How do I solve the integral from negative infinity to infinity of f(x,y)dx ?

Question

This is for a schrodinger's equation problem (normalizing a waveform) but I haven't taken calculus 3 and when I search online I get results for solving double integrals, which doesn't seem to be the same thing

edit: the equation f(x) that I need to normalize is f(x) = A exp (−λ|x|) exp (−iωt)

Answer 1

I assume what you have here is a quantum mechanical wave function, so the integrand for normalization is ψ*ψ.  Assuming the normalization factor is A, you want

 

∫_-∞^∞ ψ*ψdx = 1

 

ψ*ψ = |A|²e^-2λ|x|

 

(Note the time dependence went away, since it was in the phase.)

 

Look at e^-2λ|x|.  

 

e^-2λ|x| = e^2λx fox x < 0

e^-2λ|x| = e^-2λx for x > 0

 

Break up the integral into 2 regions:

 

-∞ < x < 0

 

and

 

0 < x < ∞.

 

Then evaluate the integrals for the two regions separately, with the correct form of e^-2λ|x| in each region, add the results, and solve for A.  Try it.  Be careful with signs when you evaluate the definite integrals.   Post a Comment or send me an email if you still have questions.