Starting only from and the definitions

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 harmonic oscillator

Problem

Starting only from [x^,p^]=iℏ[\hat x,\hat p]=i\hbar and the definitions a±=12ℏmω(∓ip^+mωx^)a_\pm = \dfrac{1}{\sqrt{2\hbar m\omega}}(\mp i\hat p + m\omega\hat x), derive H^\hat H in terms of a+a−a_+a_- by multiplying out a+a−a_+a_- directly, and separately derive H^\hat H in terms of a−a+a_-a_+ by multiplying out a−a+a_-a_+. Use the two results together to re-derive [a−,a+]=1[a_-,a_+]=1 as a consistency check, then find the zero-point energy for ω=2.5×1014 rad s−1\omega = 2.5\times10^{14}\ \mathrm{rad\,s^{-1}}.

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