# Hamiltonian (quantum mechanics)

In quantum mechanics, the Hamiltonian is the physical operator which corresponds to the total energy (i.e. both the kinetic energy and the potential energy) of the physical system. In 1833 Sir William Rowan Hamilton introduced the Hamiltonian in classical mechanics as a reformulation of the Lagrangian in classical mechanics.

## Quotes

* Gauge symmetry: Whenever the Hamiltonian is such as to conserve the total number of particles of a particular sort—or, more generally, where there is a conserved "charge"-like quantity, such as lepton or baryon number, or electric charge itself—we shall find that the Hamiltonian will exhibit a gauge invariance property.
** Philip W. Anderson, "Chapter 2. Basic Principles I: Broken Symmetry". Basic Notions of Condensed Matter Physics. CRC Press. 9 March 2018. ISBN 9780429973741. 2nd part of quote (1st edition, 1984, Westview Press)
* In their theory of superconductivity, Bardeen, Cooper, and Schrieffer made use of a reduced Hamiltonian which included only scattering of pairs of particles of opposite momentum and spin. It is shown that the solution they obtained by a variational method is correct to O(${\tfrac {1}{n}}$) for a large system. The single particle Green's function is derived and used to calculate the interaction energy.
** John Bardeen and Gerald Rickayzen, (1960). "Ground-state energy and Green's function for reduced Hamiltonian for superconductivity". Physical Review 118 (4): 936–937. DOI:10.1103/PhysRev.118.936.
* This work is a part of an effort to analyze the physical limitations of computers due to the laws of physics. For example, Bennett ... has made a careful study of the free energy dissipation that must accompany computation. He found it to be virtually zero. He suggested to me the question of the limitations due to quantum mechanics and the uncertainty principle. I have found that, aside from the obvious limitation to size if the working parts are to be made of atoms, there is no fundamental limit from these sources either. We are here considering ideal machines; the effects of small imperfections will be considered later. This study is one of principle; our aim is to exhibit some Hamiltonian for a system which could serve as a computer.
** Richard P. Feynman, (1986). "Quantum mechanical computers". Foundations of Physics 16 (6): 507-531.
* Some time in 1960 or early 1961, I learned of an idea which had originated earlier in solid-state physics and had been brought into particle physics by Heisenberg, Nambu, and Goldstone, who had worked in both areas. It was the idea of "broken symmetry," that the Hamiltonian and commutation relations of a quantum theory could possess an exact symmetry, and the physical states might nevertheless not provide neat representations of the symmetry. In particular, a symmetry of the Hamiltonian might turn out to be not a symmetry of the vacuum.
** Steven Weinberg, (12 December 1980)"Conceptual foundations of the unified theory of weak and electromagnetic interactions". Science 210 (4475): 1212–1218.

## External links

* Encyclopedic article on Hamiltonian (quantum mechanics) on Wikipedia
* The dictionary definition of Hamiltonian on Wiktionary