Sakurai.  Problem 5.11 (c).  Verify hand calculations (diagonalization)

ClearAll[ u, h, v, delta, lambda,e, display, d]

display = {e -> "(Subscript[E, 1])",lambda -> λ, delta -> Δ} ;
u = (1/Sqrt[2]) {{1,1},{1,-1}} ;
h = {{e,0}, {0,e}} ;
v = lambda delta {{0,1},{1,0}} ;
d := (# /. display // MatrixForm) & ;

{#1," = ",#2//d } &@@@{
{"(Subscript[H, 0])", h},
{"V", v},
{"U", u},
{"\!\(\*SuperscriptBox[\(U\), \(\(†\)\(\\ \)\)]\)(Subscript[H, 0]) U", u . h . u},
{"\!\(\*SuperscriptBox[\(U\), \(\(†\)\(\\ \)\)]\)V U", u . v . u}
} // Grid