Evolution of a Gaussian Wave Packet

A Gaussian wave packet centered around x=0 at time t=0 with an average initial momentum (Subscript[p, 0]) can be represented by the wavefunction ψ(x,0)=((((Sqrt[2π ])(Subscript[σ, 0])))^(-1/2))exp(-((x^2)/(4((Subscript[σ, 0])^2)))+i (Subscript[p, 0])x). (For convenience, we take ℏ=m=1.) The solution of the free-particle Schrödinger equation with this initial condition works out to ψ(x,t)=(((((Subscript[σ, 0])/(Sqrt[2π]))))^(1/2))(((((Subscript[σ, 0])^2)+((i t)/2)))^(-1/2))exp(-((Subscript[σ, 0])^2)((Subscript[p, 0])^2))exp(-((((x-2i ((Subscript[σ, 0])^2)(Subscript[p, 0])))^2)/(4(((Subscript[σ, 0])^2)+i t/2)))). The probability density is then given by ρ(x,t)=|ψ(x,t)((|)^2)=((((Sqrt[2π])σ(t)))^(-1))exp(-((((x-(Subscript[p, 0])t))^2)/(2((σ(t))^2)))), where σ(t)=(((Subscript[σ, 0])(1+((t^2)/(4((Subscript[σ, 0])^4)))))^(1/2)), shown as a black curve. The wave packet remains Gaussian as it spreads out, with its center moving to x=(Subscript[p, 0])t, thereby following the classical trajectory of the particle. The corresponding momentum probability distribution is given by ρ(p,t)=|ϕ(p,t)((|)^2)=(Sqrt[(2/π)])(Subscript[σ, 0])(e^(-2((((Subscript[σ, 0])^2)(p-(Subscript[p, 0])))^2))), shown in red. The rms uncertainties are given by Δx=(Sqrt[∫(x^2) ρ(x,t) dx])=σ(t), Δp=(Sqrt[∫(((p-(Subscript[p, 0])))^2)ρ(p,t)dp])=(1/(2(Subscript[σ, 0]))), which is independent of t. This is consistent with the fact that p is a constant of the motion for a free particle.

Thus, with ℏ put back in, the uncertainty product is given by Δx Δp=(ℏ/2)(((1+((t^2)/(4((Subscript[σ, 0])^4)))))^(1/2))≥(ℏ/2), in accord with Heisenberg's uncertainty principle. At t=0, the Gaussian probability distribution represents a minimum uncertainty wave packet with Δx Δp=ℏ/2, but the product increases when t>0.

In this Demonstration, you can drag the time slider to simulate the simultaneous time evolution of the probability and momentum distributions. Note that the x distribution broadens with time while the p distribution maintains its original width. The numerical values of x, p, and t are illustrative only and have no absolute significance.

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Snapshot 1-3: the position probability distribution broadens with increasing t, while the momentum distribution moves with (Subscript[p, 0]) but retains its original width

Reference: S. M. Blinder, "Evolution of a Gaussian Wavepacket," Am J Phys 36(6), 1968 pp. 525–526.

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Contributed by: S. M. Blinder

With corrections by: Stefano Rigolin and Michael Trott

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