Limit[((Sin[x])/x),x->0]

D[(x^2),x]

D[Exp[x],t]

D[Exp[a x], t]

D[Exp[a x],{x,4}]

D[HermiteH[6,x],{x,2}]

D[BesselJ[3,x],x]

Integrate[(x^3),x]

Integrate[(x^3),{x,0,1}]

Integrate[BesselJ[3,x],x]

Integrate[Exp[-(x^2)],{x,-∞,∞}]

Integrate[Exp[-a (x^2)],{x,-∞,∞}]

Integrate[Exp[-a (x^2)],{x,-∞,∞},Assumptions->{a>0}]

Integrate[Exp[-r Cos[θ]]Sin[θ](r^2),θ,r]

Integrate[Exp[-r Cos[θ]]Sin[θ](r^2),{θ,0,π},r]

Solve[2x+4==3,x]

Solve[{x+y==3,x-y==4},{x,y}]

Solve[{(x^2)-4==0},x]

sol=Solve[{x+(y^2)==2,(x^2)-(y^2)==0},{x,y}]

x+(y^2)/.sol[[1]]
(x^2)-(y^2)/.sol[[1]]

DSolve[D[y[x],x]==y[x],y[x],x]

DSolve[y'[x]==y[x],y[x],x]

sol=DSolve[{D[y[x],x]==y[x],y[0]==1},y[x],x]

y[x]

f=y[x]/.sol
f=y[x]/.sol[[1]]

Plot[f,{x,0,1}]

sol=DSolve[{vx'[t]==-ωc vy[t],vy'[t]==ωc vx[t],vz'[t]==0,vx[0]==vx0,vy[0]==vy0,vz[0]==vz0},{vx[t],vy[t],vz[t]},t]

sol=DSolve[{vx'[t]==-2 vy[t],vy'[t]==2 vx[t],vz'[t]==0,vx[0]==1,vy[0]==0,vz[0]==5},{vx[t],vy[t],vz[t]},t]

Plot[{vx[t]/.sol[[1]],vy[t]/.sol[[1]]},{t,0,20}]

ParametricPlot[{vx[t]/.sol[[1]],vy[t]/.sol[[1]]},{t,0,20}]

sol=DSolve[{x''[t]==-2 y'[t],y''[t]==2 x'[t],z''[t]==0,x[0]==0,y[0]==0,z[0]==0,x'[0]==1,y'[0]==0,z'[0]==0.5},{x[t],y[t],z[t]},t]

ParametricPlot3D[{x[t]/.sol[[1]],y[t]/.sol[[1]],z[t]/.sol[[1]]},{t,0,20}]

σz=PauliMatrix[3] ;
σx=PauliMatrix[1];
σy=PauliMatrix[2];

H=((ℏ ω0)/2)σz;
MatrixForm[H]
(* Ec. de Schrodinger es I (d/dt)ψ = H ψ  o I ℏ (d/dt)ψ - H ψ =0 *)

ψ={ψ1[t],ψ2[t]};

mat=I ℏ D[ψ,t]-H.ψ

eqs=Table[el==0,{el,mat}]

eqsn=eqs/.{ℏ->1,ω0->1}

ci={ψ1[0]==1,ψ2[0]==0}

deqsn=Join[eqsn,ci]

sol=DSolve[deqsn,ψ,t]

(* Sx=<ψ|((σx)/2)|ψ> *)

Sx=Conjugate[ψ].σx.ψ/.sol[[1]]
Sy=Conjugate[ψ].σy.ψ/.sol[[1]]
Sz=Conjugate[ψ].σz.ψ/.sol[[1]]

Simplify[Sz,Assumptions->{t>0}]

H=((ℏ ω0)/2)σz;
MatrixForm[H]
ψ={ψ1[t],ψ2[t]};
mat=I ℏ D[ψ,t]-H.ψ;
eqs=Table[el==0,{el,mat}];
eqsn=eqs/.{ℏ->1,ω0->1};
ci={ψ1[0]==(1/(Sqrt[2])),ψ2[0]==(1/(Sqrt[2]))};
deqsn=Join[eqsn,ci];
sol=DSolve[deqsn,ψ,t];
Sx=Simplify[Conjugate[ψ].σx.ψ/.sol[[1]],Assumptions->{t>0}]
Sy=Simplify[Conjugate[ψ].σy.ψ/.sol[[1]],Assumptions->{t>0}]
Sz=Simplify[Conjugate[ψ].σz.ψ/.sol[[1]],Assumptions->{t>0}]

ParametricPlot3D[{Sx,Sy,Sz},{t,0,8}]