Summary: It is known that many physical systems which do not exhibit deterministic chaos when treated classically may exhibit such behaviour if treated from the quantum mechanics point of view. In this paper, we will show that an annihilation operator of the unforced quantum harmonic oscillator exhibits distributional chaos as introduced by B. Schweizer and J. Smítal [Trans. Am. Math. Soc. 344, No. 2, 737–754 (1994; Zbl 0812.58062)]. Our approach strengthens previous results on chaos in this model and provides a very powerful tool to measure chaos in other (quantum or classical) models.