The Fundamental Axioms of Quantum Mechanics

Quantum mechanics is built on a few simple yet profound axioms. These axioms describe how quantum systems behave and how we connect mathematics to physical measurements.

Axiom 1: The State Axiom

A quantum system is fully described by its wave function $\\psi(\\mathbf{r},t)$, a complex function of position and time:

$$i\\hbar \\frac{\\partial \\psi(\\mathbf{r}, t)}{\\partial t} = \\left[-\\({\\frac{\\hbar^2}{2m}}\\nabla^2 + V(\\mathbf{r}, t))\\right]\\psi(\\mathbf{r}, t)$$

It encodes the probability amplitude of finding a particle at a point and evolves according to the Schrödinger equation.
Example: A free particle along the x-axis:

$$\\psi(x,t) = A e^{i(kx - \\omega t)}$$

Axiom 2: Linear Superposition Principle

If a system can exist in states $\\psi_1$ and $\\psi_2$, it can also exist in any linear combination:

$$\\psi = c_1 \\psi_1 + c_2 \\psi_2$$

This explains phenomena like interference. The system can be in “both states” until measured.
Example: Electron through a double slit forms an interference pattern because it is in a superposition of “slit 1” and “slit 2” states.

Axiom 3: Observable–Operator Correspondence

Every measurable quantity corresponds to a linear operator:

  • Position: $\\hat{\\mathbf{r}}} = \\mathbf{r}$
  • Momentum: $\\hat{\\mathbf{p}}} = -i\\hbar\\nabla$

Any function of observables $F(\\mathbf{r},\\mathbf{p})$ becomes an operator:

$$\\hat{F} = F(\\mathbf{r}, -i\\hbar\\nabla)$$

Axiom 4: Expectation-Value Axiom

The average value of an observable $A$ in a state $\\psi$ is:

$$\\langle A \\rangle = \\int \\psi^ *(\\mathbf{r}, t) \\hat{A} \\psi(\\mathbf{r}, t) \\, d^3r$$

This is the quantum analogue of classical averages and connects operators with measurable outcomes.

Axiom 5: Expansion Postulate

For a Hermitian operator $\\hat{A}$, its eigenfunctions $\\psi_1, \\psi_2, \\dots$ form a complete orthonormal set. Any state can be written as:

$$\\psi = \\sum_n c_n \\psi_n$$

Coefficients $c_n$ give the probability amplitudes for each eigenvalue. This is a structured decomposition into measurement-relevant basis states.

Summary of Quantum Axioms

  • State Axiom: Wave function defines the system.
  • Superposition: States can combine linearly.
  • Observables ↔ Operators: Measurable quantities have operator counterparts.
  • Expectation Value: Average measurements from wave functions.
  • Expansion Postulate: Any state can be expressed in terms of eigenstates.