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Uncertainty in the position of an electron (mass\[ = 9.1 \times {10^{ - 28}}gm)\] moving with a velocity \[300m{s^{ - 1}}\] accurate upon \[0.001\% \] will be:
A.\[19.2 \times {10^{ - 2}}m\]
B.\[5.76 \times {10^{ - 2}}m\]
C.\[1.92 \times {10^{ - 2}}m\]
D. \[3.84 \times {10^{ - 2}}m\]

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Last updated date: 23rd Jul 2024
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Answer
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Hint: The uncertainty principle (also known as Heisenberg's uncertainty principle) is a collection of mathematical inequalities asserting a fundamental limit to the precision with which the values for certain pairs of physical quantities of a particle, such as position, x, and momentum, p, can be predicted from initial conditions in quantum mechanics.

Complete answer:
According to Heisenberg’s uncertainty principle,
\[{{\Delta x \times m \times \Delta V = }}\dfrac{{\text{h}}}{{{{4\pi }}}}\]
Then we should look into,
\[{{\Delta x = }}\] uncertainty in the position
m=\[9.1 \times {10^{ - 31}}kg = \]mass of electron
\[{{\Delta V = 300}} \times \dfrac{{0.001}}{{100}} = 0.003m/s\]
Uncertainty in the velocity
\[h = 6.62 \times {10^{ - 36}}Js = \]Planck’s constant
Substitute values in the above formula
\[
  {{\Delta x}}\, \times \,9.1 \times {10^{ - 31}} \times 0.003 = \dfrac{{6.62 \times {{10}^{ - 34}}}}{{4 \times 3.1416}} \\
  {{\Delta x = 1}}{\text{.92}}\, \times \,{\text{1}}{{\text{0}}^{ - 2\,}}\,m \\
 \]
So we found that the correct option is C.
Complementary variables or canonically conjugate variables are such variable pairs; and, depending on interpretation, the uncertainty principle limits the extent to which such conjugate properties retain their approximate meaning, as quantum physics' mathematical framework does not support the notion of simultaneously well-defined conjugate properties expressed by a single value.
Even if all initial conditions are defined, the uncertainty principle states that it is impossible to predict the value of a quantity with arbitrary certainty. On the macroscopic dimensions of daily experience, the uncertainty principle is not readily apparent. As a result, demonstrating how it relates to more easily comprehended physical conditions is beneficial. The uncertainty theory is explained differently in two separate quantum mechanics frameworks.

Note:
Since the uncertainty principle is such a fundamental result of quantum mechanics, it is consistently observed in quantum experiments. However, as part of their key research programme, some experiments could purposefully test a specific type of the uncertainty principle.