In order to introduce my question we consider first the free electron, in a box width L, with eigenfunctions and eingenvalues given by $$\psi_k(x)=\frac1{\sqrt{L}}e^{ikx}$$ and $$E_k=\hbar^2k^2/2m$$ The plane waves are eigenfunctions of the operator $$p=-i\hbar d/dx$$ with eigenvalue $$\hbar k$$ which represents the momentum of the free electron. Obviously the $$\lt p\gt_k=p=\hbar k$$ Now, I have some difficulty with the following
- The Ehrenfest theorem tells us that: $$\lt p\gt=m\frac d{dt}\lt x\gt = m\frac d{dt} (L/2)$$ but on a stationary state, this derivative is vanishing! So: $$\lt p\gt_k=0\neq\hbar k$$
- If $$\psi(x)_k$$ is an eigenstate of operator p, then $$\Delta p=0$$ This implies total incertainty on x, i.e. $$\Delta x=\infty$$ ... but we are in a box large L, so I know that the maximum standard deviation for position is $$\Delta =L/2$$
Could someone extricate me these inconsistencies that I can not explain? I thought that these states are not physically achievable, therefore it doesn't make much sense to ask these questions, but they conflict with my knowledge of quantum mechanics. I found this doubt by studying the band theory and even then I cannot understand how the average speed of an electron in a steady state of Bloch, (n (band index), k (vector wave in fbz)), is not zero. The only solution I come up with is that they are not really states of the moment, but only of the modulus of the moment and so each of these waves must consider both p and -p ... this would solve both questions.
Thank you for your kind reading and attention!
