Pluck a guitar string fixed at both ends. It can only hum at certain notes — a lowest one, then a higher overtone, then a higher one still, with nothing allowed in between. The Schrödinger equation says the tiniest pieces of nature, like electrons, behave the same way. Trap an electron in a small space and it can no longer have just any energy; it is stuck with a ladder of allowed "notes." That single idea — that confinement forces energy into discrete steps — is the heart of quantum mechanics.
From a picture to a number
To make this exact we stop talking about where the electron is and instead track a wave called the wavefunction, written $\psi$. The wave itself is not the electron; the meaningful quantity is $|\psi|^2$, which tells you the probability of finding the electron at each spot. The Schrödinger equation is simply the rule that decides which wave shapes are allowed and what energy each one carries. For the simplest case — a particle trapped in a box of length $L$ — the allowed shapes are exactly the standing waves of a guitar string, and the energies come out as
Put a single electron in a box about $1\ \text{nm}$ wide and the lowest rung is $E_1\approx0.38\ \text{eV}$. The next rungs are not evenly spaced — they climb as $n^2$, so they sit at $4E_1$, $9E_1$, $16E_1$. Notice the lowest energy is not zero: a confined quantum particle can never sit perfectly still. This leftover "zero-point" jiggle is forced by the uncertainty principle.
The precise statement
The full law has a time-dependent form, $i\hbar\,\partial\psi/\partial t=\hat{H}\psi$, where the Hamiltonian $\hat{H}=-\frac{\hbar^2}{2m}\nabla^2+V$ is just kinetic plus potential energy written as an operator. For a fixed energy this collapses to the time-independent equation $\hat{H}\psi=E\psi$, an eigenvalue problem whose solutions are the allowed states. In the sim, the n slider picks which rung you view, L sets the box width (energies scale as $1/L^2$), mass rescales the whole ladder, and in tunnelling mode V₀, a and E control a barrier the wave can leak through even when $E
Try this in the sim above
Raise n in "Particle in Box" and count the bumps in $|\psi|^2$ — rung $n$ always has $n$ humps. Then shrink L and watch $E_n$ shoot up as the box tightens. Finally switch to "Tunnelling," set the energy $E$ below the barrier height $V_0$, and widen the barrier $a$ a little at a time — the transmission $T$ readout collapses almost vertically, showing how ferociously tunnelling depends on barrier width.
Particle in a box (infinite well). $\psi=0$ at walls (boundary conditions) → standing wave solutions: $\psi_n\propto\sin(n\pi x/L)$. Energy quantised: $E_n=n^2\pi^2\hbar^2/(2mL^2)\propto n^2/L^2$. Zero-point energy $E_1>0$: particle can never be at rest — consequence of uncertainty principle ($\Delta x=L$, $\Delta p\geq\hbar/(2L)$, $K\geq(\Delta p)^2/(2m)$).
Stationary states. For $\hat{H}\psi_n=E_n\psi_n$: time-dependent solution $\Psi_n=\psi_n e^{-iE_nt/\hbar}$. Probability density $|\Psi_n|^2=|\psi_n|^2$ — time-independent (stationary). Superposition of stationary states oscillates: $|\alpha\psi_1+\beta\psi_2|^2$ has frequency $(E_2-E_1)/h$ — corresponds to photon emission frequency.
Zero-point energy follows from the uncertainty principle: confining a particle (small $\Delta x$) requires large $\Delta p$, giving kinetic energy $\geq(\Delta p)^2/(2m)>0$. The ground state energy is a minimum, not zero. You cannot extract it — you would need somewhere lower to "fall to." You can lower it by enlarging the confining region (isothermal expansion of ideal quantum gas), but that requires doing work.
Key takeaway: Zero-point energy is irreducible — from uncertainty principle. Cannot be extracted — no lower state.
STM (scanning tunnelling microscope) — images individual atoms. Flash memory (electrons tunnel through oxide layer). Nuclear fusion in stars (protons tunnel through Coulomb barrier at energies below classical threshold — enables the Sun to shine). Alpha decay (alpha particle tunnels out of nucleus). Enzyme catalysis (proton tunnelling). Josephson junctions in superconducting qubits.
This follows from the commutation relation $[\hat{a},\hat{a}^\dagger]=1$ for ladder operators. Each application of $\hat{a}^\dagger$ raises energy by exactly $\hbar\omega$ — the same amount each time. This equal spacing is special to the harmonic potential $V=\frac{1}{2}kx^2$; anharmonic potentials (e.g., Morse potential for molecules) have unequal spacings. Infrared spectroscopy exploits this to identify molecular bonds.
Key takeaway: QHO equal spacing: from $[\hat{a},\hat{a}^\dagger]=1$. Anharmonic potentials break this — molecule IR spectra.
$E_1=\pi^2\hbar^2/(2m_eL^2)=\pi^2\times(1.055\times10^{-34})^2/(2\times9.109\times10^{-31}\times(10^{-9})^2)=6.02\times10^{-20}$ J $=0.376$ eV. Halving box size: $E_1\to4\times0.376=1.50$ eV (scales as $1/L^2$). This $\sim$ eV scale matches typical quantum confinement in semiconductor quantum dots (1–3 eV).
Key takeaway: Electron in 1 nm box: $E_1=0.376$ eV. Halve L → 4× energy. Quantum dot confinement energies.
The simulation shows four quantum scenarios. Free particle: Gaussian wavepacket $|\psi|^2$ with de Broglie wavelength. Particle in box: standing wave $|\psi_n|^2=\frac{2}{L}\sin^2(n\pi x/L)$ with quantised energy $E_n=n^2E_1$. Harmonic oscillator: Hermite-Gaussian probability densities. Tunnelling: exponential decay through barrier and transmitted amplitude on far side.
Key takeaway: Four scenarios: free wavepacket, box standing waves, harmonic oscillator states, tunnelling through barrier.
Resources: Khan Academy; HyperPhysics (hyperphysics.phy-astr.gsu.edu); MIT OCW 8.962.
Section 05
Common Misconceptions
❌ Higher quantum number means the particle is more localised.
✅ The opposite is true: higher $n$ states have more oscillations in $\psi_n$ (more nodes) and, by the correspondence principle, approximate classical behaviour — delocalized over the full box. The ground state ($n=1$) has probability density concentrated near the centre. As $n\to\infty$, $|\psi_n|^2$ approaches the classical uniform distribution (particle spending equal time everywhere).
📖 Griffiths — Introduction to Quantum Mechanics, §2.2.
❌ Quantum tunnelling is a rare or unusual phenomenon.
✅ Tunnelling is ubiquitous. The Sun's nuclear fusion rate is almost entirely due to tunnelling — protons cannot classically overcome the Coulomb barrier at solar core temperatures. Flash memory (billions of devices) stores bits via tunnelling through thin oxides. All alpha-emitting nuclei (radioactive sources) undergo tunnelling. Proton tunnelling occurs in enzymes at room temperature.
❌ The Schrödinger equation describes the electron as a particle at a specific location.
✅ The Schrödinger equation describes the evolution of the wavefunction $\psi(x,t)$ — a probability amplitude spread throughout space. The electron does not have a definite position until measured. The wavefunction's square $|\psi|^2$ gives the probability density. This probabilistic, non-local description is what distinguishes QM from classical mechanics.
📖 Griffiths — Introduction to Quantum Mechanics, §1.1; Bohr — Atomic Physics and Human Knowledge.
Misconception research: Johnston et al. (2002) Phys. Educ.; McKagan et al. (2010) Phys. Rev. ST PER.