Normalizing the angular wave function Φ(φ) = e imφ

Question

How do you normalize the angular wave function Φ(φ) = e imφ?

 

My kindest regards!

Answer 1

If no other information is available, I suggest normalizing it over the domain of φ.  I assume that φ is the azimuthal angle, and hence its domain is [0,2π].  

 

A = normalization factor

 

Normalization condition:

 

∫₀^2π (Ae^imφ)^*(Ae^imφ)dφ = 1

 

|A|² ∫₀^2π e^{i(-m + m)φ}dφ = 1

 

|A|² ∫₀^2π dφ = 1

 

|A|²2π = 1

 

|A| = 1/√(2π)

 

Normalization factors are positive by convention, so

 

A = |A| = 1/√(2π)