Hamiltonian

The Hamiltonian $H$ is the self-adjoint operator that generates time evolution for a closed quantum system. In the standard laboratory description it also represents the system’s energy. Specifying $H$ is part of defining a physical model: its Hilbert space, domain, degrees of freedom, interactions, external controls, boundary conditions, and parameters must also be known.

In Symbols

In the Schrödinger picture, a state obeys

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

For a time-independent Hamiltonian,

$$\lvert\psi(t)\rangle =U(t,t_0)\lvert\psi(t_0)\rangle,$$

with

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

Self-adjointness ensures that this exponential is unitary. The infinitesimal generator relation is

$$H =i\hbar \left. \frac{\partial U(t,t_0)}{\partial t} \right\rvert_{t=t_0}.$$

For a time-dependent Hamiltonian,

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

where time ordering is required unless

$$[H(t),H(t')]=0$$

for all relevant times.

Hamiltonian as Energy Observable

If a time-independent Hamiltonian has discrete spectral decomposition

$$H=\sum_nE_n\Pi_n,$$

then an ideal energy measurement returns $E_n$ with probability

$$p(E_n)=\operatorname{Tr}(\rho\Pi_n).$$

The mean energy is

$$\langle H\rangle_\rho =\operatorname{Tr}(\rho H),$$

when $\rho$ lies in the domain needed for the expectation to be finite. A state can have well-defined norm but infinite mean energy, so this condition is not automatic in infinite-dimensional systems.

For continuous or mixed spectra, the sum is replaced by the spectral integral

$$H=\int_{\mathbb R}E\,P_H(dE).$$

An energy eigenstate of a time-independent Hamiltonian evolves only by a phase:

$$H\lvert E\rangle=E\lvert E\rangle \quad\Longrightarrow\quad \lvert E,t\rangle =e^{-iE(t-t_0)/\hbar} \lvert E,t_0\rangle.$$

Its Schrödinger-picture probability distributions for time-independent observables need not all be constant if the observable itself has explicit time dependence, but the state ray is stationary.

Canonical Home

See Hamiltonians for the canonical conceptual treatment and modeling boundary. Schrödinger Equation owns the evolution equation, while Time-Evolution Operator owns propagators and composition.

Hamiltonians in Coordinate Space treats differential realizations. The Operator and Hamiltonian Library provides compact cards for standard models.

Domain and Self-Adjointness

For an unbounded Hamiltonian,

$$H:D(H)\subset\mathcal H\longrightarrow\mathcal H,$$

the domain $D(H)$ is part of the operator. The formal differential expression

$$-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}+V(x)$$

does not define a unique Hamiltonian until the configuration space, measure, regularity requirements, and boundary conditions are specified.

A symmetric operator satisfies

$$\langle\phi\vert H\psi\rangle =\langle H\phi\vert\psi\rangle$$

on its stated domain, but symmetry alone does not guarantee self-adjointness. Self-adjointness supplies a real spectral measure and unitary evolution. Different self-adjoint extensions of one formal differential expression can describe different boundary physics.

For a stable isolated system, the Hamiltonian is usually required to be bounded below. Self-adjointness alone does not imply a ground state or even a lower bound.

Building a Hamiltonian

Quantum formalism explains how to use a specified Hamiltonian; it does not uniquely derive $H$ from a verbal description of a system. Model construction requires physical input:

• chosen degrees of freedom and Hilbert space;
• kinetic terms and interactions;
• external fields and controls;
• symmetry constraints;
• approximation regime and neglected couplings;
• boundary conditions and operator domains;
• measured or computed parameters.

Replacing classical variables by noncommuting operators is not a complete universal quantization procedure. Ordering choices, constraints, coordinate measures, and singular interactions can lead to inequivalent quantum models.

Standard Forms

System	Hamiltonian	Important assumptions
Free nonrelativistic particle	$H=\mathbf p^2/(2m)$	Fixed particle number, no external potential
Particle in a potential	$H=\mathbf p^2/(2m)+V(\mathbf x,t)$	Scalar potential and chosen domain
Harmonic oscillator	$H=p^2/(2m)+m\omega^2x^2/2$	Quadratic confinement
Spin in a field	$H=-\boldsymbol{\mu}\cdot\mathbf B$	Effective spin and field model
General two-level system	$H=cI+\frac12\mathbf h\cdot\boldsymbol{\sigma}$	Chosen two-dimensional subspace

These expressions are model families, not complete specifications. Parameter signs, units, gauges, tensor-product identity factors, and boundary conditions remain essential.

Additive Energy Shifts

For a constant real number $C$,

$$H'=H+CI$$

produces

$$U'(t,t_0) =e^{-iC(t-t_0)/\hbar}U(t,t_0).$$

The extra factor is a global phase, so an isolated nonrelativistic system has unchanged state rays, transition frequencies, and expectation values. This is the freedom to choose the zero of energy.

The statement has boundaries. A term that depends on the state, subsystem, position, or external configuration is not generally a harmless constant. Comparisons between different physical configurations, coupling to gravity, and explicitly time-dependent transformations require separate care.

Time Dependence and Conservation

Even when $H(t)$ changes with time, it still generates evolution through the Schrödinger equation. Its instantaneous expectation value obeys

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

for sufficiently regular closed-system dynamics. Thus energy is conserved when the Hamiltonian has no explicit time dependence.

For another Schrödinger-picture observable $A(t)$,

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

If $A$ has no explicit time dependence and commutes with $H$, its expectation is conserved. The converse may require qualifications about the class of states and domains.

Representation and Picture Changes

Under a time-dependent unitary change of state representation,

$$\lvert\psi'(t)\rangle =W(t)\lvert\psi(t)\rangle,$$

the transformed generator is

$$H' =WHW^\dagger +i\hbar\dot W W^\dagger.$$

The second term is essential. A time-dependent basis change does more than conjugate the old Hamiltonian. This formula underlies rotating frames and interaction pictures.

Closed and Open Systems

For a closed system, $H$ generates unitary evolution. A subsystem interacting with an environment generally has reduced dynamics that cannot be described by a subsystem Hamiltonian alone. A Markovian master equation may have the form

$$\frac{d\rho}{dt} =-\frac{i}{\hbar}[H,\rho] +\mathcal D(\rho),$$

where $H$ generates the coherent part and $\mathcal D$ is a dissipative superoperator. The full generator is not itself merely a Hamiltonian.

Minimal Example

For a two-level system,

$$H=\frac{\hbar\Omega}{2}\sigma_z,$$

with

$$H\lvert0\rangle =\frac{\hbar\Omega}{2}\lvert0\rangle, \qquad H\lvert1\rangle =-\frac{\hbar\Omega}{2}\lvert1\rangle.$$

An initial superposition

$$\lvert\psi(0)\rangle =\alpha\lvert0\rangle+\beta\lvert1\rangle$$

evolves as

$$\lvert\psi(t)\rangle =\alpha e^{-i\Omega t/2}\lvert0\rangle +\beta e^{i\Omega t/2}\lvert1\rangle.$$

The populations in the energy basis remain fixed, while the relative phase evolves at angular frequency $\Omega$.

Common Aliases

• energy operator, in the standard closed-system setting;
• time-evolution generator;
• system Hamiltonian, when distinguished from controls, baths, or total Hamiltonians;
• effective Hamiltonian, when high-energy or fast degrees of freedom have been eliminated.

An effective Hamiltonian must be accompanied by its validity regime. It need not be a fundamental microscopic energy operator.

Common Confusions

• A Hamiltonian formula such as $-\hbar^2d^2/(2m\,dx^2)+V(x)$ still needs a Hilbert space, domain, and boundary conditions.
• The Hamiltonian is not always time independent.
• Energy zero can be shifted by adding a constant to $H$.
• A self-adjoint Hamiltonian need not have a discrete spectrum or a normalizable ground state.
• A symmetric differential operator is not automatically self-adjoint.
• Quantization is not always achieved by a unique substitution $q,p\mapsto\hat q,\hat p$.
• A time-dependent unitary change of representation adds a derivative term to the transformed Hamiltonian.
• The Hamiltonian of a subsystem is not the complete generator of generic open-system reduced dynamics.
• A conserved expectation value should not be confused with every state being an energy eigenstate.
• The classical Hamiltonian function and quantum Hamiltonian operator are related but are not the same mathematical object.

Related Entries

• Observable
• Wavefunction
• Energy Eigenstates
• Conservation Laws
• Time-Dependent Hamiltonians
• Hermitian versus Self-Adjoint
• Most-Used Hamiltonians
• Core Formulas Index

References

• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018, chs. 2 and 3.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994, chs. 4 and 5.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020, chs. 1 and 2.
• B. C. Hall, Quantum Theory for Mathematicians, Springer, 2013, chs. 7 and 9.