A Modern Approach to Quantum Mechanics · Problem 6.4

Wave Mechanics in One Dimension — John S. Townsend

Physics — Quantum Mechanics · gaussian-wave-packet-time-evolution

Problem

Show that for the Gaussian wave packet (6.62) with the Hamiltonian for a free particle H^=p^x2/2m\hat{H} = \hat{p}_x^2/2m, the time-evolved wave function is ψ(x,t)=1π[a+(iℏt/ma)]e−x2/{2a2[1+(iℏt/ma2)]}\psi(x, t) = \frac{1}{\sqrt{\sqrt{\pi}[a + (i\hbar t/ma)]}} e^{-x^2/\{2a^2[1+(i\hbar t/ma^2)]\}} by carrying out the Gaussian integral in (6.75).

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