Finding psi(x) from A(k) = N/(k² + a²)

[Felicity]

Homework Statement

this problem is from (chapter 2 problem 1, gasiorowicz, quantum physics) where I am given
A(k) = N/(k2+a2) and must calculate psi(x)

Homework Equations

I am using the equation
psi(x,t) = integral from - infinity to + infinity A(k) ei(kx-wt) dk

which when t=0 goes to

psi(x,t) = integral from - infinity to + infinity A(k) eikx dk

The Attempt at a Solution

integral from - infinity to + infinity of
N/(k2+a2) * eikx dk

is this a good approach? How can I solve the integral?

any help would be greatly appreciated

thank you

 

[nrqed]

Homework Statement

this problem is from (chapter 2 problem 1, gasiorowicz, quantum physics) where I am given
A(k) = N/(k2+a2) and must calculate psi(x)

Homework Equations

I am using the equation
psi(x,t) = integral from - infinity to + infinity A(k) ei(kx-wt) dk

which when t=0 goes to

psi(x,t) = integral from - infinity to + infinity A(k) eikx dk

The Attempt at a Solution

integral from - infinity to + infinity of
N/(k2+a2) * eikx dk

is this a good approach? How can I solve the integral?

any help would be greatly appreciated

thank you

yes it is the right approach.
just write
k2 + a2 as (k+ia)(k-ia)
and then do the integral in the complex k plane, closing above the real k axis so that eikx goes to zero and use the residue theorem

 

[Felicity]

Thank you so much! would the residue then be e^-ax/2ai ?