Quantum Mechanics Calculators
Free quantum mechanics calculators: de Broglie, Bohr radius, uncertainty principle, particle in a box, quantum tunneling, spin, Fermi energy, and more.
Quantum Mechanics Calculators - Waves, Photons, and Atoms
Quantum mechanics replaced the smooth, continuous physics of everyday life with waves, quantised energy levels, and probabilities. These calculators cover the foundational results every physics student meets first: matter waves, the photoelectric effect, and the Bohr model of the atom, each with the constants and working laid out.
Foundations: Waves, Photons, and Uncertainty
The Hydrogen Atom
Bound States and Confined Systems
Angular Momentum and Spin
Many-Particle and Statistical Quantum Systems
Dynamics and Measurement
What These Calculators Cover
Foundations: waves, photons, and uncertainty. The de Broglie Wavelength Calculator and Photon Energy from Wavelength Calculator cover the wave-particle duality of matter and light respectively. The Photoelectric Effect and Compton Scattering Calculators are the two historical experiments that forced physics to accept light as quantized. The Heisenberg position-momentum and energy-time uncertainty calculators quantify the fundamental limit on simultaneously knowing conjugate variable pairs - not a measurement limitation, but a property of nature itself.
The hydrogen atom. The Bohr Radius, Rydberg Formula, and Hydrogen Atom Energy Levels Calculators form a connected trio - orbit radius, transition wavelength, and level energy are three views of the same Bohr model. The Fine Structure Constant and Stark Effect Calculators go beyond the simple Bohr model to the small relativistic and electric-field corrections that split and shift those levels in real spectroscopy. The Lamb Shift Estimator goes one step further into pure quantum electrodynamics, estimating the famous splitting between the otherwise-degenerate 2S(1/2) and 2P(1/2) levels that first demonstrated QED effects beyond the Dirac equation.
Bound states and confined systems. The particle-in-a-box and quantum harmonic oscillator calculators are the two exactly-solvable model systems every quantum course starts with. The Finite Square Well and WKB Quantization Calculators tackle problems with no closed-form algebraic solution, requiring numerical root-finding or the semiclassical WKB approximation instead. The Variational Method Estimator demonstrates the approximation technique used throughout quantum chemistry when no exact solution exists at all. The Quantum Tunneling Calculator quantifies the purely quantum phenomenon behind scanning tunneling microscopy and alpha decay.
Angular momentum and spin. The Spin Angular Momentum Calculator and Clebsch-Gordan Calculator handle single and combined angular momenta respectively. The Zeeman Effect Calculator shows how angular momentum couples to an external magnetic field, and the Selection Rules and Parity Checker Calculators determine which atomic transitions are allowed - and which “forbidden” transitions still occur at a much lower rate, producing the famous forbidden emission lines seen in astrophysical nebulae.
Many-particle and statistical quantum systems. The Fermi Energy and Density of States Calculators describe how a Fermi gas of electrons fills available states up to the Pauli exclusion limit, the basis of the free electron model of metals. The Quantum Degeneracy Pressure Calculator extends the same physics to its most dramatic consequence: the pressure that holds up a white dwarf star against gravitational collapse.
Dynamics and measurement. The Schrödinger Equation Time Evolution Estimator shows how a quantum superposition of two energy eigenstates oscillates in time - the basis of Rabi oscillations and qubit dynamics. The Expectation Value Calculator computes the measurable average position and momentum for a stationary state, directly verifying that the Heisenberg uncertainty bound is satisfied.
Who Uses These Calculators
Undergraduate physics and chemistry students use the foundations and hydrogen atom calculators for introductory quantum mechanics coursework - de Broglie wavelength, the photoelectric effect, and Bohr model problems are universal first-semester topics. Graduate students use the bound-state, angular momentum, and many-particle calculators for advanced quantum mechanics and solid-state physics coursework, including the WKB approximation and Clebsch-Gordan coefficients rarely covered at the undergraduate level. Materials science and condensed matter students use the Fermi energy and density of states calculators for free-electron model problems. Astrophysics students use the parity and selection rule checker to understand forbidden emission lines from planetary nebulae and the quantum degeneracy pressure calculator for white dwarf structure, connecting directly to the astrophysics section’s Chandrasekhar limit calculator.
Constants Behind Quantum Mechanics
Three constants recur throughout these tools. Planck’s constant h = 6.62607 x 10^-34 J.s sets the scale of quantisation and appears in both the de Broglie relation and the photon energy E = h f. The speed of light c = 2.998 x 10^8 m/s links a photon’s wavelength and frequency. The Bohr radius a0 = 5.292 x 10^-11 m is the natural size of the hydrogen atom. Because these numbers are so small, atomic results are quoted in nanometres, picometres, and electronvolts.
Frequently Asked Questions
What is the de Broglie wavelength?
It is the wavelength associated with a moving particle, given by lambda = h / (m v). Electrons at typical speeds have wavelengths of about 0.1 to 1 nanometre, which is why electron microscopes can resolve atoms. The de Broglie Wavelength Calculator works it out for any particle.
What is the work function in the photoelectric effect?
The work function is the minimum energy needed to free an electron from a metal surface, measured in electronvolts. Light only ejects electrons if each photon carries more energy than the work function. The Photoelectric Effect Threshold Calculator shows the threshold and the electron's kinetic energy.
What is the Bohr radius?
The Bohr radius, about 5.29 x 10^-11 metres, is the radius of the ground-state electron orbit in a hydrogen atom in the Bohr model. Higher orbits scale as n squared and shrink with atomic number Z. The Bohr Radius Calculator gives the radius for any orbit.
Why are quantum energies given in electronvolts?
Joules are inconveniently small for single atoms and photons, so physicists use the electronvolt, the energy an electron gains crossing a one-volt potential, equal to 1.602 x 10^-19 joules. Photon energies and work functions are naturally a few eV, which is easy to compare.
What is the difference between the finite square well and the infinite square well (particle in a box)?
The infinite square well (particle in a box) assumes infinitely high walls, so the wavefunction must be exactly zero at the boundaries, giving the simple closed-form energies E_n = n²h²/(8mL²). The finite square well has walls of finite height, so the wavefunction leaks (tunnels) into the classically forbidden region outside the well - this means fewer bound states exist than the infinite case, energies are lower than the infinite-well prediction, and there is no simple algebraic formula for the allowed energies. The Finite Square Well Bound States Calculator solves the exact transcendental matching equations numerically since no closed form exists.
Why does the Pauli exclusion principle create a Fermi energy?
Electrons are fermions, so the Pauli exclusion principle forbids any two from occupying the same quantum state. As electrons fill a metal's available states from the bottom up (lowest energy first), they stack all the way up to a maximum energy - the Fermi energy - even at absolute zero, because no two electrons can share the ground state. This is fundamentally different from a classical or bosonic gas, which would collapse into the single lowest-energy state at T = 0. The Fermi Energy Calculator computes this energy directly from electron density for copper, silver, gold, and other common metals.
What makes a spectral line "forbidden"?
A transition is "electric-dipole forbidden" when it violates the selection rules Δl = ±1, Δm = 0 or ±1, or when the initial and final states have the same parity. Such transitions are not truly impossible, they proceed through much weaker mechanisms (magnetic dipole or electric quadrupole radiation) at rates thousands to millions of times slower than allowed transitions. In a laboratory, collisions de-excite the atom before the forbidden transition can occur, but in the near-vacuum of a planetary nebula, atoms can survive undisturbed long enough to emit these forbidden lines - which is how astronomers identify nebular gas composition. The Parity and Selection Rule Checker classifies any transition as electric dipole, magnetic dipole, or electric quadrupole.