Hamiltonian (quantum mechanics)

The Hamiltonian operator is defined as the sum of the kinetic energy operator and the potential energy operator: Ĥ = T̂ + V̂. In position representation, for a single non-relativistic particle of mass m, the kinetic operator is T̂ = −ħ²/(2m) ∇², while V̂ is a function of position. The Hamiltonian is required to be Hermitian, ensuring that its eigenvalues are real and correspond to measurable energies¹. In matrix mechanics, the Hamiltonian is represented as a square matrix whose entries are derived from the action on basis states. The explicit form depends on the chosen representation (position, momentum, or energy basis) and the nature of the system—whether it is a single particle, a collection of particles, or a field².

The Hamiltonian is the generator of time evolution in quantum mechanics. The time-dependent Schrödinger equation, iħ d|ψ(t)⟩/dt = Ĥ|ψ(t)⟩, describes how the state vector evolves. For a time-independent Hamiltonian, the solution factorizes into a spatial part and a time-dependent phase, leading to stationary states³. In the Heisenberg picture, operators evolve according to dÂ/dt = (i/ħ)[Ĥ,Â] + ∂Â/∂t, where the commutator with the Hamiltonian governs the dynamics. The Hamiltonian also determines the probabilities of energy measurements: the probabilities are given by the squared magnitudes of the coefficients of the state in the energy eigenbasis⁴.

The eigenvalues of the Hamiltonian form its spectrum, which can be discrete (for bound states) or continuous (for scattering states). The eigenstates satisfy the time-independent Schrödinger equation Ĥ|ψ⟩ = E|ψ⟩, where E is the energy eigenvalue. Degeneracy occurs when multiple linearly independent eigenstates share the same energy; this is often related to symmetries of the system¹. The spectral theorem for Hermitian operators guarantees that the eigenstates form a complete orthonormal basis, allowing any state to be expanded in energy eigenstates. The Hamiltonian's spectrum is a key predictor of experimental outcomes, such as the discrete lines in atomic spectra or the continuous energy bands in solids.

While standard Hamiltonians are Hermitian, non-Hermitian Hamiltonians with PT-symmetry (parity-time reversal) can also exhibit real spectra, a discovery that has opened new research areas in optics and quantum field theory⁵. Time-dependent Hamiltonians appear in systems driven by external fields, requiring the use of the time-ordered exponential (Dyson series) for evolution. Relativistic quantum mechanics replaces the Schrödinger Hamiltonian with the Dirac Hamiltonian, which includes spin and anti-particle solutions. In quantum field theory, the Hamiltonian is an operator on Fock space that creates and annihilates particles. The classical Hamiltonian (e.g., from classical mechanics) serves as the starting point for quantization, but subtle ordering ambiguities can arise, leading to different quantum Hamiltonians for the same classical system².

Hermitian operator

An operator that is equal to its own adjoint; its eigenvalues are real and its eigenvectors form a complete basis.

eigenvalue

A scalar λ such that Ĥ|ψ⟩ = λ|ψ⟩; represents a possible measurement outcome.

Schrödinger equation

The fundamental equation of quantum mechanics that describes how the quantum state changes in time.

stationary state

An eigenstate of the Hamiltonian with a definite energy, whose time dependence is only a phase factor.