无限深势阱中粒子的态密度

Density of states of a free particle in infinite potential well

对于自由粒子在有限容器中的能态密度,热力学统计教材一般根据半经典量子图像,由驻波条件和德布罗意关系,以动量分立值为基础出发得到;然而根据量子理论,无限深势阱中的粒子存在能量本征态,而非动量本征态.本文以能量本征态为统计对象推导了有限体积中的自由粒子的能态密度,结果与教材一致.但是我们的处理方式显得更为自然.

For a free particle in finite container,in some thermodynamic statistics textbooks,the density of energy states is deduced based on momentum discrete values due to de Broglie relations and standing wave conditions from semi-classical quantum images.However,according to quantum theory,the particle in infinite potential well has energy eigenstates rather than momentum eigenstates.In this paper,we derive the state density of free particles in finite volume based on its energy eigenstates,and the results are in agreement with textbook.But our treatment seems to be more nature.