Quantum Mechanics: Concepts and Applications · Problem 1.8

Origins of Quantum Physics — Nouredine Zettili

Physics — Quantum Mechanics · gaussian-and-square-wave-packets

Problem

Example 1.8 (Gaussian and square wave packets) (a) Find ψ(x,0)\psi(x, 0) for a Gaussian wave packet ϕ(k)=Aexp⁡[−a2(k−k0)2/4]\phi(k) = A \exp\left[-a^2(k - k_0)^2/4\right], where AA is a normalization factor to be found. Calculate the probability of finding the particle in the region −a/2≤x≤a/2-a/2 \leq x \leq a/2. (b) Find ϕ(k)\phi(k) for a square wave packet ψ0(x)={Aeik0x,∣x∣≤a,0,∣x∣>a.\psi_0(x) = \begin{cases} Ae^{ik_0x}, & |x| \leq a, \\ 0, & |x| > a. \end{cases} Find the factor AA so that ψ(x)\psi(x) is normalized.

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