Hamiltonian: meaning, definitions and examples

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Hamiltonian

[ˌhæmɪlˈtoʊniən ]

Definitions

^{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.

Interesting Facts

Etymology

The term comes from the name William Rowan Hamilton, an Irish mathematician who contributed significantly to the field of physics in the 19th century.
Hamilton created a mathematical framework called Hamiltonian mechanics, which reformulates Newton's laws of motion.
The word also relates to Hamilton's principle of least action, which simplifies how we describe motion.

Physics

In physics, a Hamiltonian represents the total energy of a system, including both kinetic (movement) and potential (stored) energy.
Hamiltonian mechanics is particularly useful in quantum mechanics, providing insights into the behavior of particles at tiny scales.
It plays a crucial role in chaos theory, helping scientists understand systems that are highly sensitive to initial conditions.

Mathematics

The Hamiltonian is typically represented as an operator in quantum mechanics, translating physical concepts into mathematical equations.
It often involves complex numbers and differential equations, showcasing the deep connection between mathematics and physical laws.
This mathematical construct allows for the exploration of phase space, where all possible states of a system are represented.

Pop Culture

The term 'Hamilton' gained fame from the Broadway musical, reminding us of the innovative nature of Hamiltonian ideas in transforming traditional narratives.
Discussions around 'Hamiltonian' also pop up in modern literature and media when addressing themes of power dynamics and change.
Movies and documentaries about mathematics and physics frequently mention Hamiltonians, bridging science and entertainment.

History of Science

Hamilton's formulation of dynamics laid the groundwork for modern physics, influencing Einstein and the development of relativity.
In the late 1800s, Hamiltonian mechanics helped shape the way physicists understood classical systems before the advent of quantum theory.
His work initiated a shift from 'particle-based' thinking to a more unified view of energy and motion in various fields of science.

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