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Section 8.6: The Quantum-mechanical Wave Packet Solution
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The time-dependent position-space wave function can be calculated from the Fourier transform
Ψ(x,t) = (2πħ)1/2 ∫ Φ(p,t) exp(ipx/ħ) dp , [integral from −∞ to +∞] (8.15)
of the time-dependent momentum-space wave function
Φ(p,t) = (α1/2π−1/4) exp[−α2(p − p0)2/2] exp (−ipx0/ħ) exp(−ip2t/2mħ) . (8.16)
The added time dependence in Eq. (8.13) is in the term exp(−ip2t/2mħ). Compare this momentum-space wave function to the one we used in Section 8.5. Note that Eq. (8.16) has another additional factor, exp(−ipx0/ħ), which sets the center of the Gaussian in position space to x0. This function is shown in the bottom panel of the animation. When we do the Gaussian integral in Eq. (8.15) using Eq. (8.16) for Φ(p,t), we get:
Ψ(x,t) = [π1/2αħ(1 + it/t0)]−1/2 exp[−ip0(x − x0)/ħ] exp(−ip02t/2mħ) exp[−(x − x0 − p0t)/m)2/(2α2ħ2(1 + it/t0))] (8.17)
where we have used the substitution5 t0 = mħα2.
Look at the animation, to see how this position-space wave function is related to the original momentum-space wave function. The bottom panel shows momentum space and the top panel shows position space. In the animation, ħ = 2m = 1. Vary x0, p0, and α and see what happens. We considered the effect of p0 and α in Section 8.5. As x0 gets larger and positive, the position-space wave function shifts to the right and is now centered on the new value of x0. At t = 0 the momentum-space wave function has bands of color which represent the exp(ip0x/ħ) factor in the wave function. Note the effect of time evolution on the position-space and momentum-space wave functions by pressing the "input values and play" button. The position-space wave function spreads over time but the momentum-space wave function does not, but φ(p,t) does change phase as a function of time. We can calculate the probability density, Ψ*(x,t)Ψ(x,t), as
ρ(x,t) = [αħ(π[1 + (t/t0)2])1/2]−1 exp[−(x − x0 − p0t)/m)2/(α2ħ2(1 + (t/t0)2)] . (8.18)
The maximum height of the probability density is governed by
[αħ(π(1 + (t/t0)2)1/2]−1 , (8.19)
since the exponential's maximum is fixed at 1. The spread in x, δx, of the probability density is proportional to the square root of the denominator in the exponential
δx is proportional to αħ[1 + (t/t0)2]1/2 ,
where αħ is the initial (t = 0) spread of the probability density and is related to the inverse of the spread in momentum. The term in the square root describes the time dependence of the spread in position of the position-space probability density. This time dependence is inversely related to the initial spread in the position-space probability density. The position of the central peak is where the numerator of the exponential is zero, or where
x = x0 + p0t/m .
Explicitly, from the calculation of expectation values, we see that
<x> = x0 + p0t/m (8.20)
<x2> = (x0 + p0t/m)2 + (α2ħ2/2) [1 + (t/t0)2] , (8.21)
<p> = p0 and <p2> = p02 + 1/2α2 . (8.22)
Δx ≡ (<x2> − <x>2)1/2 = αħ([1 + (t/t0)2]/2)1/2 (8.23)
Δp ≡ (<p2> − <p>2)1/2 = (21/2α)−1 (8.24)
ΔxΔp = ħ/2 [1 + (t/t0)2]1/2 ≥ ħ/2 , (8.25)
which it must be in order to satisfy the Heisenberg uncertainly principle.6 Since at t = 0, ΔxΔp = ħ/2, the Gaussian wave packet is called a minimum indeterminacy wave packet. Finally, the expectation value of the energy for the packet is
<E> = <T> = < p2>/2m = [p02 + 1/2α2]/2m , (8.26)
which tells us that the spread in the momentum distribution of the wave packet also contributes to its average kinetic, <T>, and average total, <E>, energies.
5This substitution is often used in the literature. See, for example, Refs. [16,17].
6By convention, we use the phrase Heisenberg uncertainly principle, but again this concept is best stated as the Heisenberg indeterminacy principle since indeterminacy better represents the concept that Heisenberg was trying to describe.