Problem 154 - A Complete Resource Book in Chemistry for JEE Main

Question

Uncertainty in position of a particle of \(25 \mathrm{~g}\) in space is \(10^{-5} \mathrm{~m}\). Hence uncertainty in velocity \(\left(\mathrm{ms}^{-1}\right)\) is (Planck constant \(\mathrm{h}=6.6 \times 10^{-34} \mathrm{Js}\) ) \(\quad\) (a) \(2.1 \times 10^{-2 \mathrm{~s}}\) (b) \(2.1 \times 10^{-34}\) (c) \(0.5 \times 10^{-34}\) (d) \(5.0 \times 10^{-24}\)

Step-by-Step Solution

Answer

The uncertainty in velocity is approximately \(8.4 \times 10^{-28} \text{ m/s}\), which is not among the given options.

Step 1: Understand the Heisenberg Uncertainty Principle

The Heisenberg Uncertainty Principle states that the product of the uncertainty in position, \(\Delta x\), and the uncertainty in momentum, \(\Delta p\), of a particle is always greater than or equal to \(\frac{\hbar}{2}\), where \(\hbar = \frac{h}{2\pi}\). The formula is given by:\[ \Delta x \cdot \Delta p \geq \frac{h}{4\pi} \]

Step 2: Express momentum in terms of velocity

Momentum \( p \) is the product of mass \( m \) and velocity \( v \). Thus, the uncertainty in momentum \( \Delta p \) can be expressed in terms of the uncertainty in velocity \( \Delta v \), as \( \Delta p = m \cdot \Delta v \).

Step 3: Substitute and rearrange the uncertainty formula

Substituting into the uncertainty relation, we have: \[ \Delta x \cdot m \cdot \Delta v \geq \frac{h}{4\pi} \]Rearranging to find the uncertainty in velocity, \( \Delta v \), gives:\[ \Delta v \geq \frac{h}{4\pi m \Delta x} \]

Step 4: Insert given values into the formula

Insert the given values: \( \Delta x = 10^{-5} \text{ m} \), \( m = 25 \text{ g} = 0.025 \text{ kg} \), and Planck's constant \( h = 6.6 \times 10^{-34} \text{ Js} \). Then calculate:\[ \Delta v \geq \frac{6.6 \times 10^{-34}}{4 \pi \times 0.025 \times 10^{-5}} \]

Step 5: Perform the calculation

Calculate the expression:\[ \Delta v \geq \frac{6.6 \times 10^{-34}}{4 \times 3.14159 \times 0.025 \times 10^{-5}} \approx 8.4 \times 10^{-28} \text{ m/s} \]

Key Concepts

Uncertainty in Position

The concept of uncertainty in position is fundamental to understanding the behavior of particles at the quantum level. In the context of quantum mechanics, uncertainty refers to the inherent limitations in measuring certain pairs of properties, such as position and momentum.
Specifically, when you attempt to measure the exact position of a particle, there is an inherent uncertainty or "fuzziness" in the measurement. This doesn't mean an error in measurement but rather a fundamental limit imposed by the nature of quantum systems.

• For example, in the exercise, we consider a particle with a small mass, where the uncertainty in its position is given as \(10^{-5} \text{ m}\).
• This level of precision reflects how finely we can pinpoint the location of the particle due to its quantum nature.

Understanding uncertainties in position is key because it sets the stage for calculating other linked uncertainties, such as momentum or velocity, using the Heisenberg Uncertainty Principle, which we will discuss shortly.

Uncertainty in Velocity

Like position, velocity of a quantum particle also has a certain level of uncertainty. In classical physics, you might expect that knowing the exact position and velocity at the same time is possible. However, due to the Heisenberg Uncertainty Principle, this isn't the case in quantum mechanics.
The calculation of uncertainty in velocity stems from rearranging the uncertainty relation that involves momentum (mass times velocity).

• According to this relationship, as certainty in the position (\(\Delta x\)) increases, \(\Delta v\) typically increases, indicating a trade-off between knowing a particle's exact position and its exact velocity.
• In our example, we determine this uncertainty by substituting known values into the formula: \(\Delta v \geq \frac{h}{4\pi m \Delta x}\), yielding a result of approximately \(8.4 \times 10^{-28} \text{ m/s}\).

This teaches us that at quantum scales, we cannot define both the position and velocity of a particle with arbitrary precision.

Quantum Mechanics

Quantum mechanics is the branch of physics that deals with the behavior of very small particles, such as atoms, electrons, and photons. This field of study reveals the odd and counterintuitive principles that govern the quantum world, including the Heisenberg Uncertainty Principle.
Unlike classical physics, quantum mechanics embraces the intrinsic uncertainties and indeterministic nature of particle states.

• The theory is groundbreaking because it explains phenomena at microscopic scales that classical theories can't.
• The Heisenberg Uncertainty Principle is one such quantum concept indicating that there's a limit to how much we can know about a particle's exact position and momentum simultaneously.

Grasping quantum mechanics is pivotal for studying modern physics, as it underpins many technological advances in fields like quantum computing and cosmology.

Planck's Constant

Planck's constant (denoted as \( h \)) is a fundamental constant in physics. It plays a crucial role in the theories of quantum mechanics. This constant represents the scale at which quantum effects become significant.
Planck's constant is very small, valued at approximately \(6.62607015 \times 10^{-34} \text{ Js}\), which is why quantum effects are not usually observed in everyday life but become noticeable at atomic and subatomic levels.

• In our exercise, Planck's constant is used to calculate the uncertainty relationship between position and velocity, emphasizing its pivotal role in defining the limits of precision in the quantum world.
• Functions like a bridge, connecting the macroscopic physics we observe with the microscopic quantum phenomena.

Understanding Planck's constant is fundamental for exploring subjects like quantum theory, as it aids in predicting how particles behave when the smallest scales of time and space are involved.