Density Function Thermal Scaling and Physical Significance of Shape Functions

The solid-state concept of thermal voltage is used to define a homothetic scaling of the quantum mechanical one-particle reduced density function. This scaling might be used to construct a temperature dependent quantum density. Once defined such scaling, named here as thermal scaling, it is simple to use it with a precise temperature, adapting such scaling matching the associated shape function. The temperature achieving this equality is termed Shape Temperature TS and, if N is the number of particles of a given quantum object, one can demonstrate that the simple equality: 12 T N S  holds. Furthermore, Shape Temperature can be associated to a characteristic Shape Frequency max  S , via Wien's law, which yields equality: max 0.7 ,  S  N THz linking number of particles with frequency.