No, a wave function that is normalized in the position basis is not automatically normalized in the momentum basis. The position and momentum bases are related by a mathematical transformation called Fourier transform, and the normalization of a wave function in one basis does not necessarily imply normalization in the other basis.
The Fourier transform is defined as follows for a wave function ψ(x) in the position basis:
ψ(p) = (1/√(2πħ)) ∫ ψ(x) e^(-ipx/ħ) dx
where ψ(p) is the wave function in the momentum basis, p is the momentum, x is the position, and ħ is the reduced Planck's constant.
The normalization condition in the momentum basis is given by:
∫ |ψ(p)|^2 dp = 1
To determine if a wave function normalized in the position basis is also normalized in the momentum basis, you would need to calculate the integral of the squared magnitude of the wave function in the momentum basis and verify if it equals 1. Simply normalizing a wave function in one basis does not guarantee its normalization in the other basis, as the transformation between the bases involves additional factors that can affect the normalization.