Question

Consider the Sch ·· 0dinger equation 

i ∂Ψ/∂t = − ∂²Ψ/∂x², Ψ = Ψ(x, t), 

which holds for all values of x and all positive values of t. Let the initial value of Ψ(x, t) be 

Ψ(x, 0) = f(x) 

and let the boundary conditions for Ψ be 

Ψ(±∞, t) = 0. 

(a) Make a Fourier transform of the Sch ·· 0dinger equation with respect to x and solve the resulting ordinary differential equation. 

(b) Express the solution Ψ(x, t) satisfying the initial condition (B) in the form of a Fourier integral. 

Answer 1

a. After making the Fourier transform, we have