MW SysArcTools

Learn by calculating

Start with numbers. Build toward quantum mathematics.

Follow a focused route through the mathematics quantum mechanics actually depends on. Every step has calculators, visible working, assumptions, meaning and limits.

Start the QM path

The learning loop

Predict → calculate → explain → change

  1. Read the problem and predict what should happen.
  2. Enter values and calculate locally in your browser.
  3. Inspect the formula and every calculation step.
  4. Explain the answer's sign, size, units and assumptions.
  5. Change one input and predict the new result before recalculating.

The quantum-mechanics spine

71 essential calculator lessons in dependency order

Finish Step 10 and you will have practised the algebra, probability, calculus, linear algebra, waves, complex numbers and physics used by introductory quantum mechanics.

Step 1

Numbers and algebra for physics

Rearrange equations and work confidently with fractions, powers, roots and logarithms.

Before starting: Comfort reading whole numbers and decimals.
  1. Step 1 · Lesson 1Fraction simplifierReduce a fraction to lowest terms using the greatest common divisor.
  2. Step 1 · Lesson 2Ratio simplifierSimplify a two-number whole-number ratio and calculate its decimal relationship.
  3. Step 1 · Lesson 3ExponentRaise a base to a selected power and see the repeated-multiplication meaning.
  4. Step 1 · Lesson 4Square rootFind the principal square root and verify it by squaring the result.
  5. Step 1 · Lesson 5LogarithmCalculate a logarithm with any valid positive base.
  6. Step 1 · Lesson 6Linear equationSolve a linear equation in the form ax + b = c and see the isolation steps.
  7. Step 1 · Lesson 7Quadratic formulaSolve ax² + bx + c = 0 and inspect the discriminant.
Step 2

Functions and graphs

Read linear, quadratic, exponential and periodic relationships as changing physical models.

Before starting: Understand equations, powers, roots and coordinates.
  1. Step 2 · Lesson 1SlopeCalculate the slope of the line through two coordinate points.
  2. Step 2 · Lesson 2Linear functionEvaluate f(x) = mx + b and identify its slope and vertical intercept.
  3. Step 2 · Lesson 3Quadratic functionEvaluate ax² + bx + c and calculate the parabola's vertex coordinates.
  4. Step 2 · Lesson 4Exponential functionEvaluate exponential growth or decay in the form f(x)=abˣ.
  5. Step 2 · Lesson 5Sine, cosine and tangentCalculate sine, cosine and tangent for an angle entered in degrees.
  6. Step 2 · Lesson 6Sine waveEvaluate a sinusoid from amplitude, frequency, time and phase.
Step 3

Geometry, angles and trigonometry

Resolve lengths, angles, rotations and oscillations using radians, sine and cosine.

Before starting: Understand algebra, functions and square roots.
  1. Step 3 · Lesson 1Pythagorean theoremFind the hypotenuse of a right triangle from its two legs.
  2. Step 3 · Lesson 2Degrees and radiansConvert an angle entered in degrees to radians and common turn measures.
  3. Step 3 · Lesson 3Right triangleSolve a right triangle from its two perpendicular legs.
  4. Step 3 · Lesson 4Cosine law sideFind a triangle side from two sides and their included angle.
  5. Step 3 · Lesson 5Polar to CartesianConvert a radius and angle into x and y coordinates.
  6. Step 3 · Lesson 6Cartesian to polarConvert x and y coordinates into radius and direction.
  7. Step 3 · Lesson 7Unit circleConvert an angle into its unit-circle coordinates, sine, cosine and reference angle.
  8. Step 3 · Lesson 8Inverse trig functionsCalculate arcsine, arccosine and arctangent for a real ratio and show their principal angles.
  9. Step 3 · Lesson 9Trig identityEvaluate sine, cosine and the identity sin²θ+cos²θ=1 at any angle.
Step 4

Probability and distributions

Interpret averages, spread and probability before treating a wavefunction as a probability amplitude.

Before starting: Understand fractions, percentages, functions and square roots.
  1. Step 4 · Lesson 1Mean and medianCalculate the arithmetic mean and median of six values.
  2. Step 4 · Lesson 2Standard deviationCalculate population and sample standard deviation for six values.
  3. Step 4 · Lesson 3Z-scoreStandardise a value by measuring how many standard deviations it lies from the mean.
  4. Step 4 · Lesson 4ProbabilityCalculate event probability from favourable and total equally likely outcomes.
  5. Step 4 · Lesson 5Expected valueCalculate the weighted mean of three possible outcomes.
  6. Step 4 · Lesson 6Normal CDFEstimate the probability below a value in a normal distribution.
Step 5

Calculus and continuous change

Use derivatives and integrals to describe motion, accumulation and changing wavefunctions.

Before starting: Understand algebra, functions, graphs, exponents and trigonometry.
  1. Step 5 · Lesson 1Power-rule derivativeDifferentiate a monomial axⁿ and evaluate its derivative at a selected x-value.
  2. Step 5 · Lesson 2Quadratic derivativeDifferentiate ax²+bx+c and evaluate its slope.
  3. Step 5 · Lesson 3Exponential derivativeDifferentiate Ae^(kx) at a selected x.
  4. Step 5 · Lesson 4Trigonometric derivativeDifferentiate A sin(kx+φ) and evaluate both the function and derivative at x.
  5. Step 5 · Lesson 5Chain ruleCombine an outer derivative and inner derivative to evaluate a composite function's derivative at a point.
  6. Step 5 · Lesson 6Product ruleEvaluate the derivative of a product from f, g and their derivatives at the same point.
  7. Step 5 · Lesson 7Quotient ruleEvaluate the derivative of f/g from both function values and derivatives at the same point.
  8. Step 5 · Lesson 8Power-rule integralIntegrate a monomial axⁿ and evaluate its antiderivative at a selected x-value.
  9. Step 5 · Lesson 9Definite power integralEvaluate the definite integral of axⁿ between lower and upper bounds.
  10. Step 5 · Lesson 10Simpson's ruleApproximate an integral from endpoint and midpoint function values.
Step 6

Vectors and linear algebra

Treat states and transformations as vectors, matrices, inner products and eigenvalue problems.

Before starting: Understand algebra, coordinates, trigonometry and functions.
  1. Step 6 · Lesson 1Vector magnitudeCalculate the length of a three-dimensional vector from its components.
  2. Step 6 · Lesson 2Vector additionAdd two three-dimensional vectors component by component.
  3. Step 6 · Lesson 3Dot productCalculate the scalar dot product of two three-dimensional vectors.
  4. Step 6 · Lesson 4Angle between vectorsFind the angle between two three-dimensional vectors using their dot product.
  5. Step 6 · Lesson 5Matrix–vector productApply a 2×2 linear transformation to a two-component vector.
  6. Step 6 · Lesson 62×2 matrix multiplicationMultiply two 2×2 matrices in order A times B.
  7. Step 6 · Lesson 72×2 eigenvaluesCalculate the real eigenvalues of a 2×2 matrix from its trace and determinant.
Step 7

Differential equations and waves

Connect rates of change to growth, oscillation, damping, frequency and wavelength.

Before starting: Understand calculus, exponentials, trigonometry and vectors.
  1. Step 7 · Lesson 1Exponential growth ODESolve y′=ky from an initial value and evaluate the solution at time t.
  2. Step 7 · Lesson 2Euler method stepApproximate one step for y′=ay+b using the slope at the current point.
  3. Step 7 · Lesson 3Harmonic oscillatorEvaluate displacement and velocity for x″+ω²x=0 with amplitude and phase.
  4. Step 7 · Lesson 4Damped oscillatorEvaluate an underdamped oscillation x(t)=Ae⁻ᵝᵗcos(ωt+φ) and its decaying envelope.
  5. Step 7 · Lesson 5Wave speed and wavelengthCalculate wave speed and period from frequency and wavelength.
  6. Step 7 · Lesson 6Angular to linear speedConvert angular speed at a radius into tangential speed.
Step 8

Complex numbers and Fourier ideas

Represent phase and interference with complex amplitudes, then decompose sampled waves into frequencies.

Before starting: Understand radians, trigonometry, vectors, exponentials and oscillation.
  1. Step 8 · Lesson 1Complex additionAdd two complex numbers by combining their real and imaginary components.
  2. Step 8 · Lesson 2Complex multiplicationMultiply two complex numbers and display the rectangular-form result.
  3. Step 8 · Lesson 3Complex divisionDivide one complex number by another using the denominator's conjugate.
  4. Step 8 · Lesson 4Complex polar formConvert z=a+bi into magnitude and phase angle.
  5. Step 8 · Lesson 5Euler formulaEvaluate eⁱᶿ as cosine and sine components on the unit circle.
  6. Step 8 · Lesson 6Phasor additionAdd two magnitude-phase phasors and return the combined magnitude and phase.
  7. Step 8 · Lesson 7Four-sample DFTCalculate one discrete Fourier transform bin from four real-valued samples.
Step 9

The physics bridge

Connect the mathematics to motion, momentum, energy, forces, photons and matter waves.

Before starting: Understand calculus, vectors, waves and differential equations.
  1. Step 9 · Lesson 1Constant accelerationCalculate final velocity and displacement under constant acceleration.
  2. Step 9 · Lesson 2Kinetic energy and momentumCalculate classical linear momentum and kinetic energy from mass and velocity.
  3. Step 9 · Lesson 3Work and force angleCalculate mechanical work done by a constant force over a displacement.
  4. Step 9 · Lesson 4Newton's second lawCalculate net force and weight from mass, acceleration and gravitational field strength.
  5. Step 9 · Lesson 5Coulomb forceCalculate the signed electrostatic force magnitude between two point charges.
  6. Step 9 · Lesson 6Photon energyConvert a photon wavelength into frequency, energy in joules and energy in electronvolts.
  7. Step 9 · Lesson 7De Broglie wavelengthCalculate the matter wavelength associated with a non-relativistic particle's momentum.
Step 10

Quantum-mechanics foundations

Calculate with uncertainty, complex probability amplitudes, normalized states, qubits and quantized energies.

Before starting: Complete the preceding physics, probability, linear-algebra, wave and complex-number steps.
  1. Step 10 · Lesson 1Uncertainty principleCalculate the minimum momentum and velocity uncertainty implied by a position uncertainty.
  2. Step 10 · Lesson 2Wavefunction probabilityConvert a complex wavefunction amplitude into magnitude, phase and probability density.
  3. Step 10 · Lesson 3Two-state normalizationNormalize two real state amplitudes and calculate their measurement probabilities.
  4. Step 10 · Lesson 4Bloch sphere stateConvert Bloch-sphere angles into qubit amplitudes and measurement probabilities.
  5. Step 10 · Lesson 5Infinite well energyCalculate a particle's quantized energy level in a one-dimensional infinite square well.
  6. Step 10 · Lesson 6Hydrogen transitionCalculate hydrogen energy levels and the photon energy and wavelength between two principal quantum numbers.

Optional mathematics library

Explore without blocking your QM progress

106 additional calculators remain available for deeper study, schoolwork and practical problems. They are useful, but they are not required checkpoints on the main path.

Number foundations 5 optional lessons
Absolute valueGCF and LCMPercentage of a numberPercentage changeCube root
Equations and algebra 2 optional lessons
Proportion solverTwo-equation system
Coordinate and practical geometry 13 optional lessons
MidpointDistance between pointsRectangleTriangle areaCircleTrapezoid areaEllipseCube dimensionsRectangular prismCylinderCone volumeSphereSquare pyramid
Statistics and probability 4 optional lessons
Weighted meanData rangeOdds to probabilityCombinations and permutations
Precalculus and functions 4 optional lessons
Arithmetic sequenceArithmetic seriesGeometric sequenceGeometric series
Linear algebra 3 optional lessons
Cross product2×2 determinant2×2 matrix inverse
Differential equations 4 optional lessons
Half-life decayLogistic growth ODENewton coolingFirst-order linear ODE
Complex and Fourier mathematics 1 optional lessons
Principal complex root
Mathematical physics 2 optional lessons
Projectile motionGravitational force
Number patterns and practical algebra 10 optional lessons
FactorialFibonacci numberTriangular numberSum 1 to nModuloPercentage differenceNth rootChange-of-base logarithmDistance, rate and timeRatio allocation
Extended geometry 10 optional lessons
Annulus areaSector areaArc lengthHeron's triangle areaParallelogram areaRhombus areaRegular polygonPolygon interior angles3D distanceTorus geometry
Extended trigonometry 3 optional lessons
Cosine law angleSine law sideBearing components
Extended statistics 10 optional lessons
Population varianceQuartilesInterquartile rangeRoot mean squareGeometric meanHarmonic meanCoefficient of variationCovariancePearson correlationLinear regression
Extended probability 7 optional lessons
Probability complementIndependent event unionConditional probabilityBinomial exact probabilityBinomial mean and variancePoisson probabilityBayes theorem
Applied derivatives and integrals 6 optional lessons
Logarithm derivativeQuadratic tangent lineQuadratic average rateDefinite linear integralDefinite quadratic integralTrapezoidal rule
Growth, decay and compounding 6 optional lessons
Simple interestCompound interestContinuous compoundingDoubling timeExponential half-lifeRequired growth rate
Computational number theory 5 optional lessons
Prime checkerDivisor countSum of divisorsEuler totientModular exponentiation
Analytic algebra and geometry 5 optional lessons
Quadratic vertexQuadratic discriminantInverse linear functionEllipse eccentricityGreat-circle distance
Matrix computation 3 optional lessons
2×2 matrix additionMatrix traceMatrix scalar multiplication
Statistical inference and numerical calculus 3 optional lessons
Normal interval probabilityMean confidence intervalNewton square-root step

Clear answers

Learning mathematics with calculators

Where should I start?

Start at Step 1 unless its concepts already feel familiar. This path includes the prerequisites that directly support quantum mechanics; other mathematics is available separately below.

How do I learn with a calculator?

Predict first, calculate second, inspect the working, then change one input and explain why the result changed.

Can this pathway lead toward quantum mechanics?

Yes, by building the required mathematics in order: algebra, trigonometry, probability, calculus, linear algebra, differential equations, complex numbers and Fourier methods before quantum models.