Hamiltonian Operator

The capital Latin letter $H$, read “Hamiltonian,” denotes the operator that generates time translations. For a closed system it also represents total energy. Authors write either $H$ or $\hat H$; the hat is a typography convention, not a mathematical operation.

Default Meaning

In the Schrödinger picture, the state obeys

$$i\hbar\frac{d}{dt} \lvert\psi(t)\rangle = H(t)\lvert\psi(t)\rangle.$$

For a time-independent Hamiltonian,

$$U(t,t_0) = \exp\left[ -\frac{i}{\hbar} H(t-t_0) \right].$$

For a general time-dependent Hamiltonian,

$$U(t,t_0) = \mathcal T \exp\left[ -\frac{i}{\hbar} \int_{t_0}^{t}H(s)\,ds \right],$$

where $\mathcal T$ orders later-time operators to the left. The time-ordering symbol can be omitted only when the relevant Hamiltonians commute at different times, or when the exponential is otherwise understood as shorthand for the evolution equation.

Mathematical Type

For closed-system quantum mechanics, $H$ is a self-adjoint linear operator on the Hilbert space:

$$H:\mathcal D(H) \subseteq\mathcal H \longrightarrow\mathcal H, \qquad H=H^\dagger.$$

In finite dimensions, this means that the Hamiltonian is represented by a Hermitian matrix. In infinite dimensions, the domain $\mathcal D(H)$ is part of the operator’s definition. A formally Hermitian differential expression need not determine a self-adjoint operator until boundary conditions and the domain have been specified.

Self-adjointness gives real spectral values and, under the standard hypotheses, unitary time evolution. Physical Hamiltonians are also commonly bounded below, but boundedness below is a stability property rather than the definition of a Hamiltonian.

The dimensions are

$$[H]=[\text{energy}] =ML^2T^{-2}.$$

In SI units, Hamiltonian matrix elements and eigenvalues are quoted in joules or electronvolts. Some communities divide by $\hbar$ and report $H/\hbar$ in angular-frequency units. An equation written with $\hbar=1$ can therefore hide an energy-versus-frequency convention that must be restored before numerical work.

Energy Spectrum

An energy eigenstate satisfies

$$H\lvert E_n\rangle =E_n\lvert E_n\rangle.$$

For discrete, nondegenerate notation one often writes

$$H = \sum_n E_n\lvert E_n\rangle \langle E_n\rvert.$$

Degenerate eigenspaces require an additional label or spectral projectors. Continuous spectra require a spectral measure rather than an ordinary countable sum. The compact eigenket formula should not be read as a universal spectral decomposition.

Matrix and Coordinate Representations

In an orthonormal basis $\{\lvert n\rangle\}$,

$$H_{mn} = \langle m\rvert H\lvert n\rangle.$$

The same abstract operator may look very different in another basis. A basis change by a unitary matrix $V$ gives

$$H' =VHV^\dagger.$$

For a one-dimensional particle with a local potential,

$$(H\psi)(x) = \left[ -\frac{\hbar^2}{2m} \frac{d^2}{dx^2} +V(x) \right]\psi(x).$$

This line states the coordinate representation, not the complete operator. The allowed wavefunctions and their boundary conditions still determine the domain.

Common Hamiltonian Forms

A nonrelativistic particle in a scalar potential is commonly written

$$H = \frac{\mathbf p^2}{2m} +V(\mathbf r,t).$$

For a particle of charge $q$ in electromagnetic potentials,

$$H = \frac{ \left[ \mathbf p-q\mathbf A(\mathbf r,t) \right]^2 }{2m} +q\phi(\mathbf r,t).$$

Here $\mathbf p$ is canonical momentum. The kinetic momentum is $\mathbf p-q\mathbf A$, so replacing one by the other changes the physical meaning.

Every two-level Hamiltonian can be written

$$H =cI +\frac12 \mathbf h\cdot\boldsymbol{\sigma},$$

where $c$ and the components of $\mathbf h$ have energy units. The identity term shifts both energies equally, while $\mathbf h$ determines the level splitting and eigenvectors.

Common Subscripts and Superscripts

Symbol	Usual meaning	Important qualification
$H_0$	Free or unperturbed Hamiltonian	The split into “free” and “interaction” parts is chosen for the problem
$H_1$ or $V$	Perturbation	$V$ may also mean a potential-energy function
$H_{\mathrm{int}}$	Interaction Hamiltonian	May mean coupling between subsystems or the interaction-picture operator
$H(t)$	Explicitly time-dependent Hamiltonian	Energy need not be conserved
$H_A$, $H_B$	Subsystem Hamiltonians	Tensor identity factors are often suppressed
$H_{\mathrm{tot}}$	Total Hamiltonian	Often includes subsystem and coupling terms
$H_{\mathrm{eff}}$	Effective Hamiltonian	Its validity depends on scale, subspace, and approximation order
$H_I(t)$	Interaction-picture Hamiltonian	Defined relative to a selected $H_0$
$H_F$	Floquet Hamiltonian	Quasienergies are branch-dependent modulo the drive frequency
$H_{\mathrm{cl}}$	Classical Hamiltonian function	A function on phase space, not a Hilbert-space operator

The decomposition

$$H=H_0+V$$

is not unique. It is a bookkeeping choice whose usefulness depends on whether $H_0$ is solvable and $V$ is controlled by an approximation.

Composite-System Convention

For $\mathcal H_A\otimes\mathcal H_B$, a noninteracting composite Hamiltonian is

$$H_{\mathrm{tot}} = H_A\otimes I_B +I_A\otimes H_B.$$

With coupling,

$$H_{\mathrm{tot}} = H_A\otimes I_B +I_A\otimes H_B +H_{AB}.$$

Authors often abbreviate the first two terms as $H_A+H_B$. The omitted identity operators must be restored when checking dimensions, commutators, or matrix representations.

Picture and Frame Changes

Suppose states are transformed according to

$$\lvert\psi'(t)\rangle = V(t)\lvert\psi(t)\rangle.$$

The Hamiltonian in the transformed frame is

$$H'(t) = V(t)H(t)V^\dagger(t) +i\hbar \dot V(t)V^\dagger(t).$$

The second term is essential for a time-dependent transformation. Omitting it is a common source of incorrect rotating-frame and interaction-picture Hamiltonians.

For $H=H_0+V$ and

$$U_0(t,t_0) = \exp\left[ -\frac{i}{\hbar} H_0(t-t_0) \right],$$

the interaction-picture perturbation is

$$H_I(t) = U_0^\dagger(t,t_0) V(t) U_0(t,t_0).$$

Thus $H_{\mathrm{int}}$ in the Schrödinger picture and $H_I(t)$ in the interaction picture are related but generally not identical expressions.

Additive Energy Shifts

Replacing the Hamiltonian by

$$H'(t)=H(t)+c(t)I$$

multiplies every state by the same phase:

$$U'(t,t_0) = \exp\left[ -\frac{i}{\hbar} \int_{t_0}^{t}c(s)\,ds \right] U(t,t_0).$$

For an isolated nonrelativistic system this global phase leaves expectation values and transition probabilities unchanged. Energy differences remain invariant. The statement does not license arbitrary shifts when energy couples to additional dynamical structures or when comparing Hamiltonians across differently defined effective descriptions.

Conservation and Expectation Values

For an operator $A(t)$ in the Schrödinger picture,

$$\frac{d}{dt} \langle A\rangle = \frac{i}{\hbar} \langle[H,A]\rangle +\left\langle \frac{\partial A}{\partial t} \right\rangle.$$

Setting $A=H$ gives

$$\frac{d}{dt} \langle H\rangle = \left\langle \frac{\partial H}{\partial t} \right\rangle.$$

Therefore the expected energy is conserved when $H$ has no explicit time dependence, subject to the regularity and domain assumptions needed for the derivatives and products to exist. A state need not be an energy eigenstate for its mean energy to be conserved.

Typography and Symbol Collisions

Form or context	Meaning
$H$ or $\hat H$ in dynamics	Hamiltonian operator
$\mathcal H$	Hilbert space, not the Hamiltonian
$H_{mn}$	Hamiltonian matrix element in a chosen basis
$H_n(x)$	Hermite polynomial
$H$ in electromagnetism	Magnetic-field intensity
$H$ in quantum circuits	Hadamard gate
$H_{\mathrm{cl}}(q,p)$	Classical Hamiltonian function

Context usually distinguishes the Hamiltonian from the Hadamard gate: the latter is a particular dimensionless $2\times2$ unitary matrix,

$$H_{\mathrm{Had}} = \frac{1}{\sqrt2} \begin{pmatrix} 1&1\\ 1&-1 \end{pmatrix}.$$

Writing $H_{\mathrm{Had}}$ on a notation-sensitive page avoids overloading $H$ twice in the same calculation.

Hat Policy

Both

$$H\lvert\psi\rangle \qquad\text{and}\qquad \hat H\lvert\psi\rangle$$

are standard. A text that reserves hats for operators may contrast $\hat H$ with the classical function $H(q,p)$. A text that omits hats relies on context and typography. Do not insert or remove hats selectively inside one derivation unless the notation policy is being changed consistently.

Canonical Home

Hamiltonians owns the operator’s physical and mathematical role. Time-Evolution Operator develops the propagator, and Time-Dependent Hamiltonians develops time ordering.

Hermitian versus Self-Adjoint explains the domain distinction. Coordinate Representation and Hamiltonians in Coordinate Space own the wave-mechanical realizations. The Hamiltonian glossary entry gives a shorter conceptual lookup.

Convention Warnings

• $H$ has energy units; $H/\hbar$ has angular-frequency units.
• Hermitian matrix and self-adjoint operator are interchangeable only in finite-dimensional settings.
• A differential expression without a domain and boundary conditions is not a complete Hamiltonian.
• $H_0+V$ is a chosen decomposition, not an invariant separation.
• Subsystem Hamiltonians often carry suppressed tensor-product identities.
• A time-dependent frame change adds $i\hbar\dot V V^\dagger$.
• $H_{\mathrm{int}}$ and $H_I(t)$ can denote operators in different pictures.
• Explicit time dependence generally prevents conservation of $\langle H\rangle$.
• Adding a scalar multiple of the identity changes global phase but not isolated-system energy differences.
• $\mathcal H$ conventionally denotes Hilbert space, while $H$ denotes the Hamiltonian.

References

• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020, chs. 1 and 2.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994, chs. 4 and 7.
• A. Messiah, Quantum Mechanics, Dover, 1999, vol. I, chs. IV and VIII.
• M. Reed and B. Simon, Methods of Modern Mathematical Physics I: Functional Analysis, rev. ed., Academic Press, 1980, secs. VIII.4 and VIII.7.