For a particle in a finite potential well, is it correct to say that each bound state of definite energy is also a state of definite wavelength? Is it a state of definite momentum? Explain.

In order to have a definite wavelength of particle, the wave function of particle has to be purely sinusoidal in nature.

According to Heisenberg uncertainty principle, the uncertainty in position for a particle inside the finite well of width L is leads to a corresponding uncertainty in momentum $\left(\Delta p_x\right)$via relation

$$\begin{gathered} L\Delta p_x\ge \frac{h}{4\pi } \\ \Delta p_x\ge \frac{h}{4\pi L} \end{gathered}$$

As, the wave function of a particle in finite potential well is sinusoidal inside the well and exponential outside the well.

Therefore it does not form a definite wavelength of a particle inside a finite potential well.

As most of the part of wave function is confined inside the well that leads to the uncertainty of momentum of the particle.

Hence, there is neither a definite state of momentum or definite state of energy for each bound state of definite energy.