Chapter 4: Quantum Mechanics in Three Dimensions

(a) From the definition (Equation 4.46), construct ${\mathbf{n}}_{\mathbf{1}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}$and${\mathbf{n}}_{\mathbf{2}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}$.

(b) Expand the sines and cosines to obtain approximate formulas for${\mathbf{n}}_{\mathbf{1}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}$androle="math" localid="1656329588644" ${\mathbf{n}}_{\mathbf{2}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}$, valid when$\mathbf{x}\mathbf{\ll }\mathbf{1}\mathbf{.}\mathbf{}$. Confirm that they blow up at the origin.

a) Check that $\mathit{A}\mathit{r}{\mathit{j}}_{\mathbf{1}}\left(kr\right)$satisfies the radial equation with $\mathit{V}\left(r\right)\mathbf{=}\mathbf{0}$and $\mathit{I}\mathbf{=}\mathbf{1}$.

(b) Determine graphically the allowed energies for the infinite spherical well, when $\mathit{I}\mathbf{=}\mathbf{1}$. Show that for large $\mathit{n}\mathbf{,}{\mathit{E}}_{\mathbf{n}\mathbf{1}}\mathbf{\approx }\left({\sout{h}}^2{\pi }^2/2{\mathrm{ma}}^2\right){\left(n+1/2\right)}^{\mathbf{2}}$. Hint: First show that ${\mathit{j}}_{\mathbf{1}}\left(x\right)\mathbf{=}\mathbf{0}\mathbf{\Rightarrow }\mathit{x}\mathbf{=}\mathbf{tan}\mathit{x}$. Plot xand$\mathbf{tan}\mathit{x}$on the same graph, and locate the points of intersection.

A particle of mass m is placed in a finite spherical well:

$\mathbf{V}\mathbf{\left(}\mathbf{r}\mathbf{\right)}\mathbf{=}\{\begin{array}{l}-V_0,r\le a;\\ 0,r>a;\end{array}$

Find the ground state, by solving the radial equation with$\mathcal{l}\mathbf{=}\mathbf{0}$. Show that there is no bound state if ${\mathit{V}}_{\mathbf{0}}{\mathit{a}}^{\mathbf{2}}\mathbf{<}{\mathbf{\pi }}^{\mathbf{2}}{\mathcal{k}}^{\mathbf{2}}\mathbf{/}\mathbf{8}\mathit{m}$.