Bohr Model · n = 1–5
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Where Physics Becomes Interactive Thought.
Explore quantum wavefunctions, tunnelling, orbitals, and uncertainty states.
An interactive journey from classical crisis to quantum reality. Each lesson builds on the last, guiding you from intuition to formalism.
Research Archive
Interactive quantum physical models built from numerical first principles.
3D interactive simulation of Niels Bohr's atomic model. Visualise electron shells, orbital energies, and quantum transitions in real time.
Explore 3D probability density clouds of hydrogen orbitals computed from Laguerre polynomials and spherical harmonics.
Solves 1D finite difference Hamiltonian matrices for particle-in-a-box energy eigenstates and wavefunction amplitudes.
Static bound state penetration & dynamic Gaussian wavepacket reflection and transmission through potential walls.
Visualise trade-offs between position and velocity precision with spectral expansion and live stats overlay.
The Project
This is an evolving experimental project — a personal lab notebook rendered interactive. Each simulation is built from mathematical first principles, not borrowed libraries or pre-baked demos.
The goal is simple: to let curious minds encounter physics as something you can touch, rotate, break, and rebuild — not just read about.
Every concept in quantum mechanics connects to others. Click any node to jump to that lesson.
In 1913, a young Danish physicist proposed something radical: electrons can only exist at specific, quantized distances from the nucleus — and nothing in between.
Overview
The Bohr Model was a pivotal step in the development of quantum mechanics. It introduced the idea that electrons occupy discrete energy levels and that transitions between these levels produce or absorb specific wavelengths of light.
Historical Context
By 1900, classical physics had hit a wall. The prevailing model of the atom — Thomson's "plum pudding" — and later Rutherford's nuclear model could not explain why atoms emit light in discrete spectral lines instead of a continuous rainbow.
J.J. Thomson discovers the electron, shattering the idea of the indivisible atom.
Ernest Rutherford proves the atom has a dense positive nucleus. But why don't electrons spiral inward?
Niels Bohr proposes that electrons occupy fixed, quantized orbits and emit photons when jumping between them.
Core Idea
Bohr's radical postulate was that electrons don't just orbit anywhere — they can only exist in specific, allowed orbits. Each orbit corresponds to a discrete energy level, labelled by the principal quantum number n.
The key insight: Between orbits, there is no in-between. An electron does not smoothly drift from one level to another — it makes an instantaneous quantum leap.
When an electron falls from a higher orbit (n=2) to a lower one (n=1), the energy difference is released as a photon with a precise wavelength. This is why hydrogen emits specific spectral lines, not a smeared spectrum.
Interactive Demonstration
Adjust the principal quantum number n and atomic number Z to see how the orbital radius and energy change — live, with the electron orbit and energy ladder updating as you drag.
Mathematics
The negative sign indicates a bound state — energy must be added to remove the electron. As \(n\) increases, the electron is farther from the nucleus and less tightly bound.
Common Misconceptions
Misconception: Electrons physically travel between orbits during a quantum leap.
Reality: There is no trajectory. The electron simply “is” in one state, then “is” in another. There is no in-between path to describe.
Misconception: The Bohr Model is correct quantum mechanics.
Reality: It is a historical approximation. It fails for multi-electron atoms and cannot predict spectral fine structure. The Schrödinger equation supersedes it.
Real-World Applications
Spectroscopy: Every element emits a unique spectral fingerprint. Astronomers use the Bohr Model’s predictions to identify elements in distant stars.
Lasers: The population inversion mechanism exploits electron transitions between discrete energy levels — a direct consequence of quantized orbits.
Interactive Lab
Now that you understand the theory, explore the Bohr atom in a fully interactive 3D simulation. Adjust electron shells, atomic number, and rotation speed.
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The user should remember this specific point before proceeding.
Electrons do not orbit the nucleus like planets around a star.
Niels Bohr proposes that electrons travel in quantized orbits.
Erwin Schrödinger formulates wave mechanics.
Interactive Demonstration Container
Solve 1D potentials numerically.
Interactive 3D simulation of Niels Bohr's atomic model. Electrons orbit the nucleus in quantised shells — each shell at radius r₀ · n². Hover over orbits for quantum data. Scroll to zoom. Drag to rotate.
In Bohr's 1913 model, electrons don't spiral into the nucleus — they occupy discrete, stable orbits. Each orbit corresponds to a specific angular momentum, and the electron cannot exist between shells. This was the first quantisation postulate in atomic physics.
The radius of each orbit grows quadratically, \(r_n = n^2 a_0\). The innermost shell \((n=1)\) sits at the Bohr radius \(a_0 \approx 0.529\,\text{Å}\) — half an Ångstrom from the nucleus — while the energy of shell \(n\) follows the Rydberg energy law below.
When an electron jumps from a higher shell \(n_i\) to a lower one \(n_f\), the atom emits a photon whose energy equals the difference between the two levels, \(\Delta E = E_f - E_i\). Because only certain transitions are allowed, hydrogen produces line spectra — discrete wavelengths rather than a continuous rainbow.
The key insight: between orbits there is no in-between — an electron does not smoothly drift from one level to another, it makes an instantaneous quantum leap. The Bohr model predicts hydrogen's spectrum with striking accuracy, but it is a historical stepping stone: the Schrödinger equation, which replaces orbits with probability clouds, supersedes it for real atoms.
3D probability density clouds of hydrogen-like atomic orbitals computed from Laguerre polynomials and spherical harmonics using Monte Carlo sampling.
In quantum mechanics the electron in a hydrogen-like atom is not a particle on an orbit — it is described by a wavefunction \(\psi(r,\theta,\phi)\), a solution to the Schrödinger equation that encodes every measurable property of the electron's state. The cloud of points you see here is a Monte Carlo sample of that wavefunction's probability density.
The squared magnitude gives the probability density — the likelihood of finding the electron at any given point in space. Dense regions of the cloud mean high probability; nodes, where the density vanishes, mean the electron is never found there.
Three quantum numbers label each orbital: n sets the energy shell, \(E_n = -13.6/n^2\,\text{eV}\), and the overall size; l fixes the shape (\(s, p, d, f, \dots\)); m controls the orientation in space. The allowed values form a strict ladder.
The radial part \(R_{nl}\) is built from associated Laguerre polynomials and decays exponentially with distance from the nucleus. Its zeros are radial nodes — spherical shells where the wavefunction, and therefore the probability, vanishes.
The key insight: an orbital is not a path — it is a probability cloud. The shape of the cloud comes from the spherical harmonics (labelled by \(l\) and \(m\)); its size comes from the radial function (labelled by \(n\)). What you rotate in this lab is the wavefunction itself.
1D Finite Difference Eigensolver & Wavefunction Visualizer. Computes exact discrete Hamiltonian matrix eigenvalues and eigenstate amplitudes for particle in a box.
This lab solves the time-independent Schrödinger equation — the eigenvalue problem of quantum mechanics — for a particle confined to a one-dimensional box. The solver discretises the Hamiltonian into an \(N\times N\) matrix and diagonalises it numerically, so the states you see are exact eigenstates of the finite-difference system.
With infinite walls, the boundary conditions force \(\psi(0) = \psi(L) = 0\), and only the standing waves \(\sin(n\pi x/L)\) survive. That single constraint — a confined wave must hold a whole number of half-wavelengths — quantises the energy, exactly as a guitar string rings only at certain pitches.
Note the \(n^2\) spacing: energy levels grow quadratically, so the gaps between levels widen as \(n\) increases. The nodes of each state — points inside the box where the particle can never be found — appear live in the visualisation.
The key insight: confinement quantises energy. It is the boundary conditions — not the equation itself — that turn a continuous spectrum into a discrete ladder of eigenstates.
Static bound state penetration & dynamic wavepacket reflection/transmission through potential barriers.
Classically, a particle with energy below a barrier's height can never cross it. Quantum mechanically the wavefunction does not stop at the barrier — inside the classically forbidden region it decays exponentially rather than vanishing, and if the barrier is thin enough a remnant emerges on the far side. That is tunnelling.
Inside the barrier the wavefunction is evanescent — it does not oscillate, it leaks, \(\psi(x)\propto e^{-\kappa x}\). Matching the wavefunction and its derivative at both edges of the barrier yields the transmission probability.
Everything about tunnelling lives in that exponent: mass, barrier height, and width all multiply there, which is why transmission is exponentially sensitive to every parameter — and why electrons tunnel easily while heavier particles barely do. For a smooth barrier the same physics integrates barrier-by-barrier into the WKB approximation.
The key insight: no energy is borrowed or violated — the particle stays below \(V_0\) the whole time. It is the wavefunction, not the energy, that penetrates the forbidden region. Tunnelling powers scanning tunnelling microscopy, flash memory, alpha decay, and fusion in the Sun.
Visualise trade-offs between position and velocity precision with spectral expansion and live stats overlay.
The more precisely a wavefunction is localised in space, the more momentum components it must contain — and vice versa. Heisenberg's uncertainty principle makes this trade-off exact: for any state, the product of the position and momentum spreads is bounded from below.
The inequality is not a limitation of instruments; it is the mathematical shadow of the fact that position and momentum operators do not commute.
The general form applies to any pair of incompatible observables, and a related energy–time relation governs how precisely energy can be defined for short-lived states.
The key insight: a narrow wave packet (small \(\Delta x\)) is a superposition of many plane waves, so its momentum spread \(\Delta p\) grows — the product never dips below \(\hbar/2\). The same principle sets the size and stability of atoms.