Given in the figures are examples of some potential energy functions in one dimension. The total energy of the particle is indicated by a cross on the ordinate axis. In each case,specify the regions,if any,in which the particle cannot be found for the given energy. Also,indicate the minimum total energy the particle must have in each case. Think of simple physical contexts for which these potential energy shapes are relevant.

(N/A) The total energy of a system is given by $E = P.E. + K.E.$,which implies $K.E. = E - P.E.$. Since kinetic energy $(K.E.)$ must be non-negative,the particle cannot exist in regions where $P.E. > E$.
$(i)$ For the first figure: The particle cannot exist in the region $x > a$ because $V(x) = V_0 > E$. The minimum total energy required is $0$.
(ii) For the second figure: The potential energy $V(x)$ is greater than $E$ in all regions shown. Thus,the particle cannot exist in any of these regions. The minimum total energy required is $V_0$.
(iii) For the third figure: The particle cannot exist in regions where $V(x) > E$. Here,$V(x) = V_0$ for $x < a$ and $x > b$. Thus,the particle is confined to $a < x < b$. The minimum total energy required is $-V_1$.
(iv) For the fourth figure: The particle cannot exist where $V(x) > E$. Based on the graph,this occurs for $x < -b/2$,$-a/2 < x < a/2$,and $x > b/2$. The particle can only exist in the regions $-b/2 < x < -a/2$ and $a/2 < x < b/2$. The minimum total energy required is $-V_1$.