Some additional explanations of the uncertainty relation (UR)

The considerations for the single slit suggest that an ensemble with a large position spread automatically has a small momentum spread. In general, this will not be the case. For an ensemble of quantum objects, both the position and the momentum can be poorly prepared.

Heisenberg’s uncertainty relation only represents a lower bound for the simultaneous preparation of position and momentum. This is expressed by the greater-than sign in the general equation:

$\Delta y \cdot \Delta p_y \geq \dfrac{h}{4 \pi}$

Compared with the single-slit version of the uncertainty relation ( $\Delta y \cdot \Delta p_y \approx h$ ), the modified prefactor $\dfrac{1}{4 \pi}$ results from the approximations made in deriving this equation.

The considerations so far regarding Heisenberg’s uncertainty relation applied only to components of position and momentum in the same direction (here the y-direction). It turns out that the impossibility of simultaneous preparation applies only to these components. This means: although it is impossible to prepare $x$ and $p_x$ or $y$ and $p_y$ simultaneously, position and momentum components in different directions — such as $y$ and $p_x$ or $x$ and $p_y$ — can indeed be prepared simultaneously.

Back