Question

Compare the ground-state energy of a particle in a one-dimensional box of length l with that of a particle in the three-dimensional box.

Answer 1

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The simplest form of the particle in a box model considers a one-dimensional system. Here, the particle may only move backwards and forwards along a straight line with impenetrable barriers at either end.[1] The walls of a one-dimensional box may be visualised as regions of space with an infinitely large potential energy. Conversely, the interior of the box has a constant, zero potential energy.[2] This means that no forces act upon the particle inside the box and it can move freely in that region. However, infinitely large forces repel the particle if it touches the walls of the box, preventing it from escaping. The potential energy in this model is given as

{\displaystyle V(x)={\begin{cases}0,&x_{c}-{\tfrac {L}{2}}<x<x_{c}+{\tfrac {L}{2}},\\\infty ,&{\text{otherwise,}}\end{cases}},} {\displaystyle V(x)={\begin{cases}0,&x_{c}-{\tfrac {L}{2}}<x<x_{c}+{\tfrac {L}{2}},\\\infty ,&{\text{otherwise,}}\end{cases}},}

where L is the length of the box, xc 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 (xc = 0 ) and the shifted box (xc = L/2 ).

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Answer 2

Answer:

The potential energy is 0 inside the box (V=0 for 0<x<L) and goes to infinity at the walls of the box (V=∞ for x<0 or x>L). We assume the walls have infinite potential energy to ensure that the particle has zero probability of being at the walls or outside the box

Explanation:

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