Postulates of Quantum Mechanics

Why is Axiomatics Needed?

Quantum mechanics arose from a series of experimental discoveries that appeared mutually contradictory: quantization of energy (Planck), wave-particle duality (de Broglie), Heisenberg's uncertainty principle. In the 1930s, Dirac and von Neumann assembled all this into a unified axiomatic formalism in the language of Hilbert spaces.

The axiomatic approach guarantees: we know exactly what we are assuming about nature, and we can systematically derive predictions. Without an axiomatics, quantum theory would be a set of recipes without explanations.

Postulate 1: State of the System

The state of a quantum system at any given moment is completely described by a normalized vector |ψ⟩ in a complex Hilbert space H, ‖|ψ⟩‖ = 1.

What this means: classically, a particle has definite position and momentum. In quantum mechanics, the particle is in "superposition"—not in any one definite state until a measurement is made. The vector |ψ⟩ contains all possible information about the system. Analogy: instead of a specific point on a map—a cloud of probability.

In coordinate representation: ψ(x) = ⟨x|ψ⟩ is the wave function. |ψ(x)|² is the probability density of finding the particle near x.

Postulate 2: Observables and Operators

Each physical observable A corresponds to a self-adjoint (Hermitian) operator = A† in H.

Self-adjointness (A = A†) guarantees two crucial properties: eigenvalues are real (measurement results are real!), and eigenvectors form a complete orthonormal basis.

Key operators: x̂|x'⟩ = x'|x'⟩ (position operator); p̂ψ(x) = −iℏ ∂ψ/∂x (momentum operator in coordinate representation); Ĥ = p̂²/(2m) + V(x̂) (Hamiltonian—energy operator).

The commutator [x̂, p̂] = x̂p̂ − p̂x̂ = iℏ is a "quantum anomaly," showing that x and p cannot be simultaneously defined.

Postulate 3: Probabilities of Measurements

Upon measuring A in the state |ψ⟩ the probability to obtain the eigenvalue aₙ is:

P(aₙ) = |⟨φₙ|ψ⟩|²

Here, |φₙ⟩ is the normalized eigenvector corresponding to aₙ; ⟨φₙ|ψ⟩ is the scalar product (probability amplitude). The squared modulus of the amplitude is the probability. This is the Born rule (1926).

Expectation value: ⟨A⟩ = ⟨ψ|Â|ψ⟩ = Σₙ aₙ |⟨φₙ|ψ⟩|²

Uncertainty principle: ΔAΔB ≥ (1/2)|⟨[Â, B̂]⟩|. For x and p: ΔxΔp ≥ ℏ/2.

Postulate 4: Collapse of the Wave Function

After a measurement yielding the result aₙ, the state of the system instantaneously "collapses" to the eigenvector |φₙ⟩.

This is the most mysterious postulate. Before measurement—superposition. After—definite state. A repeated immediate measurement yields the same result aₙ with probability 1.

"Schrödinger's cat paradox" illustrates: the cat can be in a superposition "alive + dead." Only a measurement (observation) "chooses" one of the options. Different interpretations (Copenhagen, many-worlds, objective collapse) explain this postulate in different ways.

Postulate 5: Schrödinger Equation of Evolution

Between measurements, the state evolves deterministically:

iℏ ∂|ψ⟩/∂t = Ĥ|ψ⟩

Formal solution: |ψ(t)⟩ = e^{−iĤt/ℏ}|ψ(0)⟩—unitary evolution. Unitarity means: the norm is preserved, probabilities are not "lost." This is a deterministic, reversible evolution—in contrast to the collapse upon measurement.

Full Numerical Example: Two-Level System

Spin-1/2 (electron in a magnetic field): |↑⟩ and |↓⟩ are eigenstates of Ŝz. Hamiltonian: Ĥ = ω₀ Ŝz, E↑ = +ℏω₀/2, E↓ = −ℏω₀/2.

Initial state: |ψ(0)⟩ = (1/√2)(|↑⟩ + |↓⟩)—a superposition.

Evolution: |ψ(t)⟩ = (1/√2)(e^{−iω₀t/2}|↑⟩ + e^{+iω₀t/2}|↓⟩)

Average Sz: ⟨Sz⟩ = ⟨ψ|Ŝz|ψ⟩ = (ℏ/2) cos(ω₀t)—precession around the z axis with frequency ω₀. This is nuclear magnetic resonance (NMR/MRI), which underpins magnetic tomography!

Decoherence and the Measurement Problem

Why are quantum superpositions not observed in the macroscopic world? Decoherence is the scattering of the quantum superposition due to interaction with the environment (thermal photons, air molecules). For an object of N ~ 10²³ atoms, decoherence occurs within ~10⁻³⁰ s. Quantum computers combat decoherence: qubits are isolated at temperatures T ~ 15 mK (20 times colder than the cosmic microwave background). The coherence time of modern superconducting qubits is tens of microseconds; this specifically limits the duration of quantum computations.

Quantum Mechanics in Semiconductors, Lasers, and Quantum Computers

The postulates of quantum mechanics are the foundation of most modern technologies. Semiconductor electronics is based on the quantum theory of band structure in solids: the MOSFET transistor operates due to quantum-mechanical tunneling and band structure, and modern transistors with channel thicknesses of just a few atomic monolayers demand a quantum-mechanical description. The laser is a direct consequence of the postulate of quantum transitions: Einstein's stimulated emission produces coherent light under population inversion, described by the rate equations. Quantum dots—nanocrystals with discrete energy levels determined by their size—are used in QLED displays and biomarkers. In medical diagnostics, NMR spectroscopy and MRI are based on quantum transitions between nuclear spin states. Quantum computers use the superposition and entanglement of qubits: Shor's algorithm factors numbers in polynomial time, Grover's algorithm accelerates search. It is precisely the postulate of the wave function and the principle of superposition that create the quantum advantage. Quantum cryptography (the BB84 protocol) provides theoretically proven key security through the laws of quantum mechanics, making undetectable eavesdropping impossible.

Quantum computation relies precisely on the postulate of superposition: a qubit in the state α|0⟩ + β|1⟩ processes both values simultaneously. Quantum entanglement—a violation of classical correlations—has been experimentally confirmed by Aspect (Nobel Prize 2022) via violation of Bell's inequalities, closing all "loopholes" for locally-hidden variables and confirming the postulates of quantum mechanics with fundamental precision.

Assignment: (a) At time t=0: |ψ(0)⟩ = (1/√2)(|E₁⟩ + |E₂⟩). Find |ψ(t)⟩ and ⟨Ĥ⟩(t). Is ⟨Ĥ⟩ constant? (b) Show: if [Â, Ĥ] = 0, then d⟨A⟩/dt = 0. (c) Parity operator Π: Πψ(x) = ψ(−x). If [Π, Ĥ] = 0, can one choose eigenfunctions of Ĥ with definite parity? Prove using the example of the harmonic oscillator.