Core Concepts

This library provides mathematical abstractions commonly used in one-dimensional quantum mechanics simulations. At its core are:

  • complex-valued arithmetic,
  • wave function representations,
  • normalization utilities,
  • probability sampling tools,
  • and built-in analytical solutions.

Complex numbers

Quantum wave functions are complex valued and therefore require complex arithmetic for representation and evaluation. Mathematically, complex numbers are made using the imaginary unit: i = sqrt(-1)

ComplexF represents a complex number using two float values: a real component and an imaginary component.

ComplexF z = new ComplexF(2f, 5f);

The value represents: z = real + imaginary * i

Polar form

Complex numbers can also be represented using a radius and angle:

(float radius, float angle) = MathC.ToPolar(z);
ComplexF restored = MathC.FromPolar(radius, angle);

This representation is usually referred to as polar form or Euler form.

Using complex valued functions, we can represent physical particle states.

Wave Functions

In quantum mechanics, the state of a particle is represented by a wave function. WaveFunction1D models a one dimensional wave function dependent on position and time: ψ(x, t)

The probability density is: |ψ(x, t)|²

Separable Wave Function

Many analytical solutions to the Schrödinger equation can be split into independent spatial and time dependent components.

SeparableWaveFunction1D represents a wave function by storing the two components separately.

Weighted Wave Function

Systems rarely exist in pure eigenstates, and being able to write a wave function as a sum of eigenstates is very important. WeightedWaveFunction1D represents a wave function as a linear combination of other states, typically eigenstates.

Normalization

A key aspect of the wave function is that its absolute square is equal to the probability density of finding a particle there. The integral of the probability density over all space must equal one.

To help with this WaveFunction1D contains an Amplitude field, effectively used for normalization. If it is not passed in the constructor explicitly the function will attempt to normalize itself. Simpson integration is used for this purpose.

Sampling

Sampling allows probabilistic measurement of particle positions from the wave function. WaveFunction1D includes a built in sampler which approximates the cumulative density function (CDF) and randomly selects positions.

An example of sampling can be found here.

Built-in Solutions

To simplify testing, the library includes several analytical one dimensional quantum systems:

  • Infinite well
  • Harmonic oscillator
  • Free particle plane wave
  • Gaussian packet

These implementations can be used directly or serve as building blocks for more advanced simulations. See example usage.

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