Ψ(x,t)=Aei(kx−ωt)cap psi open paren x comma t close paren equals cap A e raised to the i open paren k x minus omega t close paren power (wavenumber) (angular frequency) The problem is that
[ \Delta x , \Delta k = \sqrt\alpha \cdot \frac12\sqrt\alpha = \frac12 ] wave packet derivation
Combine the exponentials. Let’s set ( \beta = \frac\hbar t2m ) for brevity. Then the exponent is: Ψ(x,t)=Aei(kx−ωt)cap psi open paren x comma t close
Why a Gaussian? Because its Fourier transform is also a Gaussian, making the wave packet integrals solvable exactly. wave packet derivation