Question

For a particle confined to an infinite square well, is it correct to say that each state of definite energy is also a state of definite wavelength? Is it also a state of definite momentum? Explain. (Hint:Remember that momentum is a vector.)

Short answer

Yes, it is correct to say that each state of definite energy is also a state of definite wavelength.

No, it is not correct to say that each state of definite energy is not the state of definite momentum.

Step-by-step solution

Momentum of particle confined in an infinite square well of width L

According to Heisenberg uncertainty principle, the uncertainty in position for a particle inside the square well of width L is leads to a corresponding uncertainty in momentum $\mathbf{\left(}\mathit{\Delta }{\mathit{p}}_{\mathbf{x}}\mathbf{\right)}$via relation

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

Particle inside the box in a state of definite energy has two sates of definite momentum corresponding to left and right moving momentum states

 Step 2: Wavelength of a particle in confined in infinite square well of width L

According to de Broglie’s hypothesis a matter particle in motion is also associated with waves.

The wavelength of associated wave with matter particle of momentum is given by relation

$\lambda =\frac{h}{p}$

Here, wavelength $\lambda$ is related to the magnitude of momentum p which is definite for both left and right moving particle.

For definite energy Eigen state n the wavelength of moving particle is given by relation

$\lambda =\frac{2L}{n}$

Conclusion

As, wavelength of left or right moving particle is depend on the magnitude of momentum of moving particle in infinite square well therefore wavelength of moving particle is definite for any definite energy state.

Hence, it is correct to say that each state of definite energy is also a state of definite wavelength

There is always a uncertainty exit in momentum of particle according to Heisenberg uncertainty rule and also it is direction dependent for left or right moving particle.

Hence, it is not correct to say that each state of definite energy is not the state of definite momentum.