Statistical mechanics is supposedly built entirely on the fundamental assumption that any system has some definite number of "quantum states" available to it, and that it's equally likely to be found in any one of them. My book states this, and then goes into some examples with artificial systems specified to have some small number of available states. This is fine, but I have not been able to make the conceptual connection to a real macroscopic system. Does such a system (say, for example, a few cc's of nitrogen gas in a small insulated bottle at standard temperature and pressure) really have a theoretically countable number of discrete quantum states available to it? If so, what is each state? Is there any way of physically visualizing what changes as you move from one state to the next, or understanding why there is not an intermediate state between two "adjacent" ones?
1 Answer
Statistical mechanics is more abstract than this and doesn't require the assumption of quantum states. It can be applied to both classical and quantum systems.
Does such a system (say, for example, a few cc's of nitrogen gas in a small insulated bottle at standard temperature and pressure) really have a theoretically countable number of discrete quantum states available to it?
Yes. Not just countable, but finite, assuming some bound on allowable energy. Of course, the number of states is extremely large, on the order of $e^{1/k_B}$.
If so, what is each state?
It is very hard to visualize quantum many-body states. Mathematically, a pure state is 'just' an eigenfunction of the Hamiltonian. For systems containing many particles, the Hamiltonian is a complex multi-dimensional function and visualizing it is difficult.
For a single particle such as e.g. a quantum harmonic oscillator, the pure states look like standing waves in one dimension. One can then imagine functions that look like standing waves in a very large number of dimensions (number of particles times number of degrees of freedom of each particle). These are the pure states.
A closed system will then be a superposition of pure states.
The difficulties involved in trying to visualize and understand such large-dimensional problems is one of the factors that led to the development of quantum field theory, in which the symmetries and structures of the underlying physical systems (for example, spatial isotropy for the case of fluids, and periodic symmetries in the case of solids) are used to develop simplified models of the quantum states that can then be used for calculations.
