In Heisenberg uncertainty principle why do we only talk about uncertainty in position along $x$ axis, why not along other dimensions as well?


The uncertainty principle tells us that when we make a measurement of $x$ and a measurement of $p_x$ (i.e. the $x$ component of the momentum) then no matter how ideal our measurement the inequality:

$$ \Delta x\Delta p_x \ge \frac{\hbar}{2} $$

applies. The equation contains $x$ because we have arranged our axes to make the measurement in the $x$ direction. We could just as easily have written $y$ and $p_y$ or $z$ and $p_z$, it's just convention that we tend to use the $x$ label first and only resort to $y$ and $z$ if we want to simultaneously consider more spatial dimensions.

It makes perfect sense to ask what happens if we measure $y$ and $p_x$ or $x$ and $p_y$, or indeed $x$ and $y$. In that case we find:

$$ \Delta y\Delta p_x = \Delta x\Delta p_y = \Delta x\Delta y = 0 $$

i.e. that is there is no uncertainty when measuring these pairs of variables.


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