Hamiltonian operator

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

For $Q=\hat{p}-isA(x)$ with real $A$ and positive $s$, the factorization gives potential $(s^2{A}^2-s\hslash A^{\prime })/(2m)$ and a nonnegative energy form. Its zero mode is proportional to $\exp [-s\int Adx/\hslash ]$. Taking $A=x$ and optimizing positivity over $s$ gives the Heisenberg uncertainty principle for states with finite second moments.

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 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 $O_H(t)=e^{iHt/\hslash }{O}_S{e}^{-iHt/\hslash }$.

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 $U(t)=e^{-iHt/\hslash }$.

A quantum-mechanical propagator is the position-space kernel $K(q_f,t_f;q_i,t_i)=⟨q_f|U(t_f,t_i)|q_i⟩$ of the time-evolution operator. It is the amplitude to evolve between the specified endpoint configurations.

For unit mass and $\hslash =1$, this prepoint kernel tends to the identity and generates the Time-dependent Schrodinger equation. The square-root branch follows the damped Fresnel integral. Its normalized moments are $1$, $0$, and $i\Delta t$ at orders zero, one and two; these fix both the normalization and the kinetic sign.

Expand a smooth wavefunction and potential in the normalized short-time Schrödinger kernel. Its odd moment vanishes and its second moment is $i\Delta t$, so one step changes the wavefunction by $i\Delta t{\psi }^{\prime \prime }/2-i\Delta tV\psi +O(\Delta t^2)$. Division by the time step gives the Time-dependent Schrodinger equation.

For a free particle of mass $m$ on the line,

A semiclassical propagator sums $e^{i{S}_{\mathrm{cl}}/\hslash }$ over classical paths joining its endpoints, weighted by their quadratic fluctuation determinants and phase indices. It is exact for quadratic actions.

The Van Vleck determinant is the determinant of mixed endpoint derivatives of a classical action. It measures how nearby classical trajectories focus and supplies the fluctuation prefactor of the semiclassical propagator.

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

For an operator $O_H=U^{†}{O}_{S}U$ in the Heisenberg picture, differentiating the evolution operators gives $d{O}_H/dt=i[H_H,O_H]+U^{†}(\partial O_S/\partial t)U$ in units $\hslash =1$. With no explicit operator time dependence, the second term vanishes. Applying this commutator identity to the canonical momentum and real scalar field gives their operator field equations.

For $H=H_0+H_{\mathrm{int}}$, the interaction picture moves the free evolution into operators and leaves states to evolve with $H_I(t)=e^{i{H}_{0}t}{H}_{\mathrm{int}}{e}^{-i{H}_{0}t}$. Its evolution operator is the time-ordered Dyson series.

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.

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.

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.