Hamiltonian

The Hamiltonian is a fundamental concept in quantum mechanics and plays a vital role in quantum computing. It is an operator that represents the total energy of a quantum system, including both kinetic and potential energy. The Hamiltonian provides a mathematical framework for describing the dynamics and evolution of quantum states.

Mathematically, the Hamiltonian hat{H} is a Hermitian operator that acts on the state space of a quantum system. The eigenvalues of the Hamiltonian correspond to the possible energy levels of the system, and its eigenvectors represent the corresponding stationary states. The Schrödinger equation, which governs the time evolution of quantum states, is formulated in terms of the Hamiltonian:

The right side of a quantum system’s Schrödinger equation is Hψ, where Hψ = Eψ. H is the Hamiltonian operator, ψ is the wave function, and E is the total energy of the system. Solving the Schrödinger equation calculates the allowed energy levels (eigenvalues) and wave functions (eigenvectors) of a system. This information builds an understanding of atomic and molecular behaviors.

In quantum computing, the Hamiltonian is used to describe the behavior of qubits and quantum gates. Quantum algorithms often involve finding the ground state (lowest energy state) of a given Hamiltonian, which corresponds to solving optimization problems or simulating physical systems. Techniques like adiabatic quantum computing and the Variational Quantum Eigensolver (VQE) are based on manipulating and analyzing Hamiltonians.

Working with Hamiltonians in quantum computing presents challenges, including the accurate representation of complex interactions and the efficient implementation of Hamiltonian-based algorithms. Approximations and simplifications are often necessary, and understanding the Hamiltonian's structure and properties is key to effective algorithm design and error mitigation.

## What is Hamiltonian

Though closely related, a distinction can be made between the terms "Hamiltonian" and "Hamiltonian operator." The Hamiltonian describes the total energy, consisting of the kinetic energy and the potential energy, of both classical and quantum systems. The term “Hamiltonian,” however, is often applied to the Hamiltonian operator or to the broader mathematical expression that describes the energy of quantum systems.

The Hamiltonian operator, a central concept in quantum mechanics, is actually a mathematical operator. Represented by the symbol “H,” it consists of a kinetic energy operator “T” and a potential energy operator “V,” such that H = T + V, and acts on quantum systems’ wave functions. It is used in quantum mechanics to describe the total energy of a quantum system as a key component of the Schrödinger equation, which describes quantum systems’ behaviors and properties.

## Hamiltonian in Quantum Mechanics

The Hamiltonian is a central concept that connects quantum computing with the broader framework of quantum physics. It provides a bridge between the abstract mathematical description of quantum states and the physical reality of energy, motion, and interaction. Understanding and utilizing Hamiltonians is essential for quantum algorithm development, quantum simulation, and the exploration of quantum phenomena.

Quantum simulation of physical systems requires constructing Hamiltonians that accurately represent the interactions and dynamics of the system being studied. This enables the study of complex phenomena such as magnetism, superconductivity, and quantum phase transitions. Quantum simulators that can model these Hamiltonians provide valuable insights into fundamental physics and materials science.

In quantum mechanics, the Hamiltonian is a fundamental operator, denoted as “H,” that is defined as the sum of two components, the kinetic energy and the potential energy of a quantum system. The kinetic energy operator (T) is represented as T = -ħ²/2m * ∇², where ħ (h-bar) is the reduced Planck's constant, m is the particle’s mass, and ∇² is the Laplacian operator. The potential energy operator (V) is not represented by any single equation, rather V(x) depends upon the specific physical system under consideration. The Hamiltonian operator is then the sum of the kinetic energy operator and the potential energy operator, such that H = T + V. Solving the Schrödinger equation, which describes the evolution of a physical system’s quantum state over time, with the Hamiltonian operator H then yields the allowed energy levels (eigenvalues) and wave functions (eigenvectors) for the quantum system.

## Hamiltonian Applications in Various Fields

Although quantum mechanics is the fundamental application of the Hamiltonian operator, where it is used to describe the quantum systems’ energy levels and dynamics, the Hamiltonian operator has applications in a wide range of other fields, as well:

• In astrophysics and cosmology, Hamiltonians are used to study the formation and evolution of the largest bodies in the universe, as well as the universe itself
• In atomic and molecular physics, the Hamiltonian operator is used to predict electron energy levels in their different orbitals, as well as molecular behavior during chemical reactions
• In classical mechanics, Hamiltonians are used to describe the equations of motion for macroscopic systems ranging in size from handheld to a cosmic scale
• In computational chemistry, the Hamiltonian operator is used to calculate molecular electronic structures, predict chemical reactions, and study chemical processes
• In condensed matter physics, Hamiltonians describe particle interactions, such as between electrons in a crystal lattice
• In control systems engineering, Hamiltonians are used in the analysis and design of applications ranging from robotics to aerospace to electrical engineering
• With optics and photonics, Hamiltonians are used to describe light propagation through optical elements such as lenses, prisms, and waveguides
• In particle physics, Hamiltonians are used to describe fundamental particle behaviors and fundamental particle interactions for quantum field theories
• In quantum computing, Hamiltonian manipulation is used to efficiently some complex problems;  Hamiltonian operators are used in algorithms to describe the time evolution of quantum states
• In solid-state physics, Hamiltonians such as the Hubbard Model are used to model electron behavior in crystalline solids
• In statistical mechanics, Hamiltonians are used to calculate many-particle systems’ thermodynamic properties

These dozen applications, counting quantum mechanics, highlight the versatility of the Hamiltonian formalism. Hamiltonians are a framework for understanding physical systems’ behaviors from quantum to cosmic scales, enabling predictions and analyses across a broad spectrum of research disciplines.

To avoid potential confusion, it is worth noting that the term “Hamiltonian” is not exclusive to quantum mechanics. In graph theory, for example, a “Hamiltonian path” sequentially visits each node (vertex) exactly one time. A “Hamiltonian cycle” is a Hamiltonian path that starts and ends at the same node.