problem 3

f=(((q[z]))^(-1))(E^(I k ((((x^2)+(y^2)))/(2 q[z]))))

2I k (∂_z)f+(∂_(x,x))f+(∂_(y,y))f//Simplify

problem 4

Exp[I k ((((x^2)+(y^2)))/(2 (z-I k (w0^2)/2)))]Exp[-I k ((((x^2)+(y^2)))/(2 (z+I k (w0^2)/2)))]//Simplify

Solve[-((2 (k^2) (w0^2) ((x^2)+(y^2)))/((k^2) (w0^4)+4 (z^2)))==-2 ((x^2)+(y^2))/(w^2),w]

(Sqrt[(((k^2) (w0^4)+4 (z^2))/(((k w0))^2))])//Simplify

Plot[(Sqrt[1+(z^2)]),{z,0,5},AxesLabel->{"z/(Subscript[z, R])","w/(Subscript[w, 0])"}]

problem 6 (BD 2.2b)

parts[u_,v_,n_]:=-D[v,{x,n-1}] D[u,x](*function to integrate by parts*)

part a

parts[(x^2)psic[x],psi[x],2]-parts[(x^2)psi[x],psic[x],2]//Simplify

part b

parts[x psic'[x],psi[x],2]-parts[x psi[x],psic[x],3]//Simplify

-psi'[x] psic'[x]+parts[psi[x], psic[x],2]

part c

parts[psic'[x],psi[x],3]-parts[psi'[x],psic[x],3]

problem 7 (BD 2.3)

Integrate[(((((Exp[((-(p^2))/(2 (σ^2)(ℏ^2)))])/(((π (σ^2)(ℏ^2)))^(1/4)))))^2)(p^2),{p,-∞,∞},Assumptions->{2 (σ^2)(ℏ^2)>0}]

Integrate[(((Exp[((-(p^2))/(2 (σ^2)(ℏ^2)))])/(((π (σ^2)(ℏ^2)))^(1/4))))((Exp[I p x/ℏ])/(((2π ℏ))^(1/2))),{p,-∞,∞},Assumptions->{ σ >0,ℏ>0}]

Integrate[((%)^2)(x^2),{x,-∞,∞},Assumptions->{ σ >0,ℏ>0}]

part b

Integrate[(((Exp[((-(p^2))/(2 (σ^2)(ℏ^2)))])/(((π (σ^2)(ℏ^2)))^(1/4))))((Exp[I p x/ℏ-I ((p^2)/(2 m ℏ))t])/(((2π ℏ))^(1/2))),{p,-∞,∞},Assumptions->{ σ >0,ℏ>0,m>0,t>0}]

Integrate[((E^(-((m (x^2) (σ^2))/(2 m+2 I t (σ^2) ℏ))))/((π^(1/4)) (Sqrt[(1/σ)+((I t σ ℏ)/m)])))((E^(-((m (x^2) (σ^2))/(2 m-2 I t (σ^2) ℏ))))/((π^(1/4)) (Sqrt[(1/σ)-((I t σ ℏ)/m)])))(x^2),{x,-∞,∞},Assumptions->{ σ >0,ℏ>0,m>0,t>0}]