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The time-dependent Schrödinger equation described above predicts that wave functions can form standing waves, called stationary states. These states are particularly important as their individual study later simplifies the task of solving the time-dependent Schrödinger equation for any state. Stationary states can also be described by a ...
The general solution of the above differential equation for a given value of a and q is a set of linearly independent Mathieu cosines and Mathieu sines, which are even and odd solutions respectively. In general, the Mathieu functions are aperiodic; however, for characteristic values of a n ( q ) , b n ( q ) {\displaystyle a_{n}(q),b_{n}(q ...
The Hartree–Fock method is typically used to solve the time-independent Schrödinger equation for a multi-electron atom or molecule as described in the Born–Oppenheimer approximation. Since there are no known analytic solutions for many-electron systems (there are solutions for one-electron systems such as hydrogenic atoms and the diatomic ...
2.1.1 Non-relativistic time-independent Schrödinger equation. ... To solve from the Schrödinger equation: varies with situation and number of particles
In quantum mechanics and scattering theory, the one-dimensional step potential is an idealized system used to model incident, reflected and transmitted matter waves.The problem consists of solving the time-independent Schrödinger equation for a particle with a step-like potential in one dimension.
If we solve the time-independent Schrödinger equation for an energy >, letting ′ = such that = ′ then the solution has the same form as the inside-well case: = (′) + (′) and, hence, will be oscillatory both inside and outside the well. Thus, the solution is never square integrable; that is, it is always a non-normalizable state.
One particular solution to the time-independent Schrödinger equation is = /, a plane wave, which can be used in the description of a particle with momentum exactly p, since it is an eigenfunction of the momentum operator. These functions are not normalizable to unity (they are not square-integrable), so they are not really elements of physical ...
The problem consists of solving the one-dimensional time-independent Schrödinger equation for a particle encountering a rectangular potential energy barrier. It is usually assumed, as here, that a free particle impinges on the barrier from the left.