(Solved): (a) Consider a quantum particle a one-dimensional square potential energy well of width a and depth ...

(a) Consider a quantum particle a one-dimensional square potential energy well of width $a$ and depth $V_0$ below the zero level which lies to the right of the well. At the left there is a barrier of infinite height. Beskou 'n kwantumdeeltjie in 'n vierkantige potensiële energieput met wydte a en diepte $V_0$ onder die nul-vlak wat na regs van die pute le.. Aan die linkerkant is danr ' $n$ versperring van 'n oneindige hoogte. Given: $\begin{array}{rl}\frac{d^2}{d{x}^2}?& =?k_0^2?,0<x<a;E?\frac{{?}^2{k}_0^2}{2m}?V_0\\ \frac{d^2}{d{x}^2}?& ?a^2?,a<x<+?E\\ ?& =?k_0\cot \left(k_{0}a\right).\end{array}$ 1. Obtain expressions for $k_0$ and $?$; ii. Derive the following relationship: $a^2+k_0^2=w^2$ where you should also find an expression for $w$; iii. Introduce dimensionless units expressing $k_0$ and the depth parameter $u$ in terms of $?/a_i$ iv. Show how the allowed energies for a particle in the well can be obtained through a graphical solution; v. Explain the effect of the depth of the well on the number of bound states; vi. What happens if $k_0<?/(2a)$.