Hamiltonian (quantum mechanics)

Hamiltonian (quantum mechanics)
Name	Hamiltonian
Unit	Joules (J)
Symbols	$\backslash hat\{H\}$, $H$
Dimension	M L<sup>2</sup> T<sup>−2</sup>

Hamiltonian (quantum mechanics). In quantum mechanics, the Hamiltonian is the operator corresponding to the total energy of a physical system, including both kinetic and potential contributions. Its eigenvalues represent the possible, quantized energy levels the system can occupy, while its eigenstates describe the corresponding stationary states. The operator is central to the mathematical formulation of quantum theory, most famously through the Schrödinger equation, which governs the time evolution of a quantum state.

Definition and mathematical formulation

The Hamiltonian operator, typically denoted $\backslash hat\{H\}$, is constructed by applying the canonical quantization procedure to the classical Hamiltonian function. For a single non-relativistic particle, the classical kinetic energy $T\; =\; p^2\; /\; 2m$ and potential energy $V(\backslash mathbf\{r\},\; t)$ are promoted to operators, yielding $\backslash hat\{H\}\; =\; \backslash hat\{T\}\; +\; \backslash hat\{V\}\; =\; \backslash frac\{\backslash hat\{\backslash mathbf\{p\}\}^2\}\{2m\}\; +\; V(\backslash hat\{\backslash mathbf\{r\}\},\; t)$. Here, the momentum operator is $\backslash hat\{\backslash mathbf\{p\}\}\; =\; -i\backslash hbar\; \backslash nabla$ in the position representation, a direct consequence of the canonical commutation relation with the position operator. In more complex systems, such as those involving multiple particles in an electromagnetic field or incorporating special relativity, the formulation becomes more intricate, often requiring terms from quantum electrodynamics or the Dirac equation.

Time-independent and time-dependent Hamiltonians

A crucial distinction is made between time-independent and time-dependent Hamiltonians. A time-independent Hamiltonian does not explicitly contain the time variable, implying the system's potential energy, and thus its total energy, is conserved. This is typical for isolated systems like the electron in a stationary hydrogen atom or a particle in an infinite potential well. In contrast, a time-dependent Hamiltonian explicitly includes time, often modeling systems interacting with an external, time-varying field, such as an atom subjected to a laser pulse or a spin in an oscillating magnetic field in nuclear magnetic resonance. The mathematical treatment and physical implications differ significantly between the two cases.

Eigenvalues and eigenstates

Solving the time-independent eigenvalue equation $\backslash hat\{H\}\; |\backslash psi\_n\backslash rangle\; =\; E\_n\; |\backslash psi\_n\backslash rangle$ is a fundamental task in quantum mechanics. The eigenvalues $E\_n$ are the discrete or continuous allowed energy levels of the system, as first demonstrated in Bohr's model of the atom and later solidified by Erwin Schrödinger. The corresponding eigenstates $|\backslash psi\_n\backslash rangle$ are the stationary states; if a system is prepared in such a state, its probability density remains constant in time. The set of all eigenstates typically forms a complete basis for the system's Hilbert space, a concept formalized in the work of John von Neumann, allowing any general quantum state to be expressed as a superposition of these energy eigenstates.

Relation to classical mechanics

The quantum Hamiltonian is intimately connected to Hamiltonian mechanics in classical physics, a formalism developed by William Rowan Hamilton. The classical Hamiltonian function $H(\backslash mathbf\{p\},\; \backslash mathbf\{q\})$ governs dynamics via Hamilton's equations. The transition to quantum mechanics replaces the classical Poisson bracket with the commutator of operators divided by $i\backslash hbar$, as postulated by Paul Dirac. This correspondence principle ensures that the expectation values of quantum observables obey equations analogous to Newton's laws of motion in the classical limit, as articulated in the Ehrenfest theorem.

Examples of Hamiltonians

Specific forms of the Hamiltonian operator describe foundational systems in physics. The Hamiltonian for a single particle in one dimension is $\backslash hat\{H\}\; =\; -\backslash frac\{\backslash hbar^2\}\{2m\}\; \backslash frac\{d^2\}\{dx^2\}\; +\; V(x)$. For the hydrogen atom, it includes the Coulomb potential: $\backslash hat\{H\}\; =\; -\backslash frac\{\backslash hbar^2\}\{2\backslash mu\}\; \backslash nabla^2\; -\; \backslash frac\{e^2\}\{4\backslash pi\backslash epsilon\_0\; r\}$. The quantum harmonic oscillator uses $\backslash hat\{H\}\; =\; \backslash frac\{\backslash hat\{p\}^2\}\{2m\}\; +\; \backslash frac\{1\}\{2\}\; m\backslash omega^2\; \backslash hat\{x\}^2$, leading to equally spaced energy levels. In condensed matter physics, model Hamiltonians like the Ising model or the Hubbard model describe magnetism and electron behavior in solids. The Standard Model of particle physics is built upon a highly complex Hamiltonian incorporating the Higgs mechanism.

Role in the Schrödinger equation

The Hamiltonian is the generator of time evolution in quantum mechanics. This role is enshrined in the Schrödinger equation: $i\backslash hbar\; \backslash frac\{\backslash partial\}\{\backslash partial\; t\}\; |\backslash Psi(t)\backslash rangle\; =\; \backslash hat\{H\}\; |\backslash Psi(t)\backslash rangle$. For a time-independent Hamiltonian, the formal solution is $|\backslash Psi(t)\backslash rangle\; =\; e^\{-i\backslash hat\{H\}t/\backslash hbar\}\; |\backslash Psi(0)\backslash rangle$, where the exponential defines the time-evolution operator. In the path integral formulation introduced by Richard Feynman, the Hamiltonian appears in the action integral. The equation also underpins the Heisenberg picture, where operators evolve according to $d\backslash hat\{A\}/dt\; =\; (i\backslash hbar)^\{-1\}\; [\backslash hat\{A\},\; \backslash hat\{H\}]$, directly linking the Hamiltonian to the dynamics of observables like those in the Stern–Gerlach experiment.

Category:Quantum mechanics Category:Hamiltonian mechanics Category:Concepts in physics