Quantum Mechanical Application
In quantum mechanical and quantum chemical computations matrix diagonalization is one of the most frequently applied numerical processes. The basic reason is that the time-independent Schrödinger equation is an eigenvalue equation, albeit in most of the physical situations on an infinite dimensional space (a Hilbert space). A very common approximation is to truncate Hilbert space to finite dimension, after which the Schrödinger equation can be formulated as an eigenvalue problem of a real symmetric, or complex Hermitian, matrix. Formally this approximation is founded on the variational principle, valid for Hamiltonians that are bounded from below. But also first-order perturbation theory for degenerate states leads to a matrix eigenvalue problem.
Read more about this topic: Diagonalizable Matrix
Famous quotes containing the words quantum, mechanical and/or application:
“But how is one to make a scientist understand that there is something unalterably deranged about differential calculus, quantum theory, or the obscene and so inanely liturgical ordeals of the precession of the equinoxes.”
—Antonin Artaud (1896–1948)
“A committee is organic rather than mechanical in its nature: it is not a structure but a plant. It takes root and grows, it flowers, wilts, and dies, scattering the seed from which other committees will bloom in their turn.”
—C. Northcote Parkinson (1909–1993)
“The application requisite to the duties of the office I hold [governor of Virginia] is so excessive, and the execution of them after all so imperfect, that I have determined to retire from it at the close of the present campaign.”
—Thomas Jefferson (1743–1826)