Complex fourier transform and Wave Packets

anuj — Fri, 07 Mar 2025 21:35:18 +0000

Constructing a wave that is spiked in just one small region is not easy.

When you superpose several waves, you have to do so in a way that they constructively interfere only in this small region and are essentially destructively interfering everywhere else.

Complex Plane Wave Evolution

Wave Description

A plane complex harmonic wave can be described by the function:

ψ(x,t) = A e^{i(kx – ωt)}

where:

A is the amplitude
k is the wave number
ω is the angular frequency
x is the spatial coordinate
t is time

(a) Evolution in Time

At t = 0, the wave is:

ψ(x,0) = A e^{i kx}

As time progresses, the phase evolves as:

ψ(x,t) = A e^{i(kx – ωt)}

The factor e^{-i ω t} represents a time-dependent oscillation at frequency ω.

(b) Evolution in Space (along x)

At a fixed time t, the wave is:

ψ(x,t) = A e^{i(kx – ωt)}

The term e^{i kx} represents a spatial oscillation with wavenumber k, meaning the wave has a wavelength:

λ = 2π/k

The phase velocity is:

v_p = ω/k

The entire waveform shifts in space over time at this velocity.

Graphical Representation

The following graphs illustrate the evolution of the plane wave:

Spatial Evolution: Wave as a function of x at t = 0.
Time Evolution: Oscillation at a fixed spatial point (e.g., x = 0).

complex plane wave evolution

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Complex fourier transform and Wave Packets

Complex Plane Wave Evolution

Wave Description

(a) Evolution in Time

(b) Evolution in Space (along x)

Graphical Representation