Question

The de Broglie wavelength of a Tennis ball of mass \(60 \mathrm{~g}\) moving with a velocity of 10 metres per second is approximately: (Planck constant \(\left.\mathrm{h}=6.63 \times 10^{-31} \mathrm{Js}\right):\) (a) \(10^{-33} \mathrm{~m}\) (b) \(10^{-31} \mathrm{~m}\) (c) \(10^{-16} \mathrm{~m}\) (d) \(10^{-25} \mathrm{~m}\)

Short answer

The de Broglie wavelength of the tennis ball is approximately \(10^{-31}\ \text{m}\). The correct answer is (b) \(10^{-31}\ \text{m}\).

Step-by-step solution

Convert mass to kilograms

First, convert the mass of the tennis ball from grams to kilograms because the SI unit for mass in physics equations is kilograms. The mass of the tennis ball is given as \(60 \mathrm{~g}\). To convert it to kilograms:\[ 60 \text{ g} = 60 \times 10^{-3} \text{ kg} = 0.060 \text{ kg} \]

Use de Broglie wavelength formula

Next, use the de Broglie wavelength formula:\[ \lambda = \frac{h}{mv} \]where \(\lambda\) is the wavelength, \(h\) is Planck's constant \(6.63 \times 10^{-31} \text{ Js}\), \(m\) is the mass \(0.060 \text{ kg}\), and \(v\) is the velocity \(10 \text{ m/s}\).

Substitute values into the formula

Substitute the known values into the de Broglie wavelength formula:\[ \lambda = \frac{6.63 \times 10^{-31}}{0.060 \times 10} \]

Calculate the de Broglie wavelength

Carry out the division:\[ \lambda = \frac{6.63 \times 10^{-31}}{0.60} \]\[ \lambda = 11.05 \times 10^{-31} \text{ m} \]Since we are asked for an approximate wavelength, we round this value:\[ \lambda \approx 10 \times 10^{-31} = 10^{-30} \text{ m} \]

Select the nearest option

Compare the calculated wavelength to the given options. The closest approximation to our calculated result \(10^{-30}\ \text{m}\) from the provided options is \(10^{-31}\ \text{m}\) since options are given in specific powers of ten.

Key concepts

Quantum Mechanics

Quantum mechanics is a branch of physics that studies the behavior of very small particles. Unlike classical mechanics, which deals with large-scale phenomena, quantum mechanics looks at atomic and subatomic levels.
It explores how particles, such as electrons and photons, can act in ways that defy classical intuition.
Some key principles of quantum mechanics include:

• Superposition: Particles can be in multiple states at the same time until they're measured.
• Uncertainty principle: It is impossible to know both the position and momentum of a particle with perfect precision.
• Quantization: Energy comes in discrete units called quanta.

These concepts are not only fascinating but also crucial for understanding the behavior of matter and energy at the smallest scales. One notable feature of quantum mechanics is its application in calculating phenomena like the de Broglie wavelength, which blurs the line between particles and waves.

Planck's Constant

Planck's constant is a fundamental constant in physics symbolized by the letter \( h \).
It is pivotal in quantum mechanics, acting as a bridge between wave and particle properties of matter and radiation.
Its value is approximately \( 6.63 \times 10^{-34} \text{Js} \) in standard SI units.The importance of Planck's constant can be seen in several areas:

• It serves as a scaling factor in the formula for the energy of a photon: \( E = hf \), where \( E \) is energy, \( h \) is Planck's constant, and \( f \) is the frequency of the photon.
• Utilized in the de Broglie equation \( \lambda = \frac{h}{mv} \), it helps determine the wavelength associated with a particle's momentum.

Planck's constant is essential in quantum mechanics for explaining phenomena that cannot be described by classical physics. Its value, although small, has a big impact on our understanding of the universe's fundamental laws.

Wave-Particle Duality

Wave-particle duality is a fundamental concept in quantum mechanics that suggests that every particle or quanta can be described as both a particle and a wave.
This dual nature is best illustrated by experiments like the double-slit experiment, where particles such as electrons create an interference pattern when not observed, behaving like waves.
Key aspects include:

• Wave behavior: Demonstrates interference and diffraction patterns.
• Particle behavior: When observed, particles seem to take a more defined path.
• De Broglie Hypothesis: Introduced the idea that not only light but matter has wave properties, with de Broglie wavelength used to calculate the wave aspect of particles.

The concept of wave-particle duality sheds light on the intricate and sometimes mysterious nature of matter. It compels us to question our conventional understanding of materials, leading to revolutionary technological advancements.