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For example, if an electron wave packet is initially localized in a region of atomic dimensions (i.e., 10 −10 m) then the width of the packet doubles in about 10 −16 s. Clearly, particle wave packets spread out very rapidly indeed (in free space): [ 17 ] For instance, after 1 ms, the width will have grown to about a kilometer.
The group delay and phase delay properties of a linear time-invariant (LTI) system are functions of frequency, giving the time from when a frequency component of a time varying physical quantity—for example a voltage signal—appears at the LTI system input, to the time when a copy of that same frequency component—perhaps of a different physical phenomenon—appears at the LTI system output.
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.
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 .
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Brainwave entrainment, also referred to as brainwave synchronization or neural entrainment, refers to the observation that brainwaves (large-scale electrical oscillations in the brain) will naturally synchronize to the rhythm of periodic external stimuli, such as flickering lights, [1] speech, [2] music, [3] or tactile stimuli.
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In representing the wave function of a localized particle, the wave packet is often taken to have a Gaussian shape and is called a Gaussian wave packet. [25] [26] [27] Gaussian wave packets also are used to analyze water waves. [28] For example, a Gaussian wavefunction ψ might take the form: [29]