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Hamiltonian: meaning, definitions and examples

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Hamiltonian

[ ˌhæmɪlˈtoʊniən ]

Noun

^{Context #1 | Noun}

physics

A Hamiltonian is a mathematical operator corresponding to the total energy of a system in quantum mechanics. It is used to describe the dynamics of the system and is an essential concept in theoretical physics.

Synonyms

energy operator, total energy operator.

Which Synonym Should You Choose?

Word	Description / Examples
Hamiltonian	This term is commonly used in physics, particularly in quantum mechanics, to describe the total energy operator of a system. It encompasses both the kinetic and potential energy of a system and is essential for formulating the Schrödinger equation. The Hamiltonian for a free particle is given by the kinetic energy term only. In quantum mechanics, the Hamiltonian operator plays a crucial role in determining the state evolution of a system.
energy operator	Used in the context of quantum mechanics and field theory, this general term refers to any operator that is associated with the energy of a system. Unlike 'Hamiltonian,' it doesn't necessarily imply the inclusion of both kinetic and potential energy. The energy operator for a harmonic oscillator includes both kinetic and potential energy terms. In the Heisenberg picture, the energy operator evolves with time.
total energy operator	This term specifically refers to an operator that accounts for all forms of energy within a system and is synonymous with 'Hamiltonian' in many contexts. It is less frequently used but emphasizes the inclusion of all energy components. By calculating the total energy operator, one can determine the complete energy profile of the quantum system. Researchers often compute the total energy operator when analyzing the dynamics of complex molecular systems.

Word

Description / Examples

Hamiltonian

This term is commonly used in physics, particularly in quantum mechanics, to describe the total energy operator of a system. It encompasses both the kinetic and potential energy of a system and is essential for formulating the Schrödinger equation.

The Hamiltonian for a free particle is given by the kinetic energy term only.
In quantum mechanics, the Hamiltonian operator plays a crucial role in determining the state evolution of a system.

energy operator

Used in the context of quantum mechanics and field theory, this general term refers to any operator that is associated with the energy of a system. Unlike 'Hamiltonian,' it doesn't necessarily imply the inclusion of both kinetic and potential energy.

The energy operator for a harmonic oscillator includes both kinetic and potential energy terms.
In the Heisenberg picture, the energy operator evolves with time.

total energy operator

This term specifically refers to an operator that accounts for all forms of energy within a system and is synonymous with 'Hamiltonian' in many contexts. It is less frequently used but emphasizes the inclusion of all energy components.

By calculating the total energy operator, one can determine the complete energy profile of the quantum system.
Researchers often compute the total energy operator when analyzing the dynamics of complex molecular systems.

Examples of usage

The Hamiltonian operator plays a crucial role in determining the behavior of quantum systems.
In quantum mechanics, the time evolution of a system is governed by the Hamiltonian.

^{Context #2 | Noun}

mathematics

In mathematics, a Hamiltonian is a function used to describe the state of a dynamic system in terms of position and momentum variables. It is often used in classical mechanics and differential equations.

Synonyms

Hamilton function, Hamiltonian function.

Which Synonym Should You Choose?

Word	Description / Examples
Hamiltonian	Used in physics and mathematics, particularly in mechanics and quantum mechanics. It represents the total energy of a system. The Hamiltonian for this system is given by the equation H = T + V
Hamilton function	A less common term, but it also refers to the Hamiltonian in the context of physics and mathematics. Often used interchangeably with 'Hamiltonian'. The Hamilton function represents the energy states of the system

Word

Description / Examples

Hamiltonian

Used in physics and mathematics, particularly in mechanics and quantum mechanics. It represents the total energy of a system.

The Hamiltonian for this system is given by the equation H = T + V

Hamilton function

A less common term, but it also refers to the Hamiltonian in the context of physics and mathematics. Often used interchangeably with 'Hamiltonian'.

The Hamilton function represents the energy states of the system

Examples of usage

The Hamiltonian function can be used to derive the equations of motion for a physical system.
Hamiltonian mechanics provides a powerful framework for understanding the behavior of dynamical systems.

Translations

Translations of the word "Hamiltonian" in other languages:

🇵🇹 Hamiltoniano

🇮🇳 हैमिल्टोनियन

🇩🇪 Hamiltonian

🇮🇩 Hamiltonian

🇺🇦 Гамільтоніан

🇵🇱 Hamiltonian

🇯🇵 ハミルトニアン

🇫🇷 Hamiltonien

🇪🇸 Hamiltoniano

🇹🇷 Hamiltonyen

🇰🇷 해밀토니안

🇸🇦 هاميلتوني

🇨🇿 Hamiltonián

🇸🇰 Hamiltonián

🇨🇳 哈密顿量

🇸🇮 Hamiltonian

🇮🇸 Hamiltonian

🇰🇿 Гамильтониан

🇬🇪 ჰამილტონიანი

🇦🇿 Hamiltonian

🇲🇽 Hamiltoniano

Etymology

The term 'Hamiltonian' is named after Sir William Rowan Hamilton, an Irish mathematician, and physicist who introduced the concept in the 19th century. Hamilton's work on mechanics and optics laid the foundation for the development of the Hamiltonian formalism in physics and mathematics.