Quantum Foundations Calculator
Fundamental quantum mechanics calculations. Wave-particle duality, Heisenberg uncertainty, quantum harmonic oscillator, hydrogen atom, and spin.
About this calculator
This is a five-mode toolkit for foundational quantum mechanics calculations, and the Calculation Type dropdown determines which formulas run and which inputs and outputs appear — think of it as five smaller calculators sharing one interface rather than a single unified computation. Wave-Particle Duality takes an energy (for matter) or a wavelength (for photons) and derives the de Broglie wavelength, momentum, and frequency, then flags whether that wavelength sits at atomic scale or well above it, using λ = h/p for matter and λ = hc/E for photons. Heisenberg Uncertainty takes either a position or an energy uncertainty you specify and computes the minimum uncertainty the conjugate quantity must reach to satisfy Δx·Δp ≥ ℏ/2 or ΔE·Δt ≥ ℏ/2, plus a plain-language read on what physical regime that scale represents. Quantum Harmonic Oscillator computes the discrete energy levels E_n = ℏω(n + ½) for a given mass and angular frequency, including the zero-point energy that persists even at the ground state n = 0.
Hydrogen Atom applies the Bohr model's E_n = −13.6 eV/n² and r_n = n²a₀ to a chosen set of quantum numbers, reporting orbital energy, radius, and which spectral series a transition to the ground state would fall into. Spin & Pauli Exclusion compares particle types by spin, distinguishing fermions — spin-½ particles bound by the exclusion principle — from bosons, which aren't, and computes Zeeman splitting in an applied magnetic field. Every mode leans on textbook formulas and fixed physical constants, but the hydrogen and oscillator modes assume single-particle, non-relativistic idealizations that don't extend to multi-electron atoms or relativistic corrections.
Step 1 of 2
How to Use This Calculator
- In the de Broglie section, enter Particle Mass (kg) and Momentum or Velocity to compute the matter wavelength — visible diffraction requires wavelengths comparable to lattice spacing.
- Switch to the Harmonic Oscillator tab and enter Angular Frequency (rad/s) and Quantum Number (n) to see energy levels Eₙ = ℏω(n + ½).
- Use the Hydrogen Atom section with Principal (n), Orbital (l), and Magnetic (m) quantum numbers to get atomic energy and orbital radius.
- In the Spin section, enter Magnetic Field (T) and select particle type to compute Zeeman splitting and Larmor frequency.
- Read the Energy Levels chart to visualize the spacing and compare harmonic oscillator vs. hydrogen-like energy structures.
What each input means
- Calculation Type
- Calculation mode to use.
- Energy (eV) or Wavelength (nm)
- eV for matter, nm for photon
- Particle Mass (kg)
- Mass of the object.
- Quantum Number n
- n = 0, 1, 2, ...
- Principal n
- n = 1, 2, 3, ...
- Angular l
- l = 0 to n-1
- Magnetic m
- m = -l to +l
- Magnetic Field (T)
- For Zeeman splitting
How this is calculated
Formula
λ = h/p | ΔxΔp ≥ ℏ/2 | E_n = ℏω(n+½) | E_n = -13.6/n² eVWorked example, using the default values
- Identify Input Parameters4 parametersCalculation Type = 0, Particle Type = 1, Energy (eV) or Wavelength (nm) = 1, Custom Mass (kg) = 9.109e-31 = 16 input(s) provided
- Calculate de Broglie λde Broglie λ1.2264259661581491 = 1.2264259661581491
- Calculate Momentum5.402747766957797e-25 = 5.402747766957797e-25
- Calculate Frequency241798924208491.8 = 241798924208491.8
Engine last updated . Checked against 6 independently-derived tests — how we verify calculators. Built by Paul Gunder, a software engineer, not a licensed financial, medical, or legal professional.
Frequently Asked Questions
Why do the available inputs change when I switch the Calculation Type dropdown?
Each of the five modes is really a separate physics calculation with its own formula and its own required inputs — switching from Wave-Particle Duality to Hydrogen Atom swaps the entire input set to quantum numbers, since angular frequency wouldn't mean anything there. Only the fields relevant to the selected mode's formula are shown, so the wizard never asks for a number a given calculation doesn't use.
Why does a photon need a wavelength while an electron needs an energy in the duality mode?
Photons are conventionally described by wavelength or frequency since they're massless and always travel at the speed of light, making E = hc/λ the natural relation to invert. Massive particles like electrons are more naturally described by kinetic energy, from which the calculator derives momentum via p = √(2mE) and only then computes the de Broglie wavelength.
Does the Hydrogen Atom mode give accurate results for atoms other than hydrogen?
No — it uses the exact Bohr-model solution for one electron orbiting one proton, which only holds cleanly for hydrogen and hydrogen-like single-electron ions. Multi-electron atoms have electron-electron repulsion that the E_n = −13.6/n² formula doesn't account for, so results for anything heavier than hydrogen would be off.
Does the Pauli exclusion principle mean photons can never share an energy level?
It's the opposite — photons are bosons with integer spin, so exclusion doesn't apply to them at all, and selecting photon in this mode marks the exclusion rule as inapplicable with no cap on occupancy. It's fermions, including electrons, protons, and neutrons, all spin-½, that are limited to a small number of particles per orbital.
When does the interpretation label in Heisenberg Uncertainty mode actually change?
It's tied to the scale of the position uncertainty you enter (or the resulting minimum time, for the energy-time pair) — dropping the position uncertainty toward femtometer scale shifts the label toward nuclear-scale phenomena, while values around a micrometer or larger land in the classical regime where the uncertainty becomes negligible in practice.
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