Search results
Results from the WOW.Com Content Network
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 ...
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 wave function is generally complex .
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 ...
The expanding ring of waves is the wave group or wave packet, within which one can discern individual waves that travel faster than the group as a whole. The amplitudes of the individual waves grow as they emerge from the trailing edge of the group and diminish as they approach the leading edge of the group.
Solitary wave in a laboratory wave channel. In mathematics and physics, a soliton is a nonlinear, self-reinforcing, localized wave packet that is strongly stable, in that it preserves its shape while propagating freely, at constant velocity, and recovers it even after collisions with other such localized wave packets.
Dispersion occurs when sinusoidal waves of different wavelengths have different propagation velocities, so that a wave packet of mixed wavelengths tends to spread out in space. The speed of a plane wave, v {\displaystyle v} , is a function of the wave's wavelength λ {\displaystyle \lambda } :
A portion of the wave packet passes through the barrier. The wave function of a physical system of particles specifies everything that can be known about the system. [8] Therefore, problems in quantum mechanics analyze the system's wave function. Using mathematical formulations, such as the Schrödinger equation, the time evolution of a known ...
A one-way wave equation is a first-order partial differential equation describing one wave traveling in a direction defined by the vector wave velocity. It contrasts with the second-order two-way wave equation describing a standing wavefield resulting from superposition of two waves in opposite directions (using the squared scalar wave velocity).