Hamiltonian (quantum mechanics)

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In quantum mechanics, the Hamiltonian H ,also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system. Because of its close relation to the time-evolution of a system, it is of fundamental importance in most formulations of quantum theory (see below).

Contents

Basic introduction

By analogy with classical mechanics, the Hamiltonian is commonly expressed as the sum of operators corresponding to the kinetic and potential energies of a system, in the form

(although this is not the technical definition of the Hamiltonian in classical mechanics, it is the form it most commonly takes). The potential operator V typically takes the form of a function V(r,t) of position and time, which simply acts on states as a multiplicative factor. The operator T corresponding to kinetic energy is constructed by analogy with the classical formula

with p\; the momentum

Schrödinger constructed his momentum operator using the substitution

where \nabla is the gradient operator, i is the unit imaginary number, and \hbar is the reduced Planck constant . Combining this with the potential term yields

which allows one to apply the Hamiltonian to systems described by a wave function \Psi(\mathbf{r},t). This is the approach commonly taken in introductory treatments of quantum mechanics, using the formalism of Schrödinger's wave mechanics. However, in the more general formalism of Dirac, the Hamiltonian is typically implemented as an operator on a Hilbert space in the following way:

(From a mathematically rigorous point of view, care must be taken with the above assumptions. Operators on infinite-dimensional Hilbert spaces need not have eigenvalues (the set of eigenvalues does not necessarily coincide with the spectrum of an operator). However, all routine quantum mechanical calculations can be done using the physical formulation.)

Schrödinger equation

The Hamiltonian generates the time evolution of quantum states. If  \left| \psi (t) \right\rangle is the state of the system at time t, then

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