For the normalised wave function

PHYSICS · Subjective · Fermi University Physics · Quantum Mechanics · Wave mechanics in one dimension · The formalism of quantum mechanics

Quantum Mechanics: From the Photon to the Hydrogen Atom, Ch The Schrödinger equation and the Born rule

Problem

For the normalised wave function Ψ(x)=1be−∣x∣/b\Psi(x) = \dfrac{1}{\sqrt b}e^{-|x|/b}, a worked attempt computes ⟨p2⟩\langle p^2\rangle like this:“⟨p2⟩=∫Ψ∗p^2Ψ dx=−ℏ2∫Ψ Ψ′′ dx\langle p^2\rangle = \displaystyle\int\Psi^*\hat p^2\Psi\,dx = -\hbar^2\displaystyle\int\Psi\,\Psi''\,dx. Away from the origin, Ψ′′=Ψ/b2\Psi''=\Psi/b^2, so ⟨p2⟩=−ℏ2b2∫Ψ2 dx=−ℏ2b2\langle p^2\rangle = -\dfrac{\hbar^2}{b^2}\displaystyle\int\Psi^2\,dx = -\dfrac{\hbar^2}{b^2}, using that Ψ\Psi is normalised.”Identify the error, say why the result should have been rejected on sight, and find the correct value of ⟨p2⟩\langle p^2\rangle.

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