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The Hamiltonian, denoted H, has two distinct but closely related meanings. In classical mechanics, it is a function that 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 is 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.

1 The quantum Hamiltonian

As explained in the article mathematical formulation of quantum mechanics, the physical state of a system may be characterized as a ray in an abstract Hilbert space (in the case of pure quantum states),or a countable (at most) sequence of vectors and real positive numbers ,the latter satisfying the condition (the case of mixed quantum states).Physically observable quantities are described by means of self-adjoint operators acting on these vectors.

The quantum Hamiltonian H is the observable corresponding to the total energy of the system.Mathematically speaking,it is a densely defined self-adjoint operator.

The eigenkets ( eigenvectors) of H, denoted

, 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}, solving the equation:

Since H is a Hermitian operator, the energy is always a real numberIn mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. The term "real number" is a retronym coined in response to " imaginary number". Real numbers may.

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 timeFor alternate uses of "time", see Time (disambiguation). Time quantifies or measures the interval between events, or the duration of events. Time has long been perceived as a dimension in which each event has a definite (but not necessarily unique) positi evolution of quantum states. If is the state of the system at time t, then

where is h-barPlanck's constant denoted h is a physical constant that is used to describe the sizes of quanta. It plays a central role in the theory of quantum mechanics, and is named after Max Planck, one of the founders of quantum theory. It has a value of approximat. This equation is known as the Schrödinger equationIn physics, the Schrodinger equation proposed by the Austrian physicist Erwin Schrodinger in 1925, describes the time-dependence of quantum mechanical systems. It is of central importance to the theory of quantum mechanics, playing a role analogous to New. (It takes the same form as the Hamilton-Jacobi equation, which is one of the reasons H is also called the Hamiltonian.) Given the state at some initial time (t = 0), we can integrate it to obtain the state at any subsequent time. In particular, if H is independent of time, then

where the exponential operator on the right hand side is defined by the usual seriesThe exponential function is one of the most important functions in mathematics. It is written as exp x or e x where e is the base of the natural logarithm. As a function of the real variable x the graph of e x is always positive (above the x axis) and inc. This can be shown to be a unitary operatorIn functional analysis, a unitary operator is a bounded linear operator U on a Hilbert space satisfying U U UU I where I is the identity operator. This property is equivalent to any of the following: U is a surjective isometry U preserves the inner produc, and is a common form of the time evolution operator (also called the propagator).

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