Question

The de Broglie wavelength associated with a particle of mass \(10^{-6} \mathrm{~kg}\) moving with a velocity of 10 \(\mathrm{ms}^{-1}\) is: (a) \(6.63 \times 10^{-7} \mathrm{~m}\) (b) \(6.63 \times 10^{-16} \mathrm{~m}\) (c) \(6.63 \times 10^{-21} \mathrm{~m}\) (d) \(6.63 \times 10^{-29} \mathrm{~m}\)

Short answer

The de Broglie wavelength is \(6.63 \times 10^{-29} \mathrm{~m}\), option (d).

Step-by-step solution

Understanding the Formula

The de Broglie wavelength (\( \lambda \)) of a particle is calculated using the formula: \( \lambda = \frac{h}{mv} \), where \( h \) is Planck's constant \( (6.63 \times 10^{-34} \mathrm{~Js}) \), \( m \) is the mass of the particle, and \( v \) is the velocity of the particle.

Inserting Given Values

We need to find the wavelength by substituting \( h = 6.63 \times 10^{-34} \mathrm{~Js} \), \( m = 10^{-6} \mathrm{~kg} \), and \( v = 10 \mathrm{~ms}^{-1} \) into the formula: \[ \lambda = \frac{6.63 \times 10^{-34}}{10^{-6} \times 10} \]

Simplify the Denominator

Simplify the denominator by multiplying the mass and velocity: \[ 10^{-6} \times 10 = 10^{-5} \]

Calculate the Wavelength

Divide Planck's constant by the result from Step 3: \[ \lambda = \frac{6.63 \times 10^{-34}}{10^{-5}} = 6.63 \times 10^{-34 + 5} = 6.63 \times 10^{-29} \mathrm{~m} \]

Choosing the Correct Answer

Compare the calculated wavelength with the given options to find the correct choice. Based on the calculation, the de Broglie wavelength is \( 6.63 \times 10^{-29} \mathrm{~m} \), which corresponds to option (d).

Key concepts

Planck's Constant

Planck's constant is a fundamental quantity in physics that plays a crucial role in quantum mechanics. It is denoted by the symbol \( h \) and has a value of \( 6.63 \times 10^{-34} \mathrm{~Js} \). Essentially, it sets the scale for quantum mechanical effects. An easy way to understand its significance is to see it as a conversion factor between energy and frequency. In quantum mechanics, energy is quantized, meaning it can only take on discrete values. Each quantum of energy is proportional to the frequency (\( E = hu \)) of the associated electromagnetic wave.

Planck's constant helps us connect the particle-like and wave-like properties of matter, which is where the concept of de Broglie wavelength comes into play. When you think about waves, you might recall their primary characteristics: wavelength and frequency. Planck's constant is the bridge between traditional physics (like Newton's laws) and the mysterious world of quantum mechanics.

Mass-Velocity Relation

In physics, the motion of particles is often described by considering their mass and velocity. This forms what we call the mass-velocity relation. It's a fundamental concept that can smoothly lead us into understanding kinetic energy, momentum, and even the de Broglie wavelength..

• Mass: is the amount of matter in a particle, typically measured in kilograms (kg). In our exercise, the mass is \( 10^{-6} \mathrm{~kg} \).
• Velocity: is the speed of the particle in a specified direction. Here, it's given as \( 10 \mathrm{~ms}^{-1} \).

The product of mass and velocity gives us momentum, an important concept because it combines two fundamental aspects of motion. In the context of de Broglie's hypothesis, every moving particle is associated with a wave, the wavelength of which is inversely proportional to its momentum (\( \lambda = \frac{h}{mv} \)).

Simply put, a heavier or faster particle will have a shorter wavelength. This equation allows us to compute the de Broglie wavelength of particles when we know their mass and velocity. It shows how all particles, no matter their size, have wave-like properties.

Quantum Mechanics

Quantum mechanics is the branch of physics that studies the behavior of particles at the atomic and subatomic levels. It departs from classical physics in ways that can seem mind-boggling.

In the tiny world of atoms and particles, things don't follow the clear-cut paths and interactions we observe in larger systems. Instead, particles like electrons exhibit both wave-like and particle-like properties. This dual nature is at the heart of quantum mechanics and is exemplified by concepts like the de Broglie wavelength.

Traditional classical physics couldn't explain certain phenomena, such as why electrons can only exist in certain energy levels around an atom. Quantum mechanics filled in these gaps. It explains the probabilistic nature of particles and introduces the concepts of uncertainty and quantization of physical properties — such as energy levels.

• Quantum mechanics has led to the development of technologies such as semiconductors, lasers, and even MRI machines.
• It helps us understand fundamental processes in chemistry, like bonding and reactions.

So while it can be quite abstract, quantum mechanics is vitally important for the technology and scientific understanding we rely on every day. Understanding key ideas like the de Broglie wavelength is a stepping stone into this fascinating field.