# Why doesn't the expectation of position for a plane wave obey kinematics?

Consider the plane wave: $$\Psi = Ne^{i(\vec{p}\cdot\vec{r} - Et)/\hbar}$$ with N is the normalisation factor.

The expectation value of momentum for this wave is: \begin{align} \langle\vec{p}\rangle &= \int_\text{all space} \Psi^{*} \hat{p} \Psi\ \mathrm{d}V \\ &= \int_\text{all space} Ne^{-i(\vec{p}\cdot\vec{r} - Et)/\hbar} (-i\hbar \nabla) Ne^{i(\vec{p}\cdot\vec{r} - Et)/\hbar}\ \mathrm{d}V \\ &= -i\hbar N^2\int_\text{all space} e^{-i(\vec{p}.\vec{r} - Et)/\hbar} (i\vec{p}/\hbar) e^{i(\vec{p}\cdot\vec{r} - Et)/\hbar}\ \mathrm{d}V \\ &= (-i\hbar)(i\vec{p}/\hbar) N^2\int_\text{all space} e^{-i(\vec{p}\cdot\vec{r} - Et)/\hbar}e^{i(\vec{p}\cdot\vec{r} - Et)/\hbar}\ \mathrm{d}V \\ &= (-i\hbar)(i\vec{p}/\hbar) 1 \\ \langle\vec{p}\rangle &= \vec{p} \end{align}

Everything seems to be OK by now. Lets now find the expectation value of position for this wave: \begin{align} \langle\vec{r}\rangle &= \int_\text{all space} \Psi^{*} \hat{r} \Psi\ \mathrm{d}V \\ &= \int_\text{all space} Ne^{-i(\vec{p}\cdot\vec{r} - Et)/\hbar} (\vec{r}) Ne^{i(\vec{p}\cdot\vec{r} - Et)/\hbar}\ \mathrm{d}V \end{align} By symmetry of the integration, $\langle\vec{r}\rangle = \vec{0}$.

We should have (from Ehrenfest theorem): $$\frac{\mathrm{d}\langle\vec{r}\rangle}{\mathrm{d} t} = \frac{\langle\vec{p}\rangle}{m}$$ However this is not satisfied, because $\frac{\mathrm{d} \langle\vec{r}\rangle}{\mathrm{d} t} = \frac{\delta \vec{0}}{\delta t} = \vec{0}$, which is not, in general, equal to $\langle\vec{p}\rangle/m = \vec{p}/m$. So, what is wrong here?

• What fails is that plane waves are not true vectors. In particular, your integral for $\bar{\vec{r}}$ is essentially undefined. Commented Apr 7, 2016 at 12:59
• @AccidentalFourierTransform that should be an answer Commented Apr 7, 2016 at 13:03

A plane wave $\psi_p(x) = \mathrm{e}^{\mathrm{i}px}$ is not a quantum state for a free particle. The integral $\int_\mathbb{R} \psi_p(x)\psi^\ast_p(x)\mathrm{d}x$ does not converge, and it is in particular not $1$ as you use in your calculation of the expectation value of $p$. This means the expectation value of $p$ is undefined, or rather, that you shouldn't let $p$ act on this object in the first place - it is not inside the domain of definition of the momentum operator.