In quantum mechanics, the Hamiltonian of a system is an operator corresponding to the total energy of that system, including both kinetic energy and potential energy. Its spectrum, the system's energy spectrum or its set of energy eigenvalues, is the set of possible outcomes obtainable from a measurement of the system's total energy. Due to its close relation to the energy spectrum and time-evolution of a system, it is of fundamental importance in most formulations of quantum theory.

The Hamiltonian is named after William Rowan Hamilton, who developed a revolutionary reformulation of Newtonian mechanics, known as Hamiltonian mechanics, which was historically important to the development of quantum physics. Similar to vector notation, it is typically denoted by ${\displaystyle {\hat {H}}}$, where the hat indicates that it is an operator. It can also be written as ${\displaystyle H}$ or ${\displaystyle {\check {H}}}$.

Introduction

The Hamiltonian of a system represents the total energy of the system; that is, the sum of the kinetic and potential energies of all particles associated with the system. The Hamiltonian takes different forms and can be simplified in some cases by taking into account the concrete characteristics of the system under analysis, such as single or several particles in the system, interaction between particles, kind of potential energy, time varying potential or time independent one.

Schrödinger Hamiltonian

One particle

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

where

is the potential energy operator and is the kinetic energy operator in which ${\displaystyle m}$ is the mass of the particle, the dot denotes the dot product of vectors, and is the momentum operator where a ${\displaystyle \nabla }$ is the del operator. The dot product of ${\displaystyle \nabla }$ with itself is the Laplacian ${\displaystyle \nabla ^{2}}$. In three dimensions using Cartesian coordinates the Laplace operator is

Although this is not the technical definition of the Hamiltonian in classical mechanics, it is the form it most commonly takes. Combining these yields the form used in the Schrödinger equation:

which allows one to apply the Hamiltonian to systems described by a wave function ${\displaystyle \Psi (\mathbf {r} ,t)}$. This is the approach commonly taken in introductory treatments of quantum mechanics, using the formalism of Schrödinger's wave mechanics.

One can also make substitutions to certain variables to fit specific cases, such as some involving electromagnetic fields.

Expectation value

It can be shown that the expectation value of the Hamiltonian which gives the energy expectation value will always be greater than or equal to the minimum potential of the system.

Consider computing the expectation value of kinetic energy:

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