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:

Excerpt from Greiner

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:

Excerpt from Griffith

What am I misunderstanding here?


1 Answer 1


You are missing the fact that there are more low-momentum states than high-momentum states. Therefore you are still more likely to observe a low momentum, although all states are equally probable. It’s for the same reason that a twenty-sided die is less likely to land on non-prime number even when it’s perfectly fair.

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.

  • $\begingroup$ Aha, so in other words, you're saying that: even though the particle's velocity through q-space changes in time, and thus the particle "loiters"/"lingers" at some locations longer than others, the particle's velocity through p-q-space (aka "phase space") is constant in time, thus the particle's location in p-q-space does NOT "loiter" or "linger" at any locations in p-q-space compared to other locations in p-q-space. That makes sense. $\endgroup$
    – dturn805
    Mar 17 at 22:19
  • $\begingroup$ It would also mean that the particle visits every location of phase space during each "orbit" of phase space.... And for chaotic systems (such as this youtube.com/watch?v=U39RMUzCjiU) the system randomly visits each state with equal time-statistics. Is there any proof of these ideas? Without a proof I could easily be convinced by somebody claiming to have a simple mechanical device (such as youtube.com/watch?v=U39RMUzCjiU) that spends more time in particular states than others. $\endgroup$
    – dturn805
    Mar 17 at 22:35
  • $\begingroup$ In other words, I could easily be convinced by somebody claiming to have a simple chaotic device (such as the double pendulum device in the video I posted above) that spends less time in some microstates because they are "less accessible" than other microstates -- where "less accessible" means the device has to be in a rare orientation to gain entrance to the less accessible microstates. Thoughts, anyone? $\endgroup$
    – dturn805
    Mar 17 at 22:50
  • $\begingroup$ That concept/idea/hypothesis is called ergodicity and is a subject on its own. For Hamiltonian systems (such as the double pendulum), you may also want to have a look at Liouville’s theorem. Mind that chaotic systems are not random in the sense that you appear to be using. $\endgroup$
    – Wrzlprmft
    Mar 17 at 22:52
  • $\begingroup$ Aha, thank you very much! You've turned me onto the mainstream considerations of this topic: en.wikipedia.org/wiki/Ergodic_hypothesis. Again, much thanks. $\endgroup$
    – dturn805
    Mar 17 at 23:09

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