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The solutions of the Schrödinger equation in polar coordinates in vacuum are thus labelled by three quantum numbers: discrete indices ℓ and m, and k varying continuously in [,): = (,) These solutions represent states of definite angular momentum, rather than of definite (linear) momentum, which are provided by plane waves ().
Solving the equation by separation of variables means seeking a solution of the form of a product of spatial and temporal parts [18] (,) = (), where () is a function of all the spatial coordinate(s) of the particle(s) constituting the system only, and () is a function of time only.
The case of a particle in a one-dimensional ring is an instructive example when studying the quantization of angular momentum for, say, an electron orbiting the nucleus. The azimuthal wave functions in that case are identical to the energy eigenfunctions of the particle on a ring.
The Hamiltonian of the particle is: ^ = ^ + ^ = ^ + ^, where m is the particle's mass, k is the force constant, = / is the angular frequency of the oscillator, ^ is the position operator (given by x in the coordinate basis), and ^ is the momentum operator (given by ^ = / in the coordinate basis).
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 ...
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.
Let (x, y, z) be the standard Cartesian coordinates, and (ρ, θ, φ) the spherical coordinates, with θ the angle measured away from the +Z axis (as , see conventions in spherical coordinates). As φ has a range of 360° the same considerations as in polar (2 dimensional) coordinates apply whenever an arctangent of it is taken. θ has a range ...
The equation defining a plane curve expressed in polar coordinates is known as a polar equation. In many cases, such an equation can simply be specified by defining r as a function of φ . The resulting curve then consists of points of the form ( r ( φ ), φ ) and can be regarded as the graph of the polar function r .