Past exam of the mathematics course of the University of Cambridge / 2016 / ib / Paper 2 / 17B / a

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Solution

Retain the order of the noncommuting position operator and momentum operator. Using their commutator

[\overset{x}{^}, \overset{p}{^}] = i ℏ

, expansion gives

\hat{B} \hat{A} \hat{A} \hat{B} = \overset{p}{^}^{2} + m^{2} ω^{2} \overset{x}{^}^{2} + imω [\overset{x}{^}, \overset{p}{^}] = 2 m \hat{H} - m ℏ ω, = \overset{p}{^}^{2} + m^{2} ω^{2} \overset{x}{^}^{2} - imω [\overset{x}{^}, \overset{p}{^}] = 2 m \hat{H} + m ℏ ω .

Thus the two factorizations of the quantum harmonic oscillator are

\hat{B} \hat{A} = 2 m (\hat{H} - \frac{1}{2} ℏ ω), \hat{A} \hat{B} = 2 m (\hat{H} + \frac{1}{2} ℏ ω) .

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