Hamiltonians

The Hamiltonian is the operator that generates time evolution for a closed quantum system. In the standard laboratory description, it also represents the system’s energy. These two roles are closely connected but should not be collapsed into the slogan “the Hamiltonian is just total energy.”

A Hamiltonian is part of a physical model. Quantum postulates explain how to use a specified $H$; they do not determine the correct degrees of freedom, interactions, boundary conditions, or parameter values for every system.

This page defines the Hamiltonian and develops that modeling boundary. Schrödinger Equation owns the initial-value equation, and Time-Evolution Operator owns the propagator.

Classical Hamiltonian Preview

In classical mechanics, a Hamiltonian function

$$H_{\mathrm{cl}}(q,p,t)$$

generates phase-space evolution through Hamilton’s equations:

$$\begin{aligned} \dot q_j &= \frac{\partial H_{\mathrm{cl}}}{\partial p_j},\\ \dot p_j &= -\frac{\partial H_{\mathrm{cl}}}{\partial q_j}. \end{aligned}$$

For many standard systems in inertial coordinates,

$$H_{\mathrm{cl}} = T+V$$

is the total mechanical energy. Yet even classically, explicitly time-dependent canonical transformations can change the Hamiltonian by more than a passive rewrite, and $H_{\mathrm{cl}}$ need not be conserved when it depends explicitly on time.

Classical structure often motivates a quantum Hamiltonian, but the replacement

$$q_j \longmapsto Q_j, \qquad p_j \longmapsto P_j$$

is not a complete quantization algorithm. Noncommuting operators create ordering choices, singular expressions require domains, and inequivalent quantum models can share a classical limit. Hamiltonian Mechanics Review supplies the classical background; Quantization vs Classical Limit treats the construction caveat.

Quantum Definition

For a closed system, a Hamiltonian $H(t)$ is a self-adjoint operator that appears in the Schrödinger equation

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

Its physical dimension is energy. Equivalently,

$$\frac{H}{\hbar}$$

has units of angular frequency.

In a finite-dimensional Hilbert space, $H(t)$ is represented by a Hermitian matrix. In infinite dimensions, the operator includes both a differential or algebraic expression and a domain:

$$H : \mathcal D(H) \subseteq \mathcal H \longrightarrow \mathcal H.$$

Self-adjointness means

$$H = H^\dagger$$

with equality of domains, not merely equality of formal differential expressions. This distinction ensures a real spectral measure and unitary evolution. The full domain theory lives in Hermitian vs Self-Adjoint.

Time-independent and time-dependent notation

When the Hamiltonian has no explicit time dependence, write $H$. When external controls or a changing representation introduce explicit dependence, write $H(t)$. At each time, $H(t)$ must be self-adjoint on suitable domains, and enough regularity is needed for a unitary propagator to exist.

The finite-dimensional case hides these analytic conditions because every Hermitian matrix is bounded and defined on the whole space.

Generator of Time Translation

For a short interval $\delta t$, the propagator has the form

$$U(t+\delta t,t) = I - \frac{i}{\hbar} H(t)\delta t + O(\delta t^2).$$

Thus $H(t)$ determines the infinitesimal change of the state. Conversely,

$$H(t) = i\hbar \frac{\partial U(t,t_0)}{\partial t} U^\dagger(t,t_0)$$

when the derivative and convention are defined appropriately. At $t=t_0$, where $U(t_0,t_0)=I$, this reduces to the initial-time derivative alone.

For a time-independent Hamiltonian,

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

The exponential is defined by spectral or functional calculus, not by exponentiating each matrix entry separately.

Stone’s theorem gives the precise converse: every strongly continuous one-parameter unitary group has a unique self-adjoint generator. With the time-translation parameter measured in seconds, that generator is $H/\hbar$.

Density operators

For a closed system,

$$\rho(t) = U(t,t_0) \rho(t_0) U^\dagger(t,t_0).$$

Differentiation gives the von Neumann equation

$$i\hbar \frac{d\rho}{dt} = [H(t),\rho(t)].$$

Pure and mixed states therefore share the same Hamiltonian generator.

Hamiltonian as Energy Observable

In the standard laboratory frame, the Hamiltonian also supplies the energy measurement. A self-adjoint $H$ has a projection-valued spectral measure $P_H$:

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

The probability that an ideal energy measurement lies in a measurable set $\Delta$ is

$$\Pr_\rho(E\in\Delta) = \operatorname{Tr} \left[ \rho P_H(\Delta) \right].$$

For a purely discrete spectrum,

$$H = \sum_n E_nP_n,$$

and

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

The expectation value is

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

when the mean exists. For a pure state, this is

$$\langle H\rangle_\psi = \langle\psi\rvert H\lvert\psi\rangle,$$

provided $\lvert\psi\rangle$ lies in the appropriate domain.

An energy probability distribution can be normalized even when its mean or variance diverges. Finite $\langle H\rangle$ and finite

$$(\Delta H)^2 = \langle H^2\rangle - \langle H\rangle^2$$

require additional integrability or domain conditions.

Energy Eigenstates develops definite-energy states, degeneracy, and continuous-spectrum caveats.

Why Self-Adjointness Matters

If $H$ is self-adjoint and time independent, then

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

is unitary:

$$U^\dagger(t)U(t) = I.$$

Norms and inner products are preserved, so probabilities remain normalized.

For an unbounded differential operator, checking a formal integration-by-parts identity is not enough. Boundary conditions can change:

• whether the operator is self-adjoint;
• which unitary evolution it generates;
• its spectrum and eigenfunctions;
• which probability current is allowed through a boundary.

The same expression $-\hbar^2d^2/(2m\,dx^2)$ on an interval can define different Hamiltonians under Dirichlet, periodic, or phase-twisted boundary conditions.

Self-adjointness does not imply that the spectrum is bounded below. A lower energy bound is an additional stability property expected of many fundamental nonrelativistic models. Some effective or rotating-frame Hamiltonians need not display the same lower-bound interpretation as the underlying laboratory energy.

Building a Hamiltonian Is Physical Modeling

A Hamiltonian is assembled from the chosen degrees of freedom and their interactions. A common decomposition is

$$H = H_0+V,$$

where $H_0$ is a solvable or reference part and $V$ is an interaction or perturbation. The split is useful but not unique; only the total model determines exact closed-system evolution.

For a bipartite system,

$$\begin{aligned} H_{AB} &= H_A\otimes I_B + I_A\otimes H_B\\ &\quad+ V_{AB}. \end{aligned}$$

The interaction can exchange energy between subsystems and generate entanglement. The local energies $H_A\otimes I_B$ and $I_A\otimes H_B$ need not be conserved separately even when the total time-independent $H_{AB}$ is.

Constructing $H$ requires choices and evidence:

• which coordinates, modes, spins, or effective levels are retained;
• which symmetries constrain allowed terms;
• which couplings are negligible at the target accuracy;
• which boundary conditions and domains apply;
• which parameters come from microscopic theory or calibration;
• which energy and time scales make the effective description valid.

Symmetry can forbid terms or relate coefficients, but it rarely fixes every coefficient. Fitting a Hamiltonian to one data set also does not establish its validity outside the tested regime.

Energy and Generator: The Important Caveat

For a closed, time-independent system in a fixed laboratory frame, the Hamiltonian’s energy and generator roles align cleanly. Several common settings require more careful language.

Explicit driving

If $H(t)$ depends explicitly on time, it still generates system evolution. The system’s instantaneous energy expectation obeys

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

for Schrödinger evolution under $H(t)$, assuming the differentiability and domain conditions needed for the calculation.

The right side represents power supplied by the external control in the effective description. Energy conservation can be restored in a larger autonomous model that includes the drive, but the reduced system energy need not be constant.

Moving and rotating frames

Let

$$\lvert\psi_R(t)\rangle = W(t)\lvert\psi(t)\rangle$$

for a time-dependent unitary $W(t)$. The transformed state satisfies a Schrödinger equation with

$$\begin{aligned} H_R(t) &= W(t)H(t)W^\dagger(t) \\ &\quad+ i\hbar \dot W(t)W^\dagger(t). \end{aligned}$$

The second term is required by the time-dependent frame. The resulting $H_R$ generates motion in that representation, but its eigenvalues need not be interpreted as the laboratory energy levels. Rotating-frame detunings and Floquet quasienergies are familiar examples of generator spectra with a representation-dependent energy interpretation.

Open systems

The reduced state of an open system generally does not evolve by a commutator alone. A Markovian model may have

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

where $\mathcal D$ is a dissipative superoperator. Here $H$ generates the coherent part of the reduced dynamics, but the full generator is a superoperator. Lindblad–GKSL Equation is the canonical detailed treatment.

Freedom to Choose the Energy Zero

Let $c(t)$ be real and define

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

For the same initial ray, the state changes only by a time-dependent global phase:

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

Therefore

$$\rho'(t) = \rho(t),$$

and all measurement probabilities within the fixed model are unchanged.

For constant $c$, every energy eigenvalue shifts by $c$, while energy differences and transition frequencies remain the same. This is the ordinary freedom to choose an energy zero.

The exact unitary operator, including its phase, does change. Relative phases can become observable if what looked like one global shift is actually applied differently to coherent sectors or paths. The claim of irrelevance applies to a scalar multiple of the identity on the whole modeled Hilbert space.

Time-Independent Versus Time-Dependent Evolution

For time-independent $H$, powers of the same operator commute, and

$$U(t,t_0) = e^{-iH(t-t_0)/\hbar}.$$

For time-dependent $H(t)$, the naive expression

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

is valid without time ordering only when the relevant Hamiltonians commute:

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

for all times in the interval. Otherwise, operator order matters and the propagator is a time-ordered exponential. Time-Dependent Hamiltonians develops that issue.

Example: Particle in a Potential

For one nonrelativistic particle in one dimension,

$$H = \frac{P^2}{2m} + V(X).$$

In position representation,

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

This formula is incomplete until the configuration space, properties of $V$, and domain of $H$ are specified. On the full line, confining and scattering potentials lead to different spectral structures. On an interval, boundary conditions are part of the physical Hamiltonian.

Hamiltonians in Coordinate Space owns the construction details, singular-potential caveats, and multidimensional examples.

Example: Harmonic Oscillator

The oscillator Hamiltonian is

$$H = \frac{P^2}{2m} + \frac12m\omega^2X^2.$$

Using ladder operators,

$$H = \hbar\omega \left( a^\dagger a+\frac12 \right).$$

Its discrete energies are

$$E_n = \hbar\omega \left( n+\frac12 \right), \qquad n=0,1,2,\ldots.$$

The zero-point term is a scalar shift only within this isolated fixed-frequency oscillator model. Energy differences are

$$E_{n+1}-E_n = \hbar\omega.$$

Example: General Two-Level Hamiltonian

Every Hermitian two-level Hamiltonian can be written

$$H = \bar E\,I + \frac{\hbar\Omega}{2} \hat{\mathbf n}\cdot\boldsymbol{\sigma},$$

where $\hat{\mathbf n}$ is a unit vector and $\Omega\geq0$. The energies are

$$E_\pm = \bar E \pm \frac{\hbar\Omega}{2}.$$

For time-independent parameters,

$$\begin{aligned} U(t) &= e^{-i\bar E t/\hbar} \bigg[ \cos\frac{\Omega t}{2}\,I\\ &\qquad - i\sin\frac{\Omega t}{2} \hat{\mathbf n}\cdot\boldsymbol{\sigma} \bigg]. \end{aligned}$$

The factor $e^{-i\bar E t/\hbar}$ is a global phase. The Pauli-vector term generates observable Bloch-sphere rotation about $\hat{\mathbf n}$ at angular frequency $\Omega$.

Two-State Hamiltonians develops mixing, avoided crossings, and coherent oscillation.

A Practical Hamiltonian Audit

Before using a proposed Hamiltonian, check:

1. Hilbert space: what degrees of freedom and tensor factors are included?
2. Units: does every term have dimensions of energy?
3. Self-adjointness: is the matrix Hermitian, or is the unbounded operator self-adjoint on a declared domain?
4. Boundary conditions: are they compatible with the intended probability flow and physical geometry?
5. Time dependence: is $H$ constant, driven, or written in a time-dependent frame?
6. Energy meaning: are its eigenvalues laboratory energies, effective energies, or generator parameters such as detunings?
7. Symmetries: which commutators should vanish, and do they?
8. Stability: is a lower bound expected, and does the model have one in its domain of validity?
9. Approximation scale: which neglected terms are small, and compared with what?
10. Open-system boundary: is Hamiltonian evolution alone appropriate, or is a channel or master equation required?

Common Mistakes

• Treating the Hamiltonian as a state rather than an operator defining dynamics.
• Calling any Hermitian-looking differential expression a self-adjoint Hamiltonian without specifying its domain.
• Exponentiating matrix entries instead of the operator.
• Using $e^{-iH(t-t_0)/\hbar}$ for a noncommuting time-dependent Hamiltonian.
• Assuming a subsystem Hamiltonian is conserved merely because the total Hamiltonian is time independent.
• Interpreting every effective or rotating-frame generator eigenvalue as a laboratory energy.
• Forgetting the extra $i\hbar\dot W W^\dagger$ term in a time-dependent frame.
• Treating quantization as an unambiguous symbol replacement.
• Comparing Hamiltonian parameters with frequencies without the required factor of $\hbar$.
• Ignoring additive identity terms when comparing quoted absolute energies, or overemphasizing them when only closed-system transition probabilities matter.
• Using a Hamiltonian commutator as the full generator of dissipative open-system evolution.
• Assuming symmetry determines every coupling constant.

References

• P. A. M. Dirac, The Principles of Quantum Mechanics, 4th ed., Oxford University Press, 1958.
• M. H. Stone, “On one-parameter unitary groups in Hilbert space,” Annals of Mathematics 33, 643–648, 1932.
• M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. I: Functional Analysis, revised and enlarged ed., Academic Press, 1980.
• A. Messiah, Quantum Mechanics, Dover Publications, 1999.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.

Exercises

1. Shift the energy zero

Let $H'=H+cI$ with real constant $c$. Show how the time-evolution operator changes, and prove that all density-operator predictions are unchanged.

Solution

Because $H$ commutes with $I$,

$$\begin{aligned} U'(t) &= \exp \left[ -\frac{i}{\hbar}(H+cI)t \right]\\ &= e^{-ict/\hbar} e^{-iHt/\hbar}\\ &= e^{-ict/\hbar}U(t). \end{aligned}$$

For any initial density operator,

$$\begin{aligned} \rho'(t) &= U'(t)\rho(0)U'^\dagger(t)\\ &= e^{-ict/\hbar} U(t)\rho(0)U^\dagger(t) e^{ict/\hbar}\\ &= \rho(t). \end{aligned}$$

Every effect therefore has the same probability. Energy eigenvalues shift by $c$, but their differences do not.

2. Exponentiate a two-level Hamiltonian

For

$$H = \frac{\hbar\Omega}{2} \hat{\mathbf n}\cdot\boldsymbol{\sigma},$$

use

$$\left( \hat{\mathbf n}\cdot\boldsymbol{\sigma} \right)^2 = I$$

to derive $U(t)$.

Solution

Expand the exponential in even and odd powers:

$$\begin{aligned} U(t) &= \sum_{k=0}^{\infty} \frac{1}{k!} \left( -i\frac{\Omega t}{2} \hat{\mathbf n}\cdot\boldsymbol{\sigma} \right)^k\\ &= \cos\frac{\Omega t}{2}\,I - i\sin\frac{\Omega t}{2} \hat{\mathbf n}\cdot\boldsymbol{\sigma}. \end{aligned}$$

Even powers give $I$ and odd powers give $\hat{\mathbf n}\cdot\boldsymbol{\sigma}$. The eigenvalues of $H$ are $\pm\hbar\Omega/2$.

3. Prove energy conservation for a closed system

Let $H$ be time independent and let

$$i\hbar\dot\rho = [H,\rho].$$

Show that $\langle H\rangle=\operatorname{Tr}(\rho H)$ is constant.

Solution

Differentiate:

$$\begin{aligned} \frac{d}{dt} \operatorname{Tr}(\rho H) &= \operatorname{Tr}(\dot\rho H)\\ &= \frac{1}{i\hbar} \operatorname{Tr} \left( [H,\rho]H \right). \end{aligned}$$

Using cyclicity,

$$\operatorname{Tr}(H\rho H) = \operatorname{Tr}(\rho H^2),$$

so the trace of the commutator term vanishes. Therefore

$$\frac{d}{dt}\langle H\rangle = 0.$$

4. Find the power supplied by a drive

For a differentiable $H(t)$ and a Schrödinger-picture state evolving under that same Hamiltonian, prove

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

Solution

For a pure state,

$$\begin{aligned} \frac{d}{dt} \langle\psi\rvert H\lvert\psi\rangle &= \langle\dot\psi\rvert H\lvert\psi\rangle + \langle\psi\rvert\dot H\lvert\psi\rangle\\ &\quad+ \langle\psi\rvert H\lvert\dot\psi\rangle. \end{aligned}$$

Using

$$\lvert\dot\psi\rangle = -\frac{i}{\hbar}H\lvert\psi\rangle$$

and its adjoint, the first and third terms cancel. Hence

$$\frac{d}{dt}\langle H\rangle = \langle\dot H\rangle.$$

The density-operator proof gives the same result because the commutator contribution has zero trace.

5. Derive a rotating-frame Hamiltonian

Let $\lvert\psi_R\rangle=W(t)\lvert\psi\rangle$ with unitary $W(t)$. Starting from

$$i\hbar \lvert\dot\psi\rangle = H\lvert\psi\rangle,$$

derive the Hamiltonian governing $\lvert\psi_R\rangle$.

Solution

Differentiate the transformed state:

$$\lvert\dot\psi_R\rangle = \dot W\lvert\psi\rangle + W\lvert\dot\psi\rangle.$$

Multiplying by $i\hbar$ and using $\lvert\psi\rangle=W^\dagger\lvert\psi_R\rangle$ gives

$$\begin{aligned} i\hbar\lvert\dot\psi_R\rangle &= \left( i\hbar\dot W W^\dagger + WHW^\dagger \right) \lvert\psi_R\rangle. \end{aligned}$$

Therefore

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

The second term vanishes only for a time-independent frame.

6. Track subsystem energy exchange

For

$$H_{AB} = H_A\otimes I_B + I_A\otimes H_B + V_{AB},$$

assume no explicit time dependence. Find the condition under which the expectation of $H_A\otimes I_B$ is conserved.

Solution

For

$$A = H_A\otimes I_B,$$

the expectation-value equation gives

$$\frac{d}{dt}\langle A\rangle = \frac{i}{\hbar} \langle[H_{AB},A]\rangle.$$

The free local terms commute with $A$, so

$$[H_{AB},A] = [V_{AB},H_A\otimes I_B].$$

Thus the local energy is conserved for every state if

$$[V_{AB},H_A\otimes I_B] = 0.$$

If this commutator is nonzero, the interaction can exchange energy with subsystem $A$ even though the total time-independent Hamiltonian is conserved.

7. Compare boundary conditions on an interval

For a free particle on $0\leq x\leq L$, compare the energy spectra of

$$H = -\frac{\hbar^2}{2m} \frac{d^2}{dx^2}$$

with Dirichlet conditions $\psi(0)=\psi(L)=0$ and periodic conditions $\psi(0)=\psi(L)$, $\psi'(0)=\psi'(L)$.

Solution

For Dirichlet conditions, normalized eigenfunctions are proportional to

$$\sin\frac{n\pi x}{L}, \qquad n=1,2,\ldots,$$

with energies

$$E_n^{\mathrm D} = \frac{\hbar^2\pi^2n^2}{2mL^2}.$$

For periodic conditions, eigenfunctions are proportional to

$$e^{i2\pi nx/L}, \qquad n\in\mathbb Z,$$

with energies

$$E_n^{\mathrm P} = \frac{\hbar^2}{2m} \left( \frac{2\pi n}{L} \right)^2.$$

The differential expression is the same, but the self-adjoint domains and spectra differ. Boundary conditions are part of the Hamiltonian.

8. Decompose a Hermitian two-level matrix

Let

$$H = \begin{pmatrix} \varepsilon_0 & g e^{-i\phi}\\ g e^{i\phi} & \varepsilon_1 \end{pmatrix}, \qquad g\geq0.$$

Write $H=\bar E I+\mathbf h\cdot\boldsymbol{\sigma}$ and find its energy splitting.

Solution

Define

$$\bar E = \frac{\varepsilon_0+\varepsilon_1}{2}, \qquad \delta = \frac{\varepsilon_0-\varepsilon_1}{2}.$$

Using the Pauli matrices,

$$\mathbf h = \left( g\cos\phi,\, g\sin\phi,\, \delta \right).$$

Therefore

$$H = \bar E I + \mathbf h\cdot\boldsymbol{\sigma}.$$

The eigenvalues are

$$E_\pm = \bar E \pm \sqrt{\delta^2+g^2},$$

so the splitting is

$$E_+-E_- = 2\sqrt{\delta^2+g^2}.$$

The scalar part $\bar E I$ changes only a global phase in closed-system evolution; $\mathbf h$ sets the observable level splitting and rotation axis.

Summary

A closed-system Hamiltonian is a self-adjoint operator that generates time evolution. In the standard laboratory frame, its spectral measure also defines energy measurements. The equality of these roles is central but representation-dependent caveats matter for driven, effective, rotating-frame, and open-system descriptions.

The operator’s domain and boundary conditions are part of its definition. Building $H$ is a modeling task constrained by degrees of freedom, symmetries, interactions, calibration, and approximation scales. Once the Hamiltonian is specified, the Schrödinger equation and propagator determine the dynamics.