Hamiltonian (quantum mechanics)

In quantum mechanics, the Hamiltonian, often denoted by the symbol Ĥ (pronounced "H-hat"), is an operator corresponding to the total energy of a system. It is a fundamental concept and plays a central role in describing the time evolution of quantum states.

Definition and Properties

The Hamiltonian operator is a linear, Hermitian operator acting on the Hilbert space of the quantum system. Hermiticity ensures that the eigenvalues of the Hamiltonian, which represent the possible energy levels of the system, are real.

Mathematically, the Hamiltonian Ĥ is defined such that the time evolution of the quantum state |Ψ(t)⟩ is governed by the time-dependent Schrödinger equation:

iħ ∂/∂t |Ψ(t)⟩ = Ĥ |Ψ(t)⟩

where:

• i is the imaginary unit,
• ħ is the reduced Planck constant,
• |Ψ(t)⟩ is the time-dependent wave function (state vector) representing the quantum state of the system.

Relationship to Classical Mechanics

The Hamiltonian in quantum mechanics is the quantum mechanical analogue of the Hamiltonian function in classical mechanics. In classical mechanics, the Hamiltonian function is a function of the generalized coordinates and momenta of the system, representing the total energy. The quantization procedure involves replacing these classical quantities with corresponding quantum operators. For example, the position x becomes the position operator X, and the momentum p becomes the momentum operator P (often represented as -iħ∂/∂x in the position representation).

Components of the Hamiltonian

The form of the Hamiltonian operator depends on the specific quantum system being considered. However, it typically consists of two main components:

• Kinetic Energy Operator (T): Represents the kinetic energy of the particles in the system. For a single particle of mass m moving in three dimensions, the kinetic energy operator is often given by: T = -ħ²/2m ∇², where ∇² is the Laplacian operator.
• Potential Energy Operator (V): Represents the potential energy due to interactions between the particles within the system and/or external fields. The potential energy operator V can be a function of position and, in some cases, time.

Therefore, the Hamiltonian can often be expressed as:

Ĥ = T + V

Eigenvalues and Eigenstates

The time-independent Schrödinger equation is obtained by considering stationary states, i.e., states with a definite energy E. It is given by:

Ĥ |ψ⟩ = E |ψ⟩

where:

• E is the energy eigenvalue (a real number).
• |ψ⟩ is the corresponding eigenstate (energy eigenstate or stationary state).

The eigenvalues E represent the allowed energy levels of the system, and the eigenstates |ψ⟩ describe the quantum states associated with those energy levels. Solving the time-independent Schrödinger equation to find the eigenvalues and eigenstates is a central problem in quantum mechanics.

Applications

The Hamiltonian is used extensively in various areas of quantum mechanics, including:

• Atomic and Molecular Physics: Determining the energy levels and properties of atoms and molecules.
• Condensed Matter Physics: Studying the behavior of electrons in solids and other condensed phases.
• Quantum Field Theory: Describing the dynamics of quantum fields.
• Quantum Computing: Analyzing the evolution of quantum bits (qubits).