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If (,) (and therefore ()) is a Gaussian function, the wave packet is called a Gaussian wave packet. [12] For example, the solution to the one-dimensional free Schrödinger equation (with 2Δx, m, and ħ set equal to one) satisfying the initial condition (,) = / (+), representing a wave packet localized in space at the origin as a Gaussian ...
Solving the equation by separation of variables means seeking a solution of the form of a product of spatial and temporal parts [19] (,) = (), 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 asymptotic scaling of δ in terms of ε will be determined by the equation – see the example below. Substituting the above ansatz into the differential equation and cancelling out the exponential terms allows one to solve for an arbitrary number of terms S n (x) in the expansion. WKB theory is a special case of multiple scale analysis. [5 ...
By comparison with vector wave equations, the scalar wave equation can be seen as a special case of the vector wave equations; in the Cartesian coordinate system, the scalar wave equation is the equation to be satisfied by each component (for each coordinate axis, such as the x component for the x axis) of a vector wave without sources of waves ...
For example, the wave function with =, = has the same energy as the wave function with =, =. This situation is called degeneracy and for the case where exactly two degenerate wave functions have the same energy that energy level is said to be doubly degenerate. Degeneracy results from symmetry in the system.
The superposition of several plane waves to form a wave packet. This wave packet becomes increasingly localized with the addition of many waves. The Fourier transform is a mathematical operation that separates a wave packet into its individual plane waves. The waves shown here are real for illustrative purposes only; in quantum mechanics the ...
In classical wave-physics, this effect is known as evanescent wave coupling. The likelihood that the particle will pass through the barrier is given by the transmission coefficient, whereas the likelihood that it is reflected is given by the reflection coefficient. Schrödinger's wave-equation allows these coefficients to be calculated.
The Schrödinger–Newton equation, sometimes referred to as the Newton–Schrödinger or Schrödinger–Poisson equation, is a nonlinear modification of the Schrödinger equation with a Newtonian gravitational potential, where the gravitational potential emerges from the treatment of the wave function as a mass density, including a term that represents interaction of a particle with its own ...