0

In the wikipedia article about quantum harmonic oscillator, this image describes the probability density of finding the particle at certain points for the different eigenstates :

probability densities

I am confused that the probability density is zero at certain points. Does that really mean that it is impossible to measure the particle at those points ? Can we experimentally verify this ?

Pao
  • 233

1 Answers1

2

The wave function of the quantum harmonic oscillator $\psi_n(x)$ for the system at the $n$th energy level can be used to construct a probability density function,

$$\rho_n(x) := |\psi_n(x)|^2$$

such that the probability of finding the particle in an interval, $x \in[a,b]$ is given by,

$$P(a \leq x \leq b; n) = \int_a^b \rho_n(x) \, dx.$$

Thus, it follows that the probability of finding the particle at any one point, $P(x = x_0)$ is in fact zero since it has measure zero. Thus the probability of finding it at the nodes is indeed zero, but it is a meaningless statement in the sense that the probability vanishes for any individual point.

It should be stressed that, in general, for a measurable space $(\Omega, \mathcal F)$, it is not true that $P$ vanishes for continuous variables evaluated at a single $\omega \in \Omega$ as it depends on the choice of dominating measure, as explained in the statistics SE.

Community
  • 1
JamalS
  • 19,254
  • Maybe one should add that at such a node even $\lim_{\varepsilon \to 0} \frac 1 \varepsilon \int_{x_0-\varepsilon}^{x_0+\varepsilon} \rho_n(x), \mathrm dx = 0$. While it is always impossible to measure the particle exactly at any point, I would say this is what we usually mean by "it is impossible to measure the particle at the node". – Noiralef Jan 13 '17 at 15:53