Quantum Mechanics Calculators

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.

Twenty-Eight Quantum Mechanics Calculators

de Broglie Wavelength Calculator - Finds the matter wavelength of a particle from its mass and speed using lambda = h / (m v). Pick electron, proton, or neutron, or enter a custom mass, and see the wavelength in metres, nanometres, and picometres along with the momentum.

Photoelectric Effect Threshold Calculator - Works out whether light ejects electrons from a metal and, if so, their maximum kinetic energy and stopping voltage. Enter the work function and the light’s wavelength, or choose a metal, to get the photon energy, threshold wavelength, and threshold frequency.

Bohr Radius Calculator - Calculates the radius of an electron orbit in the Bohr model, r = n squared / Z times the Bohr radius. Enter the principal quantum number and atomic number to see the orbit radius in metres, nanometres, and picometres, and as a multiple of the Bohr radius.

Rydberg Formula Calculator - Finds the wavelength of a hydrogen spectral line from the two energy levels using 1 / lambda = R Z squared times (1/n1 squared minus 1/n2 squared). Names the series (Lyman, Balmer, Paschen) and gives the photon energy and frequency.

Hydrogen Atom Energy Levels Calculator - Gives the energy of a hydrogen level with E = minus 13.6 times Z squared over n squared electronvolts, in eV and joules, plus the ionisation energy from that level. The energy form of the Rydberg formula.

Heisenberg Uncertainty Principle Calculator (Position-Momentum) - Finds the minimum momentum uncertainty for a given position uncertainty using Delta x times Delta p is greater than or equal to h-bar over 2. Electron and proton presets, plus the derived minimum velocity uncertainty.

Photon Energy from Wavelength Calculator - Converts a photon’s wavelength into its energy using E = h c / lambda. Reports the energy in electronvolts and joules, the frequency, and which part of the spectrum (UV, visible, infrared, etc.) it falls in.

Compton Scattering Wavelength Shift Calculator - Calculates how much a photon’s wavelength increases after scattering off an electron, Delta lambda = lambda-C times (1 minus cos theta). Gives the shift and the resulting scattered wavelength for any angle.

Heisenberg Energy-Time Uncertainty Calculator - Finds the minimum energy uncertainty for a given lifetime using Delta E times Delta t is greater than or equal to h-bar over 2. Reports the result in electronvolts and the equivalent natural linewidth in hertz.

Particle in a Box - 1D Energy Levels Calculator - Calculates the quantized energy levels of a particle confined to a box using E_n = n squared h squared over (8 m L squared). Electron and proton presets, plus the energy gap to the next level.

Quantum Harmonic Oscillator Energy Calculator - Finds the evenly-spaced energy levels of a quantum harmonic oscillator using E_n = (n plus one-half) h f. Reports the zero-point energy and the constant spacing between levels.

Quantum Tunneling Probability Calculator (WKB) - Finds the probability a particle tunnels through a rectangular barrier using T is approximately e to the minus 2 kappa L. Electron and proton presets, plus the decay length inside the barrier.

Spin Angular Momentum Calculator - Calculates total spin angular momentum and its z-component from the spin quantum numbers s and ms using S = h-bar times the square root of s(s+1). Validates the allowed range of ms.

Fermi Energy Calculator - Finds the Fermi energy of a metal’s conduction electrons from their density. Copper, silver, gold, aluminum, and sodium presets, plus the Fermi velocity and Fermi temperature.

Particle in a Box - 2D and 3D Energy Levels Calculator - Extends the particle-in-a-box model to two or three dimensions and flags degenerate states, different quantum numbers that share the same energy in a symmetric box.

Magnetic Moment and Zeeman Effect Calculator - Calculates a quantum magnetic moment and its Zeeman energy splitting in a magnetic field using mu = g mu_B m. Works for orbital (g=1) or electron spin (g is approximately 2.0023) moments.

Density of States Calculator - Finds how many quantum states are available per unit energy per unit volume for a free electron gas, g(E) is proportional to the square root of E. Electron and proton presets.

Angular Momentum Addition (Clebsch-Gordan) Calculator - Calculates the exact Clebsch-Gordan coefficient for combining two angular momenta into a total j using Racah’s closed-form sum formula, plus the implied total m and the measurement probability.

Finite Square Well Bound States Calculator - Finds every bound-state energy of a finite-depth potential well by numerically solving the exact even and odd parity matching equations, no algebraic formula exists for this problem.

WKB Quantization Condition Calculator (Quantum Bouncer) - Calculates WKB-quantized energy levels for a particle bouncing under a uniform force off a hard floor, the exact problem measured by the GRANIT neutron-gravity experiment.

Schrödinger Equation Time Evolution Estimator - Finds the exact oscillation period and survival probability of a two-energy-eigenstate quantum superposition using the time-dependent Schrödinger equation.

Stark Effect Energy Shift Calculator - Calculates the linear Stark effect energy shift of a hydrogen state in a uniform electric field using parabolic quantum numbers, Delta E = (3/2) n (n1 minus n2) e a0 F.

Fine Structure Constant Applications Calculator - Finds the exact relativistic fine-structure energy correction for a hydrogen-like level using the Sommerfeld formula from the Dirac equation, reproducing the measured hydrogen 2p splitting.

Variational Method Ground State Energy Estimator - Estimates a particle-in-a-box ground state energy using the exact closed-form integral for the polynomial trial wavefunction x(L-x), within about 1.3% of the true value.

Quantum Number Selection Rules Calculator - Checks whether an electric dipole transition between two hydrogen-like quantum states is allowed using the exact Delta l = plus or minus 1, Delta m = 0 or plus or minus 1 selection rules.

Quantum Degeneracy Pressure Calculator - Finds the degeneracy pressure of a non-relativistic electron gas using P = (2/5) n E_F, with metal and white dwarf core presets.

Parity and Selection Rule Checker - Classifies an atomic transition as electric dipole, magnetic dipole, or electric quadrupole from the parity and change in orbital quantum number, explaining astrophysical forbidden lines.

Expectation Value Calculator - Calculates exact position and momentum expectation values for a particle in a box or quantum harmonic oscillator stationary state, verifying the Heisenberg uncertainty bound.

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.