How Heisenberg Uncertainty Violates Bohr's Theory

Quantum object electron

For those interested: Derivation from a single slit

Note: In the chapter on interference phenomena in light, the double slit and the grating were essentially dealt with. However, interference also occurs with the single slit. Here the conditions for maxima and minima look a little different.

In front of the slit, all quantum objects in the parallel bundle have the amount of momentum py. Due to the narrowing to the location \ (\ Delta x \), the bundle expands on the way to the screen. The quantum objects now get a blurring in the transverse momentum, which is to be estimated in the following:

Most of the quantum objects are located within the main maximum of the diffraction figure that arises on the screen. The main maximum is limited by the two first-order minima.

For the 1st minimum at the diffraction at the single slit with the slit width \ (\ Delta x \) applies (different from the double slit)
\ [\ Delta x \ cdot \ sin \ left ({{\ alpha _1}} \ right) = 1 \ cdot \ lambda \ Leftrightarrow \ sin \ left ({{\ alpha _1}} \ right) = \ frac {\ lambda} {{\ Delta x}} \]
The following applies to the relationship between the transverse momentum and the original momentum (cf.
\ [\ tan \ left ({{\ alpha _1}} \ right) = \ frac {{\ Delta {p_x}}} {{{p_y}}} \]
Taking into account the small-angle approximation (\ (\ sin \ left ({{\ alpha _1}} \ right) \ approx \ tan \ left ({{\ alpha _1}} \ right) \)) one obtains by equating
\ [\ frac {{\ Delta {p_x}}} {{{p_y}}} \ approx \ frac {\ lambda} {{\ Delta x}} \ Leftrightarrow \ Delta {p_x} \ cdot \ Delta x \ approx { p_y} \ cdot \ lambda \ quad (1) \]
For the de BROGLIE wavelength \ (\ lambda \) \ (\ lambda = \ frac {h} {{{p_y}}} \) applies. If you put this in \ ((1) \), you get
\ [\ Delta {p_x} \ cdot \ Delta x \ approx {p_y} \ cdot \ frac {h} {{{p_y}}} = h \]
The uncertainty relation obtained in this way does not agree exactly with that derived by Heisenberg. But this is not so crucial: the only important thing is the knowledge that the product of the imprecision of place and momentum cannot be arbitrarily small.

This derivation resulted in a slightly different constant on the right-hand side of the uncertainty relation. But this is not so crucial: the only important thing is the knowledge that the product of the imprecision of place and momentum cannot be arbitrarily small. So you cannot determine the location and the momentum of quantum objects at the same time as precisely as you want.

This realization is synonymous with that Farewell to the classic railway concept in the case of quantum objects, since the description of a path requires precise knowledge of the position and momentum of an object at the same time.