Search results
Results from the WOW.Com Content Network
This generalizes to any number of particles in any number of dimensions (in a time-independent potential): the standing wave solutions of the time-independent equation are the states with definite energy, instead of a probability distribution of different energies.
The hydrogen atom in a spherical cavity with Dirichlet boundary conditions [4] The Mie potential [5] The Hooke's atom; The Morse potential; The Spherium atom; Zero range interaction in a harmonic trap [6] Multistate Landau–Zener models [7] The Luttinger liquid (the only exact quantum mechanical solution to a model including interparticle ...
Thus the solutions () are just the Legendre functions ( ()) with =, and =,,, =,,,,. Moreover, eigenvalues and scattering data can be explicitly computed. [ 3 ] In the special case of integer λ {\displaystyle \lambda } , the potential is reflectionless and such potentials also arise as the N-soliton solutions of the Korteweg–De Vries ...
The time-independent Schrödinger equation for the wave function is ^ = [+ ()] = (), where Ĥ is the Hamiltonian, ħ is the reduced Planck constant, m is the mass, E the energy of the particle. The step potential is simply the product of V 0 , the height of the barrier, and the Heaviside step function : V ( x ) = { 0 , x < 0 V 0 , x ≥ 0 ...
One particle: N particles: One dimension ^ = ^ + = + ^ = = ^ + (,,) = = + (,,) where the position of particle n is x n. = + = = +. (,) = /.There is a further restriction — the solution must not grow at infinity, so that it has either a finite L 2-norm (if it is a bound state) or a slowly diverging norm (if it is part of a continuum): [1] ‖ ‖ = | |.
The entire vector ξ is a solution of the Schrödinger equation (with a suitable Hamiltonian), which unfolds to a coupled system of 2s + 1 ordinary differential equations with solutions ξ(s, t), ξ(s − 1, t), ..., ξ(−s, t). The term "spin function" instead of "wave function" is used by some authors.
In quantum mechanics, the Schrödinger equation describes how a system changes with time. It does this by relating changes in the state of the system to the energy in the system (given by an operator called the Hamiltonian). Therefore, once the Hamiltonian is known, the time dynamics are in principle known.
The main effort in this approximate solution of the nuclear motion Schrödinger equation is the computation of the Hessian F of V and its diagonalization. This approximation to the nuclear motion problem, described in 3 N mass-weighted Cartesian coordinates, became standard in quantum chemistry , since the days (1980s-1990s) that algorithms for ...