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 \(\mathrm{h}=6.6 \times 10^{-34} \mathrm{Js}\) ) \(\quad\) [2002] (a) \(2.1 \times 10^{-28}\) (b) \(2.1 \times 10^{-34}\) (c) \(0.5 \times 10^{-34}\) (d) \(5.0 \times 10^{-24}\)

Step-by-Step Solution

Answer

Option (a): \( 2.1 \times 10^{-28} \text{ m/s} \).

Step 1: Understand the Uncertainty Principle

The uncertainty principle is given by \( \Delta x \cdot \Delta p \geq \frac{h}{4\pi} \), where \( \Delta x \) is the uncertainty in position and \( \Delta p \) is the uncertainty in momentum. Since momentum \( p = m \cdot v \), we can express the uncertainty in momentum as \( \Delta p = m \cdot \Delta v \), where \( m \) is the mass of the particle and \( \Delta v \) is the uncertainty in velocity.

Step 2: Substitute Values into the Formula

We know \( \Delta x = 10^{-5} \text{ m} \), \( m = 25 \text{ g} = 0.025 \text{ kg} \), and \( h = 6.6 \times 10^{-34} \text{ Js} \). Now, substitute these values into \( \Delta x \cdot m \cdot \Delta v \geq \frac{h}{4\pi} \) to find \( \Delta v \).

Step 3: Solve for Uncertainty in Velocity (\(\Delta v\))

Solve \( (10^{-5} \text{ m})(0.025 \text{ kg})(\Delta v) \geq \frac{6.6 \times 10^{-34} \text{ Js}}{4 \times 3.14} \). Calculate the right-hand side: \( \frac{6.6 \times 10^{-34}}{12.56} \approx 5.26 \times 10^{-35} \text{ Js} \). Divide both sides by \( 2.5 \times 10^{-7} \text{ kg} \) to isolate \( \Delta v \): \( \Delta v \geq \frac{5.26 \times 10^{-35}}{2.5 \times 10^{-7}} \).

Step 4: Calculate and Compare with Options

Perform the calculation: \( \Delta v \geq 2.104 \times 10^{-28} \text{ m/s} \). This value matches with option (a): \( 2.1 \times 10^{-28} \text{ m/s} \).

Key Concepts

Heisenberg Uncertainty

The Heisenberg Uncertainty Principle is a fundamental concept in quantum mechanics. It states that there is a limit to how precisely we can simultaneously know certain pairs of observable quantities, such as position and momentum. Formulated by Werner Heisenberg in 1927, this principle challenges our classical notions of precise measurement.
In the context of the exercise given, we see this principle in action through the equation:

• \( \Delta x \cdot \Delta p \geq \frac{h}{4 \pi} \)
• Here, \( \Delta x \) represents the uncertainty in position, \( \Delta p \) is the uncertainty in momentum, and \( h \) is the Planck constant, a tiny yet significant value in quantum calculations.

This inherent uncertainty is not just about measurement errors, but an intrinsic feature of the quantum world. It tells us that the more precisely we know the position of a particle, the less precisely we can know its momentum and vice versa. This relation underscores many fascinating and non-intuitive aspects of quantum behavior.

Quantum Mechanics

Quantum mechanics is the branch of physics that deals with the behavior of matter and light on the atomic and subatomic scales. Unlike classical physics, quantum mechanics reveals a universe where particles can be in multiple states simultaneously, and events are fundamentally probabilistic.
In the problem being discussed, quantum mechanics provides the framework to understand why we can't measure both position and velocity with unlimited accuracy for small particles like electrons. This comes from the dual nature of particles and waves.

• Particles such as electrons exhibit wave-like properties, meaning they don't have definite, pinpointable positions.
• This wave-particle duality is central to quantum mechanics and leads to the mathematical formulation of the uncertainty principle.

This fundamental unpredictability challenges our traditional understanding of physics and has broad implications, ranging from the development of quantum computing to our comprehension of the universe at its most basic level.

Momentum Uncertainty

Momentum uncertainty plays a crucial role in the context of the Heisenberg Uncertainty Principle. Momentum \( p \), traditionally defined as the product of mass \( m \) and velocity \( v \), becomes uncertain when observed at quantum scales.
In the exercise problem, we note that the uncertainty in momentum \( \Delta p \) translates into \( \Delta p = m \cdot \Delta v \), where \( \Delta v \) is the uncertainty in velocity. This implies that knowing the exact velocity of a particle becomes increasingly difficult as its position uncertainty decreases.

• To find the uncertainty in momentum, the equation \( \Delta x \cdot m \cdot \Delta v \geq \frac{h}{4 \pi} \) is used.
• This approach allows for the calculation of uncertainty in velocity, as seen in our solution for \( \Delta v \leq 2.1 \times 10^{-28} \text{ m/s} \).

Understanding momentum uncertainty is not only important for theoretical physics but also for practical applications in technologies like electron microscopes, where precision is limited by quantum mechanical effects.