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The Born rule depends only on transition probabilitiesbetween rays. A physical transformation must preserve these probabilities. By Wigner theorem, such a ray transformation is induced by a unitary or antiunitary operator. For ordinary continuously connected transformations one chooses the unitary branch, so

The states $U(g_1)U(g_2)|\psi ⟩$ and $U(g_1{g}_2)|\psi ⟩$ represent the same ray. They may therefore differ by a state-independent quantum phase:Thus quantum transformations form a projective unitary representation.

If $g$ is a symmetry of the Hamiltonian operator $H$, it also obeysequivalently $[U(g),H]=0$. For a differentiable one-parameter subgroup, writewith a self-adjoint generator $Q$. Differentiating the symmetry equation at $s=0$ yieldsFor a time-independent $Q$, the Heisenberg picture equation isHence the generator is a conserved observable. Conversely, a self-adjoint conserved $Q$ commutes with $H$ and its unitary group generates symmetries. This is the conserved generator of a continuous quantum symmetry.

Let $H$ be the one-dimensional Hamiltonian operator and letbe its discrete energy eigenvalues. Expanding a normalized trial wavefunction in an orthonormal energy basis,givesThus the Rayleigh-Ritz variational principle givesfor every nonzero admissible trial wavefunction. One chooses a parameterized family and minimizes the quotient.

When $V(x)=V(-x)$, parity commutes with $H$. In one dimension the nondegenerate ground state is even and the first excited state is odd. Restricting the trial family to odd wavefunctions makes every trial state orthogonal to the ground state, soThis is the odd-state variational principle for an even potential.

For a free particle, the Hamiltonian operator isSince ${\chi }^{\prime \prime }=-k^2\chi$,Thuswhere the momentum follows because $-i\hslash d\chi /dx=\hslash k\chi$.

With a nonconstant potential energy, momentum no longer diagonalizes the Hamiltonian operator. The operator derivation must use the energy eigenfunctions of the full Hamiltonian, a Dyson series, or a time-sliced Trotter product formula. In the classical derivation, the straight paths are replaced by every solution of the nonlinear Euler-Lagrange equation with the specified endpoints. The semiclassical propagator becomes a sumwhere the prefactor is the Van Vleck determinant and ${\mu }_{\mathrm{cl}}$ is a Maslov index. Unlike a quadratic theory, the classical-path sum is generally only an asymptotic approximation: the exact path integral includes fluctuations of every order. On the circle, the sum must still include all winding number sectors.

The parity of a wavefunction is its eigenvalue under spatial reflection: $\psi (-x)=+\psi (x)$ for even parity and $\psi (-x)=-\psi (x)$ for odd parity. If $V(-x)=V(x)$, the Hamiltonian commutes with reflection, so its energy eigenspaces admit a parity eigenbasis. In particular, every nondegenerate bound-state energy eigenfunction has definite parity; within a degenerate scattering eigenspace one must choose the even and odd linear combinations.

Write the expectation value asDifferentiating and retaining the possible explicit time dependence of the observable givesThe Schrodinger equation and its adjoint arewhere the Hamiltonian operator is Hermitian. Substitution yieldsThis is the proof of Ehrenfest theorem from the Schrodinger equation.

Let a projective measurement have mutually orthogonal projections $P_i$ satisfying ${\sum }_i{P}_i=I$. On a density operator $\rho$, the Born rule and the Lüders rule giveconditional on outcome $i$. On the first part of a bipartite quantum system, replace $P_i$ by $P_i\otimes I_2$.

The subsystems are isolated when there is no interaction term coupling them. Their Hamiltonian operator has the formso their subsequent unitary time evolution factorizes as $U_1\otimes U_2$.

The reduced density matrix of subsystem 1 iswhere the partial trace is characterized byfor every local observable $M_1$. Thus ${\rho }_1$ contains exactly the statistics accessible by measurements on subsystem 1.

Suppose a projective measurement $\{Q_j\}$ is performed on subsystem 2 and its outcome is not communicated. The resulting nonselective state isFor every $M_1$, the cyclic property of the trace and ${\sum }_j{Q}_j^2=I_2$ giveHence ${\rho }_1^{\prime }={\rho }_1$. No local projective measurement on subsystem 1 can reveal whether the remote unreported measurement occurred. This is quantum no-signalling, which prevents a choice made at a spacelike separation from transmitting information faster than light and makes the measurement formalism compatible with relativistic causality.

The proposed nondisturbing device would violate no information without disturbance. In an ordinary quantum instrument, let $M_{i\alpha }$ be the Kraus operators associated with classical output $i$. If every pure state $|\psi ⟩$ remains unchanged even conditional on the displayed outcome, every nonzero $M_{i\alpha }|\psi ⟩$ must be parallel to $|\psi ⟩$. A linear operator for which every vector is an eigenvector is a scalar multiple of the identity, so $M_{i\alpha }=c_{i\alpha }I$. Its output probabilityis independent of the state. It cannot equal $⟨\psi |P_i|\psi ⟩$ for arbitrary projectors. Equivalently, repeated nondisturbing samples would permit quantum state tomography of one specimen and then quantum cloning, contradicting ordinary quantum theory.

Such devices would not necessarily enable superluminal signalling. One consistent operational extension could make every sequence of outputs depend only on the local reduced density matrix and local settings. Since an unreported remote measurement leaves that matrix unchanged, all local device statistics would remain unchanged too. Other extensions could add nonlocal outcome-dependent rules and permit signalling, but that behavior is additional to the device specification rather than forced by it.

A quantum time-reversal operator is an antiunitary operator reversing spatial momentum and angular momentum. For a time-reversal-invariant Hamiltonian operator, its conjugation of $i$ makes it intertwine forward and backward unitary time evolution without reversing the energy spectrum. A scalar momentum eigenstate may transform as $\hat{T}|\mathbf{p}⟩=e^{i\chi (\mathbf{p})}|-\mathbf{p}⟩$. With unit-modulus phases and a complete normalized basis, expansion of arbitrary states proves antiunitarity.

For every normalized trial state $\psi$ in the domain of a Hamiltonian operator $H$ that is bounded below,where $E_0$ is the ground-state energy. Equality holds precisely when the trial state lies in the ground-state eigenspace.

In the Schrödinger picture, states evolve with the Hamiltonian operator while observables without explicit time dependence are fixed. For time-independent $H$, the corresponding Heisenberg picture operator is $O_H(t)=e^{iHt/\hslash }{O}_S{e}^{-iHt/\hslash }$.

A stoquastic Hamiltonian is a real Hermitian Hamiltonian operator with nonpositive off-diagonal entries in a specified basis, usually the computational basis. The property depends on that basis. Permutation matrices in a quantum circuit's propagation terms and $|-⟩⟨-|$ input penalties give a useful family of such operators.

When a Hamiltonian operator changes instantaneously by a finite potential step, the quantum state is continuous across the switching time. Its coefficients in the new energy eigenbasis are its overlaps with those eigenstates; the Born rule gives their squared moduli as probabilities. The state need not remain an energy eigenstate of the new Hamiltonian.

The time-evolution operator maps an initial quantum state to its state at a later time. For a time-independent Hamiltonian operator $H$, it is the unitary operator $U(t)=e^{-iHt/\hslash }$.

The time-ordered exponential solves $i\hslash {\partial }_{t}U=H(t)U$ with initial condition $U(t_0,t_0)=I$. Its expansion is the Dyson series. For bounded continuous Hamiltonian operators the series converges, as in differentiation of the Dyson time-ordered exponential. If the generators commute at all times, time ordering is unnecessary and an ordinary operator exponential suffices.

For a Hamiltonian operator with kinetic term ${\hat{p}}^2/2$, insert position-basis resolutions between Trotter product formula factors. The measure is the limit of $(2\pi i\Delta {)}^{-M/2}{\prod }_{j=1}^{M-1}d{q}_j$, and the exponent contains ${\sum }_j[(q_j-q_{j-1}{)}^2/(2\Delta )-\Delta V(q_{j-1})]$. Its continuum notation is $\int \mathcal{D}q{e}^{iS[q]}$, with fixed endpoint values. A damping prescription defines the oscillatory limit.

A fermionic mode with canonical anticommutation relations $\{b,b^{†}\}=1$ has the symmetrically ordered Hamiltonian operator $H=\frac{E}{2}(b^{†}b-b{b}^{†})=E(b^{†}b-1/2)$. Its empty-state zero-point contribution is $-E/2$. A quantum harmonic oscillator with bosonic canonical commutation relations contributes $+E/2$ instead. For a Dirac field, sum over both particle and antiparticle modes and over spin: there are four independent oscillators per spatial momentum in four spacetime dimensions. Comparing bosonic and fermionic vacuum energies requires a common regularization in quantum field theory, matching degrees of freedom and a stated vacuum-energy subtraction convention.