An oscillator has mass and angular frequency

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

The Structure of Quantum Mechanics, Ch The oscillator as an algebra

Problem

An oscillator has mass m=1.0×10−26 kgm = 1.0\times10^{-26}\ \mathrm{kg} and angular frequency ω=3.0×1013 rad s−1\omega = 3.0\times10^{13}\ \mathrm{rad\,s^{-1}}. Instead of solving ⟨x∣a^∣0⟩=0\langle x|\hat a|0\rangle = 0 as a differential equation in xx, work in the momentum representation, where p^→p\hat p \to p (multiplication) and x^→iℏ d/dp\hat x \to i\hbar\,d/dp (the momentum-representation rule following from [x^,p^]=iℏ[\hat x,\hat p]=i\hbar). ((a)) Using a^=mω2ℏ x^+i2mℏω p^\hat a = \sqrt{\dfrac{m\omega}{2\hbar}}\,\hat x + \dfrac{i}{\sqrt{2m\hbar\omega}}\,\hat p, solve ⟨p∣a^∣0⟩=0\langle p|\hat a|0\rangle=0 for the shape of ψ~0(p)\tilde\psi_0(p). ((b)) Compute ⟨0∣p^2∣0⟩\langle 0|\hat p^2|0\rangle from this momentum-space Gaussian, and confirm it agrees with the value obtained by writing p^\hat p in ladder form and counting, ⟨0∣p^2∣0⟩=mℏω/2\langle 0|\hat p^2|0\rangle = m\hbar\omega/2. Give the numerical value.

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