Introduction to Quantum Mechanics · Problem 3.14

Formalism — David J. Griffiths

Physics — Quantum Mechanics · commutator_identities

Problem

(a) Prove the following commutator identities: [A^+B^,C^]=[A^,C^]+[B^,C^],[\hat{A} + \hat{B}, \hat{C}] = [\hat{A}, \hat{C}] + [\hat{B}, \hat{C}], [A^B^,C^]=A^[B^,C^]+[A^,C^]B^.[\hat{A}\hat{B}, \hat{C}] = \hat{A}[\hat{B}, \hat{C}] + [\hat{A}, \hat{C}]\hat{B}. (b) Show that [xn,p^]=iℏnxn−1[x^n, \hat{p}] = i\hbar n x^{n-1}. (c) Show more generally that [f(x),p^]=iℏdfdx,[f(x), \hat{p}] = i\hbar \frac{df}{dx}, for any function f(x)f(x) that admits a Taylor series expansion. (d) Show that for the simple harmonic oscillator [H^,a^±]=±ℏωa^±.[\hat{H}, \hat{a}_\pm] = \pm \hbar\omega \hat{a}_\pm. Hint: Use Equation 2.54.

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