Quantum system

A quantum system is modeled by a complex Hilbert space; its quantum states describe preparations, while observables describe measurable quantities.

A pure quantum state is a ray in a complex Hilbert space and may be represented by a normalized state vector.

A qubit is a two-dimensional quantum system. Its pure state is a ray in a two-dimensional complex Hilbert space and may be written $\alpha |0⟩+\beta |1⟩$ with $|\alpha {|}^2+|\beta {|}^2=1$.

A quantum number is a discrete label, usually an observable's eigenvalue, used to distinguish quantum states.

Multiplying a quantum state by one global complex phase does not change any measurement probability; relative phases between components can affect interference.

A global phase multiplies an entire quantum state by one complex number of modulus one. It does not change any measurement probability, unlike a relative phase between components.

A quantum observable is represented by a self-adjoint operator; its possible measured values are spectral values of that operator.

In a normalized state $|\psi ⟩$, the expectation of an observable $A$ is $⟨\psi |A|\psi ⟩$.

The Hamiltonian operator represents the total energy and generates time evolution through the Schrödinger equation.

A unitary operator $U$ is a symmetry of a time-independent Hamiltonian $H$ whenequivalently $[U,H]=0$. A differentiable one-parameter symmetry $U(s)=e^{-isQ/\hslash }$ has a self-adjoint generator $Q$ satisfying $[Q,H]=0$, so $Q$ is conserved.

If $U(s)=e^{-isQ/\hslash }$ and $U(s{)}^{†}HU(s)=H$, differentiation at $s=0$ givesIn the Heisenberg picture,

A time-independent Hamiltonian evolves a state by the unitary operator $U(t)=e^{-iHt/\hslash }$.

In the Heisenberg picture, states are fixed and an observable evolves asFor a time-independent Schrödinger-picture observable,

A time-reversal operator $T$ intertwines forward and backward evolution:It must be antiunitary in ordinary quantum mechanics. Its conjugate linearity sends $i$ to $-i$, allowing it to commute with a time-reversal-invariant Hamiltonian rather than reversing the sign of its energy.

If a unitary $T$ satisfied $U(t)T=TU(-t)$, differentiation at zero would giveIt would map every energy $E$ to $-E$. A Hamiltonian unbounded above would consequently be unbounded below and have no stable ground state.

For antiunitary $T$, conjugate linearity gives $T(i\psi )=-iT\psi$. Differentiatingtherefore yields $HT=TH$. Time reversal maps an energy eigenspace to itself and creates no negative-energy instability.

A bound state is a normalizable energy eigenstate spatially confined by a potential.

A stationary state is an energy eigenstate whose time dependence is only an overall phase.