Final answer:
The normalization constant k for the wavefunction ψ(x)=kx is √3. The probability of finding the particle between x=0.3 and x=0.6 is 0.09.
Step-by-step explanation:
To normalize the wavefunction ψ(x)=kx for a particle confined to the x-axis between x=0 and x=1 and ψ(x)=0 elsewhere, we need to find the value of constant k such that the integral of ψ(x)*ψ*(x) from 0 to 1 is equal to 1. This integral represents the probability of finding the particle in that interval, which should add up to 1 (certainty) for the wavefunction to be normalized. The integral ∫ ψ(x)*ψ*(x) dx from 0 to 1 is equal to ∫ k^2 ⋅ x^2 dx, which when evaluated from 0 to 1, gives us ⅓ k^2. Equating this to 1 and solving for k gives us k = √3.
For part (b), the probability of finding the particle between x=0.3 and x=0.6 is given by the integral of ψ(x)*ψ*(x) over the interval [0.3, 0.6]. This equates to ∫ from 0.3 to 0.6 of (kx)^2 dx. Substituting k = √3 and integrating, we find the probability to be 0.09.