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For the infinite square well, we consider a superposition of a large number of states so as to resemble an initial Gaussian wave packet at t = 0 in order to study the dynamics of such packets.⁴  For the simple harmonic oscillator, we will outline the same approach.⁵ 

We can examine the time dependence of such an initially localized state by choosing a Gaussian of the form

Ψ_G(x,0) = (αħπ^1/2)^−1/2 exp(−(x − x₀)²/(2α²ħ²)) exp(ip₀(x − x₀)/ħ),

where by direct calculation: <x>_{t = 0} = x₀, <p>_{t = 0} = p₀, and Δx_{t = 0} = Δx₀ = αħ/2^1/2. The general expression for a wave packet solution constructed from such energy eigenfunctions is

Ψ_G(x,t) = Σ c_n ψ_n(x)  exp(−iE_nt/ħ) [sum from n = 1 to ∞] (12.15)

where the expansion coefficients satisfy Σ_n |c_n|² =1. The expansion constants are determined by the integral

c_n = ∫ψ*_n(x,0) Ψ_G(x,0) dx [sum from −∞ to +∞]

Once we determine these coefficients, we can use Eq. (12.15) to reconstruct the wave packet and study the corresponding packet dynamics. In the animation, we calculate the time dependence of packets with x₀ = 0 and an α, p₀, and ω that you can vary. We have set ħ = 2m = 1.

The time dependence of a wave packet in a harmonic oscillator is determined by all of the exp(−iE_nt/ħ) factors. For the harmonic oscillator there is only one characteristic time scale, T_cl = 2π/ω, which gives a result in agreement with classical expectations. Because the energy of the harmonic oscillator depends linearly on the quantum number, n, there are no other time scales (unlike wave packets in the infinite square well and other wells).  In fact there are only two possibilities for the wave packet's evolution: it can either be a squeezed state or a coherent state.  For a squeezed state the packet remains Gaussian shaped, but as it moves throughout the well its width grows and contracts.  You can see this effect by keeping the default ω and choosing p₀ = 3 and α = 0.5 or α = 2. The other situation, a coherent state (keeping the default ω and choosing p₀ = 3 and α = 1) occurs for special packets and wells and is easily noticeable as the wave packet retains its exact shape throughout its motion throughout the well.

⁴For a comprehensive review of this topic see: R. W. Robinett, "Quantum Wave Packet Revivals," Phys. Rep. 392, 1-119 (2004).
⁵In practice, one uses propagator methods to exactly determine the localized Gaussian wave function and its time dependence. For the details, see pages 206-208 of R. W. Robinett, Quantum Mechanics: Classical Results, Modern Systems, and Visualized Examples, Oxford, New York (1997).