Measuring Energy and Linear Momentum Consecutively

[Domnu]

Problem
Can the total energy and linear momentum of a particle moving in one dimension in a constant potential field be measured consecutively with no uncertainty in the values obtained?

Solution
Yes, this is possible. The energy and linear momentum operators are, respectively

$$\hat{H} = -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} + V$$

$$\hat{p} = -i\hbar\frac{\partial}{\partial x}$$

Now, note that these two operators commute. Because of this, they share the same eigenstates. Now, once the total energy or linear momentum is measured, the wavefunction of the state collapses into an eigenstate of the measured observable. Now, this eigenstate is also an eigenstate of the observable which hasn't been measured yet. Therefore, we can precisely measure this unmeasured observable as well. $$\blacksquare$$

Are my arguments correct?

 

[dx]

You can also directly tell from the fact that the energy in a constant potential field is a function of momentum only. If you've measured momentum, you know the energy too. And yes, your argument is correct.