Double well potential.  ps7 p1.

ClearAll[ m, omega, a, v, phi, anought, hbar, psipm, s, x , p, inner, psiPlus, psiMinus, cp, cm, averageE, diffE,hpsiPlus, hpsiMinus,  eKineticPhiPlusExpectation, eKineticPhiMinusExpectation,ePotentialPhiPlusExpectation, ePotentialPhiMinusExpectation]

(* declare various variables as Real and > 0 *)
$Assumptions=And@@(#>0&)/@{m, omega, a, anought, hbar};

(* V(x) *)
v := (m omega^2/8/a^2) (# + a)^2 (#-a)^2 &;

(* ϕ(x) : without normalization. *)
phi := (1/Pi/anought^2)^(1/4) Exp[ - #^2/2/anought^2] & ;

(* (Subscript[ψ, ±])(x) = ϕ(x-a) ± ϕ(x+a) : without normalization *)
psipm[x_, s_] := (phi[x-a] + s phi[x+a] ) ;

(* (SubPlus[ψ])(x), (SubMinus[ψ])(x) *)
psiPlus = psipm[ #, 1] & ;
psiMinus = psipm[ #, -1] & ;

(* <f | g> *)
inner[f_, g_] := Integrate[ f[x] g[x] , {x, -Infinity, Infinity}]

(* <(Subscript[ψ, ±])|(Subscript[ψ, ±])> *)
cp = inner[ psiPlus, psiPlus ] // FullSimplify ;
cm = inner[ psiMinus, psiMinus ] // FullSimplify ;

(* ((p^2)/(2 m)) (Subscript[ψ, ±])(x) *)
hpsiPlus[x_] = (-hbar^2/2/m) D[psiPlus[x], {x,2}]  ;
hpsiMinus[x_] = (-hbar^2/2/m) D[psiMinus[x], {x,2}] ;

(* <(Subscript[ψ, ±])|((p^2)/(2 m))|(Subscript[ψ, ±])> *)
eKineticPhiPlusExpectation = inner[ psiPlus, hpsiPlus[#] &] // FullSimplify;
eKineticPhiMinusExpectation = inner[ psiPlus, hpsiPlus[#] &] // FullSimplify ;

(* <(Subscript[ψ, ±])|V(x)|(Subscript[ψ, ±])> *)
ePotentialPhiPlusExpectation = inner[ psiPlus[#] psiPlus[#]&, v] // FullSimplify ;
ePotentialPhiMinusExpectation = inner[ psiMinus[#] psiMinus[#]&, v ] // FullSimplify ;

(*  (Subscript[(OverBar[E]), ±]) = <(Subscript[ψ, ±])|H |(Subscript[ψ, ±])>/<(Subscript[ψ, ±])|(Subscript[ψ, ±])>, in multiples of ℏω *)
averageE = {(eKineticPhiPlusExpectation + ePotentialPhiPlusExpectation)/cp,
(eKineticPhiMinusExpectation + ePotentialPhiMinusExpectation)/cm} /hbar/omega /. anought -> Sqrt[hbar/m/omega]// Simplify;

(* (SubPlus[(OverBar[E])]) - (SubMinus[(OverBar[E])]), in multiples of ℏω *)
diffE = ((averageE // First) - (averageE // Last)) // FullSimplify ;

ClearAll[ display ]
display = {anought -> (Subscript[a, 0]), hbar -> ℏ, omega -> ω} ;

 {#1," = ",#2/. display // TraditionalForm } &@@@{
{"<x|(SubPlus[ψ])>", psiPlus[x]  // Simplify},
{"<x|(SubMinus[ψ])>", psiMinus[x] // Simplify},
{"<(SubPlus[ψ])|\!\(\*FractionBox[SuperscriptBox[\(p\), \(2\)], \(2\\ m\)]\)|(SubPlus[ψ])>", eKineticPhiPlusExpectation  },
{"<(SubMinus[ψ])|\!\(\*FractionBox[SuperscriptBox[\(p\), \(2\)], \(2\\ m\)]\)|(SubMinus[ψ])>",  eKineticPhiMinusExpectation},
{"<(SubPlus[ψ])|V(x)|(SubPlus[ψ])>", ePotentialPhiPlusExpectation  },
{"<(SubMinus[ψ])|V(x)|(SubMinus[ψ])>",  ePotentialPhiMinusExpectation},
{"<(SubPlus[ψ])|(SubPlus[ψ])>", cp  },
{"<(SubMinus[ψ])|(SubMinus[ψ])>", cm },
{ "((SubPlus[(OverBar[E])])/(ℏω)) = \!\(\*FractionBox[\(\(\(<\)\*SubscriptBox[\(ψ\), \(+\)]\) | H\\  | \(\*SubscriptBox[\(ψ\), \(+\)]\(>\)\)\), \(\(\(<\)\*SubscriptBox[\(ψ\), \(+\)]\) | \*SubscriptBox[\(ψ\), \(+\)] > ℏω\)]\)",(averageE// First) },
{ "((SubMinus[(OverBar[E])])/(ℏω)) = \!\(\*FractionBox[\(\(\(<\)\*SubscriptBox[\(ψ\), \(-\)]\) | H\\  | \(\*SubscriptBox[\(ψ\), \(-\)]\(>\)\)\), \(\(\(<\)\*SubscriptBox[\(ψ\), \(-\)]\) | \*SubscriptBox[\(ψ\), \(-\)] > ℏω\)]\)",(averageE // Last) },
{ "(1/(ℏω))((SubPlus[(OverBar[E])]) - (SubMinus[(OverBar[E])]))", diffE }
}  // Grid

Final result (after some manual manipulation) :

(SubPlus[(OverBar[E])])-(SubMinus[(OverBar[E])]) = ((ℏ ω)/8) ((((a^2)/(Subscript[a, 0]^2))-5 )exp(((a^2)/(Subscript[a, 0]^2)))+4 ((a^2)/(Subscript[a, 0]^2))-2 ) (coth(((a^2)/(Subscript[a, 0]^2)))-1)