Postulates of Quantum Mechanics

See also: Mathematical Preliminaries for definitions of Hilbert space, observables, eigenstates, etc.

Postulate 1 — State Space

The state of an isolated physical system is completely described by a unit vector $|\psi ⟩$ (the state vector) in a complex separable Hilbert space $\mathcal{H}$, called the state space of the system. Two state vectors that differ only by a global phase $e^{i\theta }$ represent the same physical state.

Postulate 2 — Observables

Every measurable physical quantity (an observable) is represented by a self-adjoint (Hermitian) linear operator $\hat{A}$ acting on the Hilbert space $\mathcal{H}$. The possible outcomes of a measurement of $\hat{A}$ are the eigenvalues $a_n$ of the operator:

$$\hat{A}|a_n⟩=a_n|a_n⟩.$$

Because $\hat{A}$ is Hermitian, its eigenvalues are real and its eigenvectors form a complete orthonormal basis of $\mathcal{H}$.

Postulate 3 — Measurement (Born Rule)

If the system is in the normalized state $|\psi ⟩$, the probability of obtaining the eigenvalue $a_n$ when measuring the observable $\hat{A}$ is

$$P(a_n)=|⟨a_n|\psi ⟩{|}^2,$$

assuming a non-degenerate spectrum. For a degenerate eigenvalue $a_n$ with projector ${\hat{P}}_n$ onto its eigenspace,

$$P(a_n)=⟨\psi |{\hat{P}}_n|\psi ⟩.$$

Postulate 4 — Collapse (Projective Measurement)

Immediately after a measurement of $\hat{A}$ that yields the eigenvalue $a_n$, the state of the system collapses to the normalized projection onto the corresponding eigenspace:

$$|\psi ⟩⟶\frac{{\hat{P}}_n|\psi ⟩}{\sqrt{⟨\psi |{\hat{P}}_n|\psi ⟩}}.$$

Postulate 5 — Time Evolution

The time evolution of the state vector of a closed quantum system is governed by the Schrödinger equation:

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

where $\hat{H}$ is the Hamiltonian operator (the observable corresponding to the total energy of the system). Equivalently, evolution is given by a unitary operator $\hat{U}(t,t_0)$ such that $|\psi (t)⟩=\hat{U}(t,t_0)|\psi (t_0)⟩$.

Postulate 6 — Composite Systems

The state space of a composite physical system is the tensor product of the state spaces of its components. If systems $A$ and $B$ have state spaces ${\mathcal{H}}_A$ and ${\mathcal{H}}_B$, then the joint system has state space

$$\mathcal{H}={\mathcal{H}}_A\otimes {\mathcal{H}}_B.$$

If the subsystems are prepared independently in states $|{\psi }_A⟩$ and $|{\psi }_B⟩$, the joint state is $|{\psi }_A⟩\otimes |{\psi }_B⟩$. General states in $\mathcal{H}$ may be entangled and not expressible as such a product.

Postulate 7 — Symmetrization (Identical Particles)

The state vector of a system of identical particles is either fully symmetric under the exchange of any two particles (bosons, integer spin) or fully antisymmetric (fermions, half-integer spin):

$${\hat{P}}_{ij}|\psi ⟩=\pm |\psi ⟩.$$