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Question: The product of uncertainties of displacement and velocity of a moving particle of mass \[9.1\times {...

The product of uncertainties of displacement and velocity of a moving particle of mass 9.1×10−28g9.1\times {{10}^{-28}}g is:
(a) 1.54×10−4m2s−11.54\times {{10}^{-4}}{{m}^{2}}{{s}^{-1}}
(b) 5.77×10−5m2s−15.77\times {{10}^{-5}}{{m}^{2}}{{s}^{-1}}
(c) 1.54×10−5m2s−11.54\times {{10}^{-5}}{{m}^{2}}{{s}^{-1}}
(d) 5.77×10−4m2s−15.77\times {{10}^{-4}}{{m}^{2}}{{s}^{-1}}

Explanation

Solution

Heisenberg’s Uncertainty principle states that there is uncertainty in measuring a variable of a particle. It equation can be written as ΔxΔp≥h4π\Delta x\Delta p\ge \dfrac{h}{4\pi }, where Δx\Delta x=uncertainty in displacement and Δp\Delta p= uncertainty in momentum and h is the planck's constant.
It is applied to the position and momentum of a particle.

Complete step by step solution: The uncertainty principle states that the more precisely the position is known the more uncertain the momentum is and vice versa. It is a fundamental theory in quantum mechanics that defines why it is difficult to measure multiple quantum variables simultaneously. Uncertainties in the products of momentum and position were defined by Heisenberg as having a minimum value corresponding to Plank’s constant divided by4π4\pi . The relation between the uncertain momentum and displacement is given by the equation
ΔxΔp≥h4π\Delta x\Delta p\ge \dfrac{h}{4\pi }
According to Heisenberg Uncertainty rule,
ΔxΔp=h4π\Delta x\Delta p=\dfrac{h}{4\pi }
Here Δx\Delta x=uncertainty in displacement and Δp\Delta p= uncertainty in momentum and is planck's constant.
We know Δp=m×Δv\Delta p=m\times \Delta vwhere Δv\Delta v= uncertainty in velocity and m is the mass.
The given condition is that,
Δv×Δx=h4πm\Delta v\times \Delta x=\dfrac{h}{4\pi m}
Given that h=6.6×10−346.6\times {{10}^{-34}} and π=3.14\pi =3.14 and m=9.1×10−28m=9.1\times {{10}^{-28}} g
Substituting the values in the above equation,
=6.6×10−344×3.14×9.1×10−28=\dfrac{6.6\times {{10}^{-34}}}{4\times 3.14\times 9.1\times {{10}^{-28}}}
= =5.77×10−5m2s−1=5.77\times {{10}^{-5}}{{m}^{2}}{{s}^{-1}}.

Therefore, the correct answer to the question is option (b).

Additional Information:
-This principle also applies to energy and time. It is hard to know exactly where a particle is at a given moment. This is because of the wave-like nature of a particle. A particle when spread out, it occupies a range of positions. Momentum cannot be precisely known because the particle consists of a packet of waves and each of these packets has their own momentum. So, we get a range of momentum for a particle.

Note: The particle is considered to be very small or quantum size. Then only the principle is valid. For large molecules, it is easy to identify a position of the molecule and its momentum. It is because the size is very small, there is uncertainty in the calculating position and momentum.