Question

Explain how Heisenberg’s uncertainty principle is contradictory to Bohr’s theory. Explain the validity of Heisenberg’s uncertainty principle to larger objects with reason?

Answer 1

Hint: The Heisenberg’s uncertainty principle states that the position and momentum of an electron can never be measured simultaneously. According to Bohr's theory, electrons revolve around the nucleus in fixed circular paths called orbits, having fixed energies.

Complete step by step answer:
-Werner Heisenberg in $1927$, gave Heisenberg's uncertainty principle which states that it is impossible to determine precisely the position and momentum of an object simultaneously.
It is given by the relation,
$\Delta x.\Delta p \geqslant \dfrac{h}{{4\pi }}$
Where, $\Delta x = $ uncertainty in position or deviation in position measurement
$\Delta p = $ uncertainty in momentum or deviation in momentum measurement
-If we try to determine the position of the object more precisely, it will lead to an increase in the uncertainty in the momentum. While trying to determine the momentum of an object accurately, the uncertainty in the position increases.
-Bohr’s theory suggests that the structure of an atom is such that the electrons revolve around the nucleus in fixed circular orbits having fixed energies. The electron can only exist in these orbits and not anywhere in between, due to which it acquires or loses only quantized energies in order to excite to a higher energy level or a lower energy level.
-Therefore, according to Bohr’s theory the position of an electron and its trajectory can be defined because we know the exact orbits in which it is revolving, while the uncertainty principle says that they cannot be defined simultaneously.
-The Heisenberg’s uncertainty principle applies to objects that show wave-particle duality, that is, particles that also behave like a wave. The wavelength of such waves is given by the de-Broglie equation,
$\lambda = \dfrac{h}{{mv}}$
For large objects, ‘m’ is large, which makes the value of $\lambda $ very small (negligible). -Therefore, we cannot see a large ball showing wave motion (the wavelength of its wave motion is negligible compared to its size).
-Thus, for large objects the Heisenberg’s uncertainty principle doesn’t hold good and their position as well as momentum can be determined simultaneously and precisely.

Note:
As we know, momentum of an object is given by $p = mv$, therefore, the deviation in momentum measurement, considering the mass to be constant can be given as,
$\Delta p = m\Delta v$
Where, $\Delta v = $ uncertainty in velocity or deviation in velocity measurement
The uncertainty relation can also be written as,
$\Delta x.\Delta v \geqslant \dfrac{h}{{4\pi m}}$.