one at x = 0 and breaks the left-hand condition, so only a sine survives: ψ is A times the sine of kx, where A is a constant amplitude. The right-hand wall is where the drama happens.

For ψ to vanish at x = L as well, the sine of kL must be zero, which happens only when kL is a whole-number multiple of π: kL = nπ, with n = 1, 2, 3 and so on. In terms of wavelength, a whole number of half-wavelengths must fit exactly across the box, λₙ = 2L/n, the same rule that fixes the notes of a guitar string. Two candidates are struck off the list. The value n = 0 gives a wavefunction that is zero everywhere, which describes no particle at all. Negative n merely flips the sign of ψ, an unobservable overall phase, so it adds no new state.

Now chain together two ideas you already own. De Broglie's relation p = h/λ turns each allowed wavelength into an allowed momentum, pₙ = n·h/(2L). Inside the box all the energy is kinetic, p²/(2·m), so Eₙ = n²h²/(8·m·L²). Stop and admire this. Bohr, in 1913, had to decree that angular momentum comes in fixed units; here nobody decrees anything. Quantization falls out of a wave equation plus two boundary conditions, the very point Erwin Schrödinger made in 1926 when he titled his papers Quantization as an Eigenvalue Problem. An eigenvalue problem is an equation that has acceptable solutions only for special values of a parameter, here the energy.

One loose end remains: the amplitude A. Born's rule from chapter 9 requires the total probability of finding the particle somewhere in the box to be one, a condition called normalization. A squared sine averages to one half, so A² times L/2 must equal one, giving A = √(2/L). The complete solution, ψₙ(x), is √(2/L) times the sine of nπx/L inside the box and zero outside. Each ψₙ is a stationary state, or energy eigenstate: a state of definite energy whose probability density, |ψₙ|², never changes in time. The integer n is a quantum number, a label that counts the allowed states.

Sketch the first three states. ψ₁ is a single arch, largest at the centre. ψ₂ is one full wave, positive on the left, negative on the right, and zero exactly in the middle. ψ₃ is three half-arches. A point inside the box where ψ = 0 is called a node, and there the probability density is exactly zero. The state ψₙ has n − 1 interior nodes, and the count is no accident: as chapter 10 showed, kinetic energy is tied to curvature, so more nodes in the same width mean more wiggle and more energy. Each probability density is a row of n equal bumps.

The lowest-energy state, the ground state, is n = 1, with E₁ = h²/(8·m·L²). That is not zero, and it never can be. This irreducible minimum is the zero-point energy: the energy a confined quantum system keeps even at absolute zero temperature. The bookkeeping reason is that n = 0 is not a state. The physical reason is the uncertainty principle of chapter 8: pinning a particle within a width L forces a spread in its momentum, and a momentum spread means kinetic energy. For an electron in a box one nanometre wide, E₁ ≈ 0.376 eV, about fifteen times the room-temperature thermal energy scale of 0.026 eV. Cooling cannot remove it: it is not heat but the price of confinement.

Now resist the most tempting picture in the subject: a tiny ball rebounding between the walls. In a stationary state the probability density is frozen: |ψ₂|² is zero at the centre now and forever, yet a shuttling ball would cross the middle on every trip. The standing wave is better read as a right-moving and a left-moving wave present simultaneously, not alternately. A momentum measurement on a highly excited state gives values near plus or minus n·h/(2L) with equal odds, but between measurements standard quantum mechanics assigns no trajectory. Interpretations differ on what lies underneath: in the de Broglie and Bohm pilot-wave picture of chapter 22, the particle in a stationary box state is actually at rest. Every interpretation agrees that |ψₙ|² is frozen.

Two features of Eₙ = n²h²/(8·m·L²) govern everything that follows. First, the energies climb as n², in the ratio 1, 4, 9, 16, so the rungs spread apart as you climb: the gap between level n and level n + 1 is (2n + 1)E₁. Second, every energy is inversely proportional to the mass and to the square of the width. Halve the box and every level quadruples; double it and every level drops to a quarter. Light particles in small boxes have huge level spacings; heavy particles in big boxes, tiny ones. That is the quantitative reason the quantum world is the world of the small.

Take the formula on a tour of scales. An electron in a box 0.1 nm wide, roughly the size of an atom, has E₁ of about 37.6 eV, the same order of magnitude as real atomic energies: hydrogen's ionization energy, from chapter 5, is 13.6 eV. A one-dimensional box is a crude caricature of an atom, yet it lands on the right scale. A proton, about 1,836 times heavier than an electron, in a box 10 fm across, the size of a nucleus, has E₁ of about 2.0 MeV. The box is ten thousand times narrower, which raises the energy a hundred-million-fold, while the extra mass lowers it 1,836-fold. Same formula, nuclear physics.

Then there are balls in boxes. A one-gram marble in a ten-centimetre box has E₁ of about 5.5×10⁻⁶³ joules, and its levels are far too close for any instrument to tell apart. Electrons in atoms and marbles in shoeboxes both obey quantum mechanics. What differs is whether the level spacing is large compared with every other energy in play, such as thermal jiggling or the resolution of your instrument. When the spacing wins, quantization is unmissable; when it loses, energy looks continuous and classical physics works beautifully.

If every stationary state is frozen, where does motion come from? From superposition. The Schrödinger equation is linear, so any sum of solutions is a solution. Mix equal parts of n = 1 and n = 2. Each component carries a phase that rotates in the complex plane at its own rate, e to the power minus i Eₙt/ħ. A single rotating phase is invisible in |ψ|², which is why stationary states stand still. But two phases rotating at different rates drift in and out of step, and the interference term in |Ψ|² oscillates as the cosine of (E₂ − E₁)t/ħ.

Now picture it: ψ₁ is positive everywhere; ψ₂ is positive on the left and negative on the right. When the two are in step they reinforce on the left and cancel on the right, so probability piles up in the left half of the box; half a period later the lump sits in the right half. Mathematically, the average position oscillates about the centre, L/2, as a cosine with amplitude 16L/(9π²): it swings about 18 percent of the box width either side of centre, with period T = h/(E₂ − E₁). For an electron in a box one nanometre wide, T ≈ 3.7 fs. Don't panic — superpose. All motion in quantum mechanics arises this way: a wave packet crossing a box is a superposition of many energy states whose phases march in and out of step.

Two consequences deserve a drumroll. First, the sloshing charge oscillates at a frequency equal to the energy gap divided by h, exactly the frequency of the photon emitted in the jump from n = 2 to n = 1. An oscillating charge radiates at its own frequency, which gave Schrödinger a natural account of Bohr's 1913 rule linking photon frequency to the energy gap. Second, an energy measurement on the sloshing state never returns an average: it gives E₁ or E₂, each with probability one half, and leaves the system in the state found, so the sloshing stops. Measurement means a physical interaction with an apparatus, not a conscious glance. Sloshing is real, too: Ahmed Zewail won the 1999 Nobel Prize in Chemistry for using femtosecond laser pulses to follow wave packets moving inside molecules.

A quantum dot is a semiconductor nanocrystal a few nanometres across. In a semiconductor, light lifts an electron across the band gap, a forbidden range of energies, leaving behind a positively charged vacancy called a hole. In a nanocrystal both are trapped in a tiny box, so each gains a confinement energy that grows as 1/L², added to the band gap. Shrink the dot and its light shifts toward blue; grow it and the light reddens. One material, a whole rainbow, selected purely by size. Real dots are more complicated: they are three-dimensional, in typical dot materials the electron moves with an effective mass well below its free mass, and electron and hole attract each other. Even so, the confinement term still grows as 1/L² and dominates in the smallest dots.

Glassmakers knew that one additive could tint glass different colours depending on how the molten glass was heated and cooled. In 1981 Aleksey Yekimov, in the Soviet Union, showed that copper chloride crystals inside such glass ranged from about 2 to 30 nm, and the smaller the crystals, the bluer the light they absorbed. In 1983 Louis Brus at Bell Laboratories, unaware of that work, saw the same effect in cadmium sulphide particles in solution: particles 4.5 nm across absorbed bluer light than particles 12.5 nm across. In 1993 Moungi Bawendi's group at MIT learned to grow near-perfect dots of chosen size. The three shared the 2023 Nobel Prize in Chemistry; today quantum dots turn blue light into pure reds and greens in QLED screens.

Chemistry found its boxes even earlier. In a conjugated molecule, a carbon chain with alternating single and double bonds, the π electrons are not tied to any one bond but spread along the chain, like particles in a one-dimensional box about as long as the molecule. In 1948 the Swiss chemist Hans Kuhn built this into the free-electron model of dye colour. Electrons fill the levels two at a time, the Pauli exclusion principle of chapter 18, so with N π electrons the highest filled level is n = N/2, and absorbing light lifts one electron up a level, costing ΔE = (N + 1)h²/(8·m·L²).

Try it on a simple cyanine dye: a three-carbon chain between two nitrogen atoms, with six π electrons spread over about 0.85 nm. The model predicts absorption near 340 nm; the measured peak is about 309 nm, within about ten percent. Longer chains shrink the gap because L² grows faster than N + 1, which is why the fluorescent label Cy5, whose chain is two carbon atoms longer than that of Cy3, absorbs near 650 nm rather than 550 nm. Cyanines suit the model because resonance makes their chain bonds nearly equal in length. Polyenes such as beta-carotene, whose single and double bonds stay unequal, fare much worse, a failure Kuhn traced to that bond alternation.

The correspondence principle, named by Niels Bohr in 1920 for an idea he had used since 1913, is stated in textbooks as the requirement that quantum predictions merge with classical ones at large quantum numbers. A classical ball moving at constant speed spends equal time everywhere, so its probability density is flat, 1/L. At n = 1 quantum mechanics disagrees sharply: about 61 percent of the probability lies in the middle third of the box, against the classical 33 percent, and at n = 2 only about 20 percent. But as n grows, |ψₙ|² becomes a comb of n ever-narrower bumps, and any detector of finite width averages them into the flat classical value.

The energies agree. The fractional gap between neighbouring levels, (2n + 1)/n², shrinks toward zero. Our marble, crawling at one centimetre per second, sits at n of roughly 3×10²⁷, where adjacent levels differ by roughly one part in 10²⁷. Quantum mechanics has not switched off; its graininess is too fine to detect. Yet even at enormous n, a single stationary state is still frozen. A real bouncing ball corresponds to a wave packet built from many neighbouring states, sloshing like our two-state superposition but with countless components forming one compact lump that travels from wall to wall. The bouncing ball is what a well-built superposition becomes in the classical limit.