Origin of the quantum shape effect.

The quantum size and shape effects are often considered difficult to distinguish from each other because of their coexistence. Essentially, it is possible to separate them and focus solely on the shape effect by considering a size-invariant shape transformation, which changes the discrete energy spectra of strongly confined systems and causes the quantum shape effects. The size-invariant shape transformation is a geometric technique of transforming shapes by preserving the boundary curvature, topology, and the Lebesgue measure of a bounded domain. The 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. While quantum size effects can usually be obtained via bounded continuum approximation, the quantum shape effect is a direct consequence of the energy quantization in specifically designed confined geometries. 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 one-dimensional 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. The position of the partition becomes the shape variable. We investigate the quantum shape effect on the thermodynamic properties of confined particles considering their discrete spectrum. In addition, we applied an analytical model based on dimensional transitions to predict thermodynamic properties under the quantum shape effect accurately. A fundamental understanding of quantum shape effects could pave the way for employing them to engineer physical properties and design better materials at the nanoscale.