# Why are the microstates of a harmonic oscillator considered equally probable, when the particle spends more time at locations of zero momentum?

Introductory statistical mechanics for a microcanonical ensemble (a.k.a. constant energy, no exchange of heat or work with environment) claims that all microstates in the momentum–location phase space are equally probable to occur.

But for a harmonic oscillator, the particle is more likely to be found at points of lower momentum, because the particle spends more time in those locations.

Here is an excerpt from Chapter 5 of Greiner’s Thermodynamics and Statistical Mechanics showing the example of the microstates in phase space of a 1D harmonic oscillator: It seems like common sense that those microstates have a differing probability of occurrence than each other, because the probability of finding the particle at the various locations is not equal.

Here is an excerpt from Chapter 2 of Griffith’s Introduction to Quantum Mechanics showing this distribution: What am I misunderstanding here?

If you take your Figure 5.1, let $$ΔE$$ go towards $$0$$, and collapse all the area on the $$q$$ axis, you will get a similar distribution as in your Figure 2.7.