ceigenstates

Ceigenstates, short for "continuous eigenstates," are a concept in quantum mechanics that describes the states of a quantum system that evolve over time in a simple, exponential manner. Unlike discrete eigenstates, which are associated with eigenvalues and eigenvectors in linear algebra, ceigenstates are continuous and depend on a continuous parameter.

In the context of quantum mechanics, ceigenstates are often used to describe the evolution of a quantum system under a time-dependent Hamiltonian. The time evolution of a quantum state can be described by the Schrödinger equation, which, for a time-dependent Hamiltonian H(t), is given by:

iħ(∂ψ/∂t) = H(t)ψ

where i is the imaginary unit, ħ is the reduced Planck constant, and ψ is the wave function describing the quantum state. The ceigenstates are the solutions to this equation that take the form:

ψ(x,t) = exp(-iεt/ħ)φ(x)

where φ(x) is a continuous function of the spatial coordinate x, and ε is a continuous parameter that depends on the energy of the system. The ceigenstates evolve in time with a phase factor exp(-iεt/ħ), which represents a simple, exponential evolution.

Ceigenstates are particularly useful in the study of quantum chaos and quantum scattering, where the continuous spectrum of energies leads to a continuous set of ceigenstates. They provide a way to understand the statistical properties of quantum systems and to make predictions about their behavior in the presence of chaos or scattering.

In summary, ceigenstates are a powerful tool in quantum mechanics for describing the time evolution of quantum systems with a continuous spectrum of energies. They provide a way to understand the statistical properties of quantum systems and to make predictions about their behavior in the presence of chaos or scattering.