for momentum. position $\frac{2m_i}{?}(pL-L$vec$(p))-k\frac{r}{r},$ where vec$(p),$vec$(r),$ and vec$(L)$ denote the Hermitian operators or momentum. position, and angular momentum respectively. One can think of them as

vec$(r)=$hat$(x{)}_1$vec$(e{)}_1+$hat$(i{)}_2$vec$(e{)}_2+$hat$(r{)}_3$vec$(e{)}_3$

vec$(p)=$hat$(p{)}_1$vec$(e{)}_1+$hat$(p{)}_2$vec$(e{)}_2+$hat$(p{)}_3$vec$(e{)}_3$

vec$(L)=$hat$(L{)}_1$vec$(e{)}_1+$hat$(L{)}_2$vec$(e{)}_2+$vec$(L{)}_3$vec$(e{)}_3$

Here vec$(e{)}_i*vec(e{)}_j={}_{ij}.$ The components of vec$(r),$vec$(p),$ and vec$(L)$ are operators. A key property of the angular momentum operators is their commutation relation with the hat$(r{)}_i$ and hat$(p{)}_{,\text{o}\text{p}\text{e}\text{r}\text{a}\text{t}\text{o}\text{r}\text{s}\text{.}}$

Let $H=\frac{\text{v}\text{e}\text{c}(p{)}^2}{2m_c}-\frac{k}{r}$ be the Hamiltonian for the hydrogen atom, where $h=\frac{c^2}{\sqrt[\phantom{2}]{ma}},m{h}_e$ is the mass of an electron and $r=r^{}{r}^{}.$

$(1)$ Show that $[H,L_i]=0,[H,A_1]=0,i=x,y,z.$

$(2)$ Find out vec$(A)*vec(L)$ and vec$(A)*vec(A)$

$(3)$Define vec$(B)=\sqrt[\phantom{2}]{\frac{-m}{2E}}$vec$(A)$ and vec$(T)=\frac{1}{2}(vec(L)+$vec$(B)).$ Show that vec$(B)*vec(B)+$vec$(L)vec(L)=1$vec$(T)vec(T)=h^2-\frac{1^{2}me}{2E}$

$(4)$ Assuming $T^2|v$:$||,$ then using $(3)$ show that $E=\frac{-m*k^2}{2{}^2(2t+1{)}^2}$