Integrating out z direction

φ[z_,ℓ_]:=(1/((Sqrt[ℓ])(π^(1/4))))(E^(-(z^2)/(2(ℓ^2))));

Manipulate[Plot[((φ[z,ℓ])^2),{z,-2,2},PlotRange->{0,1.5}],{ℓ,0.5,2}]

Normalization

Simplify[Integrate[((φ[z])^2),{z,-∞,∞}],Assumptions->ℓ>0]

Kinetic Energy

Simplify[Integrate[φ[z]D[φ[z],z,z],{z,-∞,∞}],Assumptions->ℓ>0]

Trap Energy

Simplify[Integrate[(z^2)((φ[z])^2),{z,-∞,∞}],Assumptions->ℓ>0]

Contact Interactions

Simplify[Integrate[((φ[z])^4),{z,-∞,∞}],Assumptions->ℓ>0]

Quantum Fluctuations

Simplify[Integrate[((φ[z])^5),{z,-∞,∞}],Assumptions->ℓ>0]

Simplify[(Sqrt[(2/(5 (ℓ^3)(π^(3/2))))])==((Sqrt[(2/5)])/((π^(3/4)) (ℓ^(3/2)))),Assumptions->ℓ>0]

Energy

Simplify[Integrate[((D[φ[z],z])^2),{z,-∞,∞}],Assumptions->ℓ>0]

DDI

Fpar = -1+3(Sqrt[π])((qd^2)/q) (E^(q^2))Erfc[q];
Fperp = 2-3(Sqrt[π])q (E^(q^2))Erfc[q];

Infinite momentum limit

Large q, expand in 1/q

((qd^2)/q) (E^(q^2))Erfc[q]/.{q->(1/λ)}

Simplify[Series[(E^(1/(λ^2))) λ Erfc[(1/λ)],{λ,0,6}]//Normal,Assumptions->{λ>0}]/.{λ->(1/q)}//Expand

q (E^(q^2))Erfc[q]/.{q->(1/λ)}

Simplify[Series[(((E^(1/(λ^2))) Erfc[(1/λ)])/λ),{λ,0,6}]//Normal,Assumptions->{λ>0}]/.{λ->(1/q)}//Expand

Small momentum limit

Small q, replace qd^2= qx^2cos(η)^2+qy^2sin(η)^2, and then qx=q cos(η), qy = q sin(η)

(((Cos[η])^4)+((Sin[η])^4))Series[q (E^(q^2))Erfc[q],{q,0,4}]//Normal

Series[q (E^(q^2))Erfc[q],{q,0,4}]

At q=0

q (E^(q^2))Erfc[q]/.{q->0}

Limit[-1+3(Sqrt[π])q (E^(q^2))Erfc[q],{q->0}]
 Limit[ 2-3(Sqrt[π])q (E^(q^2))Erfc[q],{q->0}]

Integral

Integrate[(Sqrt[π])(q^1) (E^(q^2))Erfc[q],q]

NIntegrate[-3+3(Sqrt[π])q (E^(q^2))Erfc[q],{q,0,7000}]

Plot[-3+3(Sqrt[π])q (E^(q^2))Erfc[q],{q,0,25},PlotRange->All]

Two params

ψ[z_]:=(Sqrt[(ν/(2Gamma[(1/ν)]ℓ))])(E^(((-1)/2)(((((Abs[z])/ℓ)))^ν)));
ψp[z_]:=(Sqrt[(ν/(2Gamma[(1/ν)]ℓ))])(E^(((-1)/2)((((z/ℓ)))^ν)));
ψm[z_]:=(Sqrt[(ν/(2Gamma[(1/ν)]ℓ))])(E^(((-1)/2)(((((-z)/ℓ)))^ν)));

Normalization

Integrate[((ψ[z])^2),{z,-∞,∞},Assumptions->{ν>0,ℓ>0}]

Kinetic Energy

Integrate[ψp[z]D[ψp[z],z,z],{z,0,∞},Assumptions->{ν>0,ℓ>0}]+Integrate[ψm[z]D[ψm[z],z,z],{z,-∞,0},Assumptions->{ν>0,ℓ>0}]

((ℏ^2)/(2m))(Integrate[D[ψp[z],z]D[ψp[z],z],{z,0,∞},Assumptions->{ν>0,ℓ>0}]+Integrate[D[ψm[z],z]D[ψm[z],z],{z,-∞,0},Assumptions->{ν>0,ℓ>0}])

Potential Energy

(1/2)m (ω^2) Integrate[(z^2)((ψ[z])^2),{z,-∞,∞},Assumptions->{ν>0,ℓ>0}]

Contact interactions

Integrate[((ψ[z])^4),{z,-∞,∞},Assumptions->{ν>0,ℓ>0}]

Integrate[((ψ[z])^5),{z,-∞,∞},Assumptions->{ν>0,ℓ>0}]

DDI

φ[z]

Simplify[FourierTransform[φ[z],z,k],Assumptions->ℓ>0]

((((((E^(-(1/2) (k^2) (ℓ^2))) (Sqrt[ℓ]))/(π^(1/4)))))^2)

Integrate[(((E^(-(k^2) (ℓ^2))) ℓ)/(Sqrt[π])),{k,-∞,∞},Assumptions->{ℓ>0}]

Integrate[(((E^(-(k^2) (ℓ^2))) ℓ)/(Sqrt[π])),{k,-∞,∞},Assumptions->{ℓ>0}]
Integrate[((E^(-(1/4) (k^2) (ℓ^2)))/((Sqrt[2 π]) (Sqrt[(1/(ℓ^2))]) ℓ)),{k,-∞,∞},Assumptions->{ℓ>0}]

Manipulate[Plot[(((ν^2) Gamma[2-(1/ν)])/(8(ℓ^2) Gamma[(1/ν)]))+((((444/100)))^2)(((ℓ^2) Gamma[(3/ν)])/(2 Gamma[(1/ν)])),{ν,0.5,16},PlotRange->{0,10}],{ℓ,0.2,5}]

(((ν^2) Gamma[2-(1/ν)])/(8 (ℓ^2)Gamma[(1/ν)]))+gz(((ℓ^2) Gamma[(3/ν)])/(2 Gamma[(1/ν)]))/.{ℓ->4,ν->6,gz->((((167/100)))^2)}//N

D[(((ν^2) Gamma[2-(1/ν)])/(8(ℓ^2) Gamma[(1/ν)]))+(g^2)(((ℓ^2) Gamma[(3/ν)])/(2 Gamma[(1/ν)])),ν]

Thomas Fermi profile

ζ[z_,Z_]:=((Sqrt[15])/(4(Sqrt[Z])))(1-((((z/Z)))^2));

Integrate[((ζ[z,Z])^2),{z,-Z,Z},Assumptions->{Z>0}]

φ[z]

Beyond TF

BTF[z_,Z_,Q_]:=(((((4(Q^2) Z)/(1+3 Q+2 (Q^2)))))^(-1/2))(1-((((z/Z)))^Q));

Integrate[((BTF[z,Z,Q])^2),{z,0,Z},Assumptions->{Z>0,Q>0}]

Manipulate[Plot[{((ζ[z,Z])^2)(HeavisideTheta[z+Z]-HeavisideTheta[z-Z]),((BTF[Abs[z],Z,Q])^2)(HeavisideTheta[z+Z]-HeavisideTheta[z-Z])},{z,-3,3},PlotRange->{0,3}],{Z,0.2,3},{Q,2,8}]

2D profiles

Thomas-Fermi

ζ2D[x_,X_,y_,Y_]:=(Sqrt[(3/(π X Y))])(1-((((x/X)))^2)-((((y/Y)))^2));

Integrate[((ζ2D[x,X,y,Y])^2),{x,-X,X},{y,-Y(Sqrt[1-((x^2)/(X^2))]),Y(Sqrt[1-((x^2)/(X^2))])},Assumptions->{X>0,Y>0}]

Gaussian

φxy[x_,ℓx_,y_,ℓy_]:=(1/((Sqrt[ℓx])(π^(1/4))))(E^(-(x^2)/(2((ℓx)^2))))(1/((Sqrt[ℓy])(π^(1/4))))(E^(-(y^2)/(2((ℓy)^2))));

Integrate[((φxy[x,ℓx,y,ℓy])^2),{x,-∞,∞},{y,-∞,∞},Assumptions->{ℓx>0,ℓy>0}]

Quartic

Qxy[x_,ℓx_,y_,ℓy_]:=(1/(Sqrt[4 ℓx ℓy ((Gamma[(5/4)])^2)]))(E^(-(x^4)/(2((ℓx)^4))))(E^(-(y^4)/(2((ℓy)^4))));
Integrate[((Qxy[x,ℓx,y,ℓy])^2),{x,-∞,∞},{y,-∞,∞},Assumptions->{ℓx>0,ℓy>0}]

Integrate[((Qxy[x,26,y,26])^2),{x,-50,50},{y,-50,50},Assumptions->{ℓx>0,ℓy>0}]//N
Integrate[((Qxy[x,30,y,30])^2),{x,-50,50},{y,-50,50},Assumptions->{ℓx>0,ℓy>0}]//N

Manipulate[Plot[{((ζ2D[x,2.5X,0,1])^2),((φxy[x,X,0,1])^2),((Qxy[x,X,0,1])^2)},{x,-2,2},PlotRange->{0,1}],{X,1,2}]

test = (E^(-((x^4)+(x^2))/(2((ℓx)^4))))(E^(-((y^4)+(y^2))/(2((ℓy)^4))));

Integrate[(test^2),{x,-∞,∞},{y,-∞,∞},Assumptions->{ℓx>0,ℓy>0}]

500/(((12π))^2)//N

0.08*(((12π))^2)

444/137.0

74.2/6

(Sqrt[((1.054571817×(10^(-34)))/(164*1.66053906660*(10^(-27))*2π*444))])*(10^6)

((5*(10^(-6)))/(131.0*5.291772109*(10^(-11))))