(* new attempt at 24.4.3.  Get a different answer *)
Clear["Global`*"](*Setting default environment*)

$Assumptions = b > 0 && ω > 0 && ℏ > 0 && m > 0 

psi[r_] = (Sqrt[b]) ((E^(b r)) HeavisideTheta[-r]+(E^(-b r)) HeavisideTheta[r]) ;
psiA[r_] = Sqrt[b] E^(-b Abs[r]) ;

(* check normalization *)
Integrate[ psi[r]^2, {r, -Infinity, Infinity}] ;
Integrate[ psiA[r]^2, {r, -Infinity, Infinity}] ;

a = -ℏ^2/(2m) ;
hPsi[r_] = -a D[ D[ psi[r], r], r] + (1/2) m ω^2 r^2 // FullSimplify
hPsiA[r_] = hPsi[r] /. ((E^(2b r)) HeavisideTheta[-r]+HeavisideTheta[r] ) -> E^(b r)psiA[r] /Sqrt[b]

eIntegrand[r_] = psiA[r] hPsiA[r] // FullSimplify


e = Integrate[ eIntegrand[r], {r, -Infinity, Infinity}] ;
(*e = FullSimplify[e, b > 0]*)
de[b] = D[e,b]
s= Solve[de[b]==0,b]

psiA[r] /. b -> (((5^(2/9)) (m^(4/9)) (ω^(4/9)))/(ℏ^(4/9)))

e /. b -> (((5^(2/9)) (m^(4/9)) (ω^(4/9)))/(ℏ^(4/9))) // FullSimplify