  Size-invariant shape transformation gives rise to the so-called quantum shape
effect in strongly confined systems. While quantum size and shape effects are
often thought to be difficult to distinguish because of their coexistence, it
is actually possible to separate them and focus solely on the shape effect. In
fact, quantum shape effect is a quite different phenomenon from quantum size
effects, as it can have the opposite influence on the physical properties of
nanoscale systems. Here we explore the origin of the quantum shape effect by
theoretically investigating the simplest system that can produce the same
physics: quantum particles in a box separated by a moving partition. The
partition moves quasistatically from one end of the box to the other, allowing
the system to remain in equilibrium with a reservoir throughout the process.
The partition and the boundaries are impenetrable by particles, forming two
effectively interconnected regions. Position of the partition becomes the shape
variable. We investigate quantum shape effect on the thermodynamic properties
of confined particles. In addition, we applied a new analytical model based on
dimensional transitions to accurately predict thermodynamic properties under
the quantum shape effect. A fundamental understanding of quantum shape effects
could pave the way for employing them to engineer physical properties and
design better materials at nanoscale.
