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# Section 8.5: Towards a Wave Packet Solution

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Considering our failure (the lack of localization and normalization) with using only one solution to the Schrödinger equation (a single time-dependent energy eigenfunction) for the free-particle problem, what about a superposition of plane wave solutions which you have explored in Sections 8.3 and 8.4? Restart. While these constructions approach a localized solution, there are always copies of the localized solution created. Instead of a sum of individual solutions, consider an integral,

Ψ(*x*) = 1/(2π*ħ*)^{1/2} ∫ Φ(*p*) e^{ipx/ħ} *dp* [integral from −∞ to +∞] (8.11)

which is called a Fourier transform. The Fourier transform adds a continuum of plane wave solutions, e^{ipx/ħ}, weighted by a function of momentum, φ(*p*). This function of momentum is called the momentum-space wave function since it plays the same role in momentum space as ψ(*x*) does in position space. The momentum-space wave function, φ(*p*), is itself the* inverse* Fourier transform of ψ(*x*) and is given by:

Φ(*p*) = 1/(2π*ħ*)^{1/2} ∫ Ψ(*x*) e^{−ipx/ħ} *dp* [integral from −∞ to +∞] (8.12)

Now, we seek to understand the generic wave function as defined by the Fourier transform in the first equation by substituting a reasonable function for φ(*p*) and calculating the position-space wave function. Consider a normalized Gaussian distribution in momentum centered on a momentum, *p*_{0}, such that

Φ(*p*) = (α^{1/2}/π^{1/4}) exp[−α^{2}(*p* −* p*_{0})^{2}/2]. (8.13)

Note that |Φ(*p*)|^{2} goes to 1/e of its maximum value when *p* = *p*_{0} ± 1/α. Therefore 1/α tells us something about the spread of the momentum-space wave function. This momentum-space wave function is shown in the bottom panel of the animation. ** In the animation, ħ = 2m = 1**.

To find the position-space wave function, we must use Eq. (8.13) in Eq. (8.11) and evaluate the resulting integral. When we do this Gaussian integral, we get:^{4}

Ψ(*x*) = [π^{−1/4 }(α*ħ*)^{−1/2}] exp(*ip*_{0}*x*/*ħ* − *x*^{2}/2α^{2}*ħ*^{2}). (8.14)

Look at the animation to see how the 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. Vary *p*_{0} and α and see what happens. As *p*_{0} gets larger and positive, the momentum-space wave function shifts to the right and is centered on the new value of *p*_{0}. The position-space wave function now has bands of color which represent the exp(*ip*_{0}*x*/*ħ*) factor in the wave function. As α increases, the momentum-space wave function narrows and the position-space wave function widens (which is a result of the Heisenberg uncertainty principle).

Our packet has almost all of the right features we want in a packet that simulates a *particle*. However, it does not have a time dependence and it does not allow us to shift the initial position, *x*_{0}, of the packet to any value we like. We will add these features next.

^{4}Since the momentum-space wave function was normalized, so is the resulting position-space wave function. In general, due to the relationship between ψ(*x*) and φ(*p*) as expressed in the first and second equation, we have that: ∫ |ψ(*x*)|^{2}* dx* = ∫ |φ(*p*)|^{2} *dp* [integrals from −∞ to +∞], and hence if one is normalized, so is the other.}

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