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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).
The particle in a one-dimensional potential energy box is the most mathematically simple example where restraints lead to the quantization of energy levels. The box is defined as having zero potential energy inside a certain region and infinite potential energy outside.
In the case of degenerate energy levels, we can write the partition function in terms of the contribution from energy levels (indexed by j) as follows: =, where g j is the degeneracy factor, or number of quantum states s that have the same energy level defined by E j = E s.
For a quantum particle with a wave function | moving in a one-dimensional potential (), the time-independent Schrödinger equation can be written as + = Since this is an ordinary differential equation, there are two independent eigenfunctions for a given energy at most, so that the degree of degeneracy never exceeds two.
The density of states related to volume V and N countable energy levels is defined as: = = (()). Because the smallest allowed change of momentum for a particle in a box of dimension and length is () = (/), the volume-related density of states for continuous energy levels is obtained in the limit as ():= (()), Here, is the spatial dimension of the considered system and the wave vector.
Re-arranging the equation leads to =, where the energy factor E is a scalar value, the energy the particle has and the value that is measured. The partial derivative is a linear operator so this expression is the operator for energy: E ^ = i ℏ ∂ ∂ t . {\displaystyle {\hat {E}}=i\hbar {\frac {\partial }{\partial t}}.}
The energy of the particle is given by: where h is the Planck constant, m is the mass of the particle, n is the energy state (n = 1 corresponds to the ground-state energy), and L is the width of the well.
A classical particle with energy larger than the barrier height would always pass the barrier, and a classical particle with < incident on the barrier would always get reflected. To study the quantum case, consider the following situation: a particle incident on the barrier from the left side ( A r {\displaystyle A_{r}} ).