How are Gaussian and Plane Waves Represented Mathematically?
A Gaussian wave can be mathematically expressed as: \[ \psi(x,t) = A e^{-(x - vt)^2 / 2\sigma^2} e^{i(kx - \omega t)} \] where \(A\) is the amplitude, \(v\) is the velocity, \(\sigma\) is the width of the wave packet, \(k\) is the wave number, and \(\omega\) is the angular frequency.
A plane wave can be expressed as: \[ \psi(x,t) = A e^{i(kx - \omega t)} \] Here, the amplitude \(A\) is constant, and the wave number \(k\) and angular frequency \(\omega\) determine the wave's spatial and temporal properties.
