# Question #60ee9

Apr 24, 2017

The ${m}_{l}$ values for the ${p}_{x}$ and ${p}_{y}$ orbitals are $\setminus \pm 1$ and they have opposite signs, but there is no way to tell which is which. We can pick either opposite-sign combination (e.g. ${p}_{x} = + 1 , {p}_{y} = - 1$), but that is only for our bookkeeping.

#### Explanation:

Out of the three $p$ orbitals in the second and higher shells, only ${p}_{z}$ is built from a single quantum state with a well-defined angular momentum about the $z$-axis (${m}_{l} = 0$). The other ${m}_{l}$ values, $+ 1$ and $- 1$, correspond to a pair of complex conjugate wave functions whose amplitude at any location varies with time – one way for ${m}_{l} = + 1$ and the opposite way for ${m}_{l} = - 1$. These varying amplitudes are unsuitable for forming fixed chemical bonds, so the only states we can see in those bonds are linear combinations that remain fixed as a function of time. One such combination is ${p}_{x}$; the other, orthogonal combination is ${p}_{y}$.

Because both angular momentum states enter into each of these linear combinations, a specific assignment such as ${p}_{x} = + 1 , {p}_{y} = - 1$ can only be our convention and not empirical reality (although the requirement that the signs be opposite is real, because all $p$ orbitals taken together must add up to a zero angular momentum vector).

The lack of a definite ${m}_{l}$ value for either ${p}_{x}$ or ${p}_{y}$ -- which also applies to $d$ and other orbitals with nonzero ${m}_{l}$ values – is ultimately a manifestation of the Heisenberg Uncertainty Principle. Usually the uncertainty principle is stated in terms of position and momentum in straight-line motion, but it applies also to rotational motion. In either the ${p}_{x}$ or ${p}_{y}$ orbital, we constrain the longitudinal position of the electrons around the $z$-axis because the lobes are oriented in a specific pair of opposing directions within the $x y$ plane. If the position is thereby constrained, the corresponding angular momentum must have some uncertainty associated with it and thus we can’t have a single value of ${m}_{l}$ anymore.

The ${p}_{z}$ orbital, which is symmetric around the $z$-axis, has no constraint around that axis so it can be assigned a definite ${m}_{l} = 0$ value, but we would not be able determine a specific $x$ or $y$ component of angular momentum.