Quantum Mechanics - Mathematical Formalism

Summary

The rigorous mathematical structure of quantum mechanics is built on Hilbert spaces, Hermitian operators as observables, and the Born rule for probabilities. This note covers the postulates: state vectors, observables, measurement, time evolution, and composite systems including entanglement.

Overview

The mathematical formalism of QM requires tools from functional analysis, linear algebra, and complex analysis. The state of a system is a normalized vector in a complex Hilbert space; physical quantities are Hermitian operators; measurements give eigenvalues with Born-rule probabilities.

Postulates

State Space Postulate

The state of a quantum system is a normalized vector $\psi \in \mathcal{H}$, where $\mathcal{H}$ is a separable complex Hilbert space with $⟨\psi ,\psi ⟩=1$.

• States are defined up to a global phase: $\psi$ and $e^{i\alpha }\psi$ describe the same physical state
• The possible states are points in the projective Hilbert space
• For position/momentum: $\mathcal{H}=L^2(\mathbb{C})$ (square-integrable functions)
• For spin-1/2: $\mathcal{H}={\mathbb{C}}^2$ with the standard inner product

Observable Postulate

Physical quantities (position, momentum, energy, spin) are represented by Hermitian (self-adjoint) operators acting on $\mathcal{H}$.

• A quantum state $\psi$ is an eigenstate of observable $A$ with eigenvalue $a$ if $A\psi =a\psi$
• More generally, $\psi$ is a superposition of eigenstates: $\psi ={\sum }_n{c}_n{\varphi }_n$ where $A{\varphi }_n=a_n{\varphi }_n$

Born Rule (Measurement Postulate)

When observable $A$ is measured on state $\psi$:

• The outcome is one of the eigenvalues $a_n$ of $A$
• For non-degenerate $a_n$: $P(a_n)=|⟨{\varphi }_n,\psi ⟩{|}^2$ (squared inner product with eigenvector)
• For degenerate $a_n$: $P(a_n)=⟨\psi ,P_n\psi ⟩$ where $P_n$ is the projector onto the eigenspace
• After measurement giving result $a_n$, the state collapses to ${\varphi }_n$ (non-degenerate case)

Time Evolution Postulate

Between measurements, the state evolves unitarily:

$$i\hslash \frac{d}{dt}|\psi (t)⟩=H|\psi (t)⟩$$

with solution $|\psi (t)⟩=U(t)|\psi (0)⟩$ where $U(t)=e^{-iHt/\hslash }$ is unitary.

• Unitarity preserves normalization and probability
• Any observable $A$ that commutes with $H$ is conserved: $[A,H]=0\Rightarrow ⟨A⟩$ constant

Uncertainty Principle

Heisenberg Uncertainty Principle

For position $\hat{X}$ and momentum $\hat{P}$ satisfying the canonical commutation relation:

$$[\hat{X},\hat{P}]=\hat{X}\hat{P}-\hat{P}\hat{X}=i\hslash$$

it follows that:

$${\sigma }_X{\sigma }_P\ge \frac{\hslash }{2}$$

where ${\sigma }_X=\sqrt{⟨X^2⟩-⟨X{⟩}^2}$ is the standard deviation of $X$.

General form: For any two observables $A$, $B$:

$${\sigma }_A{\sigma }_B\ge \frac{1}{2}|⟨[A,B]⟩|$$

Fourier duality: Position and momentum operators are Fourier transforms of each other. In position space, $\hat{P}=-i\hslash \frac{\partial }{\partial x}$. This is why the uncertainty principle follows from the mathematical properties of Fourier pairs.

Composite Systems and Entanglement

Composite System

For two quantum systems $A$ and $B$ with Hilbert spaces ${\mathcal{H}}_A$ and ${\mathcal{H}}_B$, the combined system has:

$${\mathcal{H}}_{AB}={\mathcal{H}}_A\otimes {\mathcal{H}}_B$$

A separable (product) state has the form ${\psi }_A\otimes {\psi }_B$.

Quantum Entanglement

A state in ${\mathcal{H}}_{AB}$ is entangled if it cannot be written as a product state ${\psi }_A\otimes {\psi }_B$. Example of an entangled state:

$$\frac{1}{\sqrt{2}}({\psi }_A\otimes {\psi }_B+{\varphi }_A\otimes {\varphi }_B)$$

Properties of entangled states:

• Cannot describe component systems individually by state vectors
• Described by reduced density matrices: ${\rho }_A={\text{Tr}}_B({\rho }_{AB})$
• Measuring one subsystem instantly constrains the other, regardless of distance
• Enables quantum computing, quantum key distribution, superdense coding

Symmetries and Conservation Laws

Quantum Noether Theorem

If observable $A$ commutes with the Hamiltonian $H$, then $⟨A⟩$ is conserved under time evolution:

$$[A,H]=0⟹\frac{d}{dt}⟨A⟩=0$$

This is the quantum analog of Noether’s theorem: every differentiable symmetry of the Hamiltonian corresponds to a conservation law.

Equivalent Formulations

Formulation	Key Object	Invented By
Matrix mechanics	Infinite matrices for observables	Heisenberg (1925)
Wave mechanics	Wave function $\psi (x,t)$ and PDE	Schrödinger (1926)
Dirac transformation theory	Bra-ket notation unifying both	Dirac (1930s)
Feynman path integrals	Sum over all paths from $a$ to $b$	Feynman (1948)

All formulations are mathematically equivalent but provide different intuitions.

Connections

• Quantum Mechanics - Overview — Historical context and physical intuition
• Quantum Mechanics - Key Phenomena — Physical implications: interference, tunneling, Bell’s theorem
• Quantum Field Theory - Overview — Extends this formalism to relativistic multi-particle systems via field quantization
• Gauge Theory - Overview — Gauge symmetry is a local version of the symmetries described by Noether’s theorem

See Also

• Quantum Mechanics - Key Phenomena — Observable consequences of this formalism
• QED and Renormalization — How path integrals and perturbation theory are applied in QFT