Hamiltonian Operator · aiwiki.page

The Hamiltonian operator, usually written $\hat H$ , is the operator representing the total energy of a system in quantum mechanics. It also determines how quantum states evolve in time. Its spectrum describes the possible results of energy measurements, while its action in the Schrödinger equation governs dynamics. Unlike a numerical energy value, the Hamiltonian is a mathematical operation acting on states in a Hilbert space. Its form encodes the system’s motion, interactions, and external influences. (damtp.cam.ac.uk)

Relationship to classical mechanics

The quantum Hamiltonian develops from Hamiltonian mechanics, a formulation of classical mechanics in which a function $H(q,p,t)$ generates motion through Hamilton’s equations. For a nonrelativistic particle in Cartesian coordinates, with no velocity-dependent interactions, the classical expression is

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

where the first term is kinetic energy and $V$ is potential energy. In quantum theory, position and momentum become operators rather than ordinary variables. This replacement works directly for simple Hamiltonians, but more complicated expressions require care because operator products need not commute. (damtp.cam.ac.uk)

In the position representation, the momentum operator is $\hat{\mathbf p}=-i\hbar\nabla$ , giving

\hat H=-\frac{\hbar^2}{2m}\nabla^2+V(\mathbf r,t).

Here $m$ denotes the particle’s mass, and $\hbar$ is the reduced Planck constant. The Laplacian differentiates the wave function, whereas a local potential acts by multiplication. This familiar differential expression is a particular representation, not the general definition of a Hamiltonian. (ocw.mit.edu)

Mathematical properties and domains

In standard quantum mechanics, the Hamiltonian of a closed system is a self-adjoint operator. Self-adjointness ensures a real energy spectrum and supplies the mathematical basis for probability-preserving time evolution. In finite dimensions, it is equivalent to the condition $H^\dagger=H$ for a Hermitian matrix, where the dagger denotes conjugate transpose. (arxiv.org)

Infinite-dimensional Hamiltonians require additional precision. A differential expression alone does not completely specify an operator: its domain—the set of states on which it acts—must also be defined. Boundary conditions can determine different self-adjoint realizations of the same expression and therefore different spectra. An operator may be symmetric under the inner product on its domain without being self-adjoint; the latter requires equality with its adjoint, including their domains. These distinctions matter for particles on intervals and for singular potentials. (arxiv.org)

Energy spectra and measurement

The time-independent energy equation is

\hat H|n\rangle=E_n|n\rangle.

The quantities $E_n$ and $|n\rangle$ are eigenvalues and eigenvectors of the Hamiltonian. Different independent states can share one eigenvalue, a property called degeneracy. The spectral theorem connects a self-adjoint Hamiltonian with energy measurement through its spectral projections. (damtp.cam.ac.uk)

A spectrum need not consist solely of discrete levels. Bound-state energies are often discrete, whereas unbound motion can produce a continuous spectrum; one Hamiltonian can have both. Thus energy quantization does not mean that every quantum system has exclusively discrete energies. For a normalized state expanded in a discrete orthonormal energy basis, $|\psi\rangle=\sum_n c_n|n\rangle$ , the Born rule assigns probability $|c_n|^2$ to the corresponding nondegenerate energy outcome. The energy’s expectation value is $\langle\psi|\hat H|\psi\rangle$ , provided it exists. (damtp.cam.ac.uk)

Time evolution and conservation

The time-dependent Schrödinger equation takes the abstract form

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

For a time-independent self-adjoint Hamiltonian,

|\psi(t)\rangle =e^{-i\hat H(t-t_0)/\hbar}|\psi(t_0)\rangle.

The exponential is a unitary evolution operator, preserving normalization and total probability. An energy eigenstate acquires only a phase, so its position probability density remains stationary. A superposition of different energies generally develops changing relative phases. (ocw.mit.edu)

When Hamiltonians at different times do not commute, evolution generally requires a time-ordered exponential rather than an ordinary exponential of their integral. Energy conservation is also conditional: under suitable regularity assumptions,

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

Consequently, a Hamiltonian without explicit time dependence conserves mean energy, whereas external driving can change it. More generally, a time-independent observable commuting with the Hamiltonian has a conserved expectation value. (ocw.mit.edu)

Representative systems

For the one-dimensional quantum harmonic oscillator,

\hat H=\frac{\hat p^2}{2m} +\frac12m\omega^2\hat x^2, \qquad E_n=\hbar\omega\left(n+\frac12\right).

Its equally spaced levels and nonzero zero-point energy make it a central model for quantum vibrations. For a particle in a box with infinite walls at $0$ and $L$ , vanishing endpoint wave functions produce $E_n=\pi^2\hbar^2n^2/(2mL^2)$ , with $n=1,2,\ldots$ . These models illustrate how both potential energy and boundary conditions determine a spectrum. (damtp.cam.ac.uk)

The hydrogen Hamiltonian combines kinetic energy with Coulomb attraction between an electron and a proton. The nonrelativistic relative-coordinate treatment uses their reduced mass. Additional interactions produce corrections to the simplest energy spectrum, including fine structure and shifts caused by external magnetic fields. (damtp.cam.ac.uk)

Approximation methods

Most realistic Hamiltonians cannot be solved exactly. Perturbation theory writes $\hat H=\hat H_0+\lambda\hat V$ , starting from a solvable reference system. For an isolated, nondegenerate reference level, the first-order correction is

\Delta E_n^{(1)} =\lambda\langle n^{(0)}|\hat V|n^{(0)}\rangle.

Degenerate levels require a separate treatment, typically diagonalizing the perturbation within the degenerate subspace. The WKB approximation provides a different approach for appropriate slowly varying spatial potentials, including estimates of bound-state energies and tunneling amplitudes. The applicable method depends on the Hamiltonian’s structure and on the size and character of the neglected effects. (ocw.mit.edu)