The **Hamiltonian**, denoted *H*, has two distinct but closely related meanings. In classical mechanics, it is a function which describes the state of a mechanical system in terms of position and momentum variables (i.e. symplectic variables), which is the basis for a re-formulation of classical mechanics known as Hamiltonian mechanics. In quantum mechanics, the Hamiltonian refers to the observable corresponding to the total energy of a system. The classical Hamiltonian is described in the article on Hamiltonian mechanics. This article discusses the Hamiltonian operator in quantum mechanics.

Table of contents |

2 Energy Eigenket Degeneracy, Symmetry, and Conservation Laws 3 Hamilton's equations |

As explained in the article mathematical formulation of quantum mechanics, the physical state of a system may be characterized as a vector in an abstract Hilbert space, and physically observable quantities as Hermitian operators acting on these vectors.

The quantum Hamiltonian *H* is the observable corresponding to the total energy of the system. The eigenkets (eigenvectors) of *H*, denoted {|a⟩}, provide an orthonormal basis for the Hilbert space. The spectrum of allowed energy levels of the system is given by the set of eigenvalues, denoted {*E*_{a}}:

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Depending on the Hilbert space of the system, the energy spectrum may be either discrete or continuous. In fact, certain systems have a continuous energy spectrum in one range of energies and a discrete spectrum in another range. An example of such a system is the finite potential well, which admits bound states with discrete negative energies and free states with continuous positive energies.

The Hamiltonian generates the time evolution of quantum states. If |ψ(*t*)⟩ is the state of the system at time *t*, then

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In many systems, two or more energy eigenstates have the same energy. A simple example of this is a free particle, whose energy eigenstates have wavefunctions that are propagating plane waves. The energy of each of these plane waves is inversely proportional to the square of its wavelength. A wave propagating in the *x* direction is a different state from one propagating in the *y* direction, but if they have the same wavelength, then their energies will be the same. When this happens, the states are said to be *degenerate*.

It turns out that degeneracy occurs whenever a nontrivial unitary operator *U* commutes with the Hamiltonian. To see this, suppose that |a⟩ is an energy eigenket. Then *U*|a⟩ is an energy eigenket with the same eigenvalue, since

The existence of a symmetry operator implies the existence of a conserved observable. Let *G* be the Hermitian generator of *U*:

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Hamilton's equations in classical Hamiltonian mechanics have a direct analogy in quantum mechanics. Suppose we have a set of basis states { |*n*⟩ }, which need not necessarily be eigenstates of the energy. For simplicity, we assume that they are discrete, and that they are orthonormal, i.e.,

The instantaneous state of the system at time *t*, |*ψ(t)*⟩, can be expanded in terms of these basis states:

The expectation value of the Hamiltonian of this state, which is also the mean energy, is

Each of the *a _{n}(t)*'s actually corresponds to