Relationship between single particle partition function and V

[hokhani]

Why when the particles are nonlocalized, the single particle partition function is directly proportional to V, namely the volume of the system, and when the particles are localized, the single particle partition function is independent of V? (Pathria, Statistical Mechanics, chapter 4, section 4.4, equations 2&14)

 

[iScience]

when a particle is localized, then, well, they're localized, so no need to worry about the entire volume. but when a particle isn't localized, you have to invoke quantum mechanics and view the particles as wave functions (probability wave). we view this wave function as a standing wave and the number of antinodes in a given space would be the number of possible locations at which this 'particle' can be at. and the number of antinodes is determined by the dimensions of the container these particle can be in (the volume)