I Is Probability in Quantum States Proportional to Energy Levels?

[PeterDonis]

I understand we can only measure eigenvalues. But we can certainly know the expectation value for the energy of the system.

If what you mean is "expectation value", then you need to say "expectation value". You can't just say "energy" and expect everyone to know that you mean "the expectation value of energy in some particular state that is not an eigenstate of energy".

I'm just trying to understand what that entails

You have been told two different ways what kind of preparation is necessary for a Boltzmann distribution to apply. I did in post #15, and @f95toli did in post #17. The two descriptions we gave are equivalent.

and how it would apply to a quantum system of a particle in a box?

It wouldn't. You have already been told why (@Nugatory did that in post #14).

 

[bob012345]

The original question of this thread has been answered and I think I have synthesized in my mind all the answers given about that and the discussion of the Boltzmann distribution so I thank everyone for their answers.

 

[PeterDonis]

The original question of this thread has been answered and I think I have synthesized in my mind all the answers given about that and the discussion of the Boltzmann distribution so I thank everyone for their answers.

Thanks for the feedback! I'm glad we were able to answer your questions.