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The potential energy in this model is given as = {, < < +,,, where L is the length of the box, x c is the location of the center of the box and x is the position of the particle within the box. Simple cases include the centered box ( x c = 0) and the shifted box ( x c = L /2) (pictured).
A rather good approximation of an exciton's behaviour is the 3-D model of a particle in a box. [4] The solution of this problem provides a sole [clarification needed] mathematical connection between energy states and the dimension of space. Decreasing the volume or the dimensions of the available space, increases the energy of the states.
In quantum mechanics, the results of the quantum particle in a box can be used to look at the equilibrium situation for a quantum ideal gas in a box which is a box containing a large number of molecules which do not interact with each other except for instantaneous thermalizing collisions.
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Unlike the infinite potential well, there is a probability associated with the particle being found outside the box. The quantum mechanical interpretation is unlike the classical interpretation, where if the total energy of the particle is less than the potential energy barrier of the walls it cannot be found outside the box.
The presence of degenerate energy levels is studied in the cases of Particle in a box and two-dimensional harmonic oscillator, which act as useful mathematical models for several real world systems. Particle in a rectangular plane
The energy levels of a single particle in a quantum dot can be predicted using the particle in a box model in which the energies of states depend on the length of the box. For an exciton inside a quantum dot, there is also the Coulomb interaction between the negatively charged electron and the positively charged hole.
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