Answer
Yes. No.
Work Step by Step
A particle in a box with a definite energy E does have a unique wavelength $\lambda=\frac{2 \pi}{k}$, where the energy E is related to k by equation 40.24.
The momentum has a definite magnitude, as we see from equation 40.31, but the momentum can be either negative or positive, so the particle is not in a state of definite momentum.
