Pregunta: V(x)=∞ if ∣x∣>a+bV(x)=0 if a≤∣x∣≤a+bV(x)=V0 if ∣x∣

$\begin{array}{l}V(x)=\infty \text{ if }|x|>a+b\\ V(x)=0\text{ if }a\le |x|\le a+b\\ V(x)=V_0\text{ if }|x|<a\end{array}$ (a) Show that the energy eigenvalues for this potential are given by the equations: $\tan bk=-\frac{k}{\kappa }\coth a\kappa$, for even solutions, and $\tan bk=-\frac{k}{\kappa }\tanh a\kappa$, for odd solutions with, $k=\sqrt{\frac{2mE}{{\hslash }^2}}$ and $\kappa =\sqrt{\frac{2m\left(V_0-E\right)}{{\hslash }^2}}$ (b) Study the solutions where $V_0\to \infty$ and show that the odd and even solutions give the energy spectrum of an "ordinary box", corresponding to two atoms which are not interacting. (c) Also, for the solutions where $V_0\to E$, show that the odd and even solutions give slightly different energy levels, such that the energy spectrum will consist of pairs of close energy levels. For many such adjacent boxes this would give a band of energies. (d) Show that $a\to 0$ gives the same energy spectrum as an ordinary box with width $2b$. $\begin{array}{l}V(x)=\infty \text{ if }|x|>a+b\\ V(x)=0\text{ if }a\le |x|\le a+b\\ V(x)=V_0\text{ if }|x|<a\end{array}$ (a) Show that the energy eigenvalues for this potential are given by the equations: $\tan bk=-\frac{k}{\kappa }\coth a\kappa$, for even solutions, and $\tan bk=-\frac{k}{\kappa }\tanh a\kappa$, for odd solutions with, $k=\sqrt{\frac{2mE}{{\hslash }^2}}$ and $\kappa =\sqrt{\frac{2m\left(V_0-E\right)}{{\hslash }^2}}$ (b) Study the solutions where $V_0\to \infty$ and show that the odd and even solutions give the energy spectrum of an "ordinary box", corresponding to two atoms which are not interacting. (c) Also, for the solutions where $V_0\to E$, show that the odd and even solutions give slightly different energy levels, such that the energy spectrum will consist of pairs of close energy levels. For many such adjacent boxes this would give a band of energies. (d) Show that $a\to 0$ gives the same energy spectrum as an ordinary box with width $2b$.