Normalizing a wavefunction and probability help

[moaharris2004]

Good afternoon I need help with a question in my physical chemistry class.
A particle having mass m is described as having the (unnormalized) wavefunction ψ=k, where k is some constant, when confined to an interval in one dimension; that interval having length a (that is, from x=0 to a). What is the probability that the particle will exist in the first third of the interval, that is from x=0 to (1/3)a?

I know that the wavefunction needs to be normalized first.
ψ=Nψ
∫Nψ* x Nψdx
∫N*k (Nk)dx
limit 1/3a to 0∫N*N (k^2)dx
This is where I keep getting stuck because I don't know where to go from here since the wavefunction is only a constant. If someone could guide me I would greatly appreciate it. Thank You in advance.

 

[PaulDirac]

I can't imagine how such a problem can exist in which a particle is confined within an interval with a constant wave function. but anyway, since the WF is a constant it does imply that the probability of finding the particle within the interval (0, a) is constant. If you normalize the WF you'll get a constant like 1/sqrt(a). Integrating from 0 to 1/3a you get a 1/3.

 

[Nugatory]

To normalize the wave function, you multiply it by a particular constant, namely the square root of the reciprocal of the integral from $-\infty$ to $\infty$ of $\psi^*\psi$. (For your wave function, that integral is unusually easy ).

Determine the value of that normalization constant $N$ and your normalized wave function will look something like $\psi_N=N\psi=Nk$ in the region where it is non-zero.