Question

Calculate the de Broglie wavelength for each of the following. a. an electron with a velocity 10.% of the speed of light b. a tennis ball ( 55 g) served at 35 \(\mathrm{m} / \mathrm{s}(\sim 80 \mathrm{mi} / \mathrm{h})\)

Short answer

The de Broglie wavelength for an electron moving at 10% the speed of light is determined by using the formula \(\lambda = \frac{h}{p}\), where \(h = 6.626 \times 10^{-34} \, \text{Js}\) and \(p = (9.11 \times 10^{-31} \, \text{kg})(0.1 \times 3 \times 10^8 \, \text{m/s})\). Calculating the de Broglie wavelength yields \(\lambda_e \approx 7.29 \times 10^{-12} \, \text{m}\). For the tennis ball served at 35 m/s, we again use the formula \(\lambda = \frac{h}{p}\), where \(p = (0.055 \, \text{kg}) \times (35 \, \text{m/s})\). Calculating the de Broglie wavelength results in \(\lambda_t \approx 3.46 \times 10^{-34} \, \text{m}\).

Step-by-step solution

Case a: Electron moving at 10% of the speed of light

First, we need to find the momentum of the electron. The mass of an electron is \(9.11 \times 10^{-31} \, \text{kg}\), and its velocity is 10% of the speed of light, \(c = 3 \times 10^8 \, \text{m/s}\). Therefore, the velocity of the electron is: \[v = 0.1c = 0.1 \times 3 \times 10^8 \, \text{m/s}\] The momentum, \(p\), of the electron can be calculated as: \[p = mv = (9.11 \times 10^{-31} \, \text{kg}) \times (0.1 \times 3 \times 10^8 \, \text{m/s})\] Now, we can calculate the de Broglie wavelength as: \[\lambda = \frac{h}{p}\]

Case b: Tennis ball served at 35 m/s

In this case, we have a tennis ball with a mass of 55 g which is equal to \(0.055\) kg. The velocity of the tennis ball is given as 35 m/s. First, we find the momentum of the tennis ball: \[p = mv = (0.055 \, \text{kg}) \times (35 \, \text{m/s})\] Then, we can calculate the de Broglie wavelength as: \[\lambda = \frac{h}{p}\] By calculating the de Broglie wavelength for both cases, we will have the answers to the given exercise.

Key concepts

Momentum Calculation

To begin understanding de Broglie wavelength, it's important to know about momentum calculation. Momentum is a measure of the motion of an object and is calculated as the product of its mass and velocity. For any object, big or small, momentum can be expressed as \( p = mv \). Here:

• \( m \) is the mass of the object,
• \( v \) is its velocity.

In the exercise, the electron's momentum is determined by multiplying its mass \((9.11 \times 10^{-31} \, \text{kg})\) with its velocity \((0.1 \times 3 \times 10^8 \, \text{m/s})\) since it's moving at 10% the speed of light. This step provides the foundation for calculating the wavelength later.
For the tennis ball, its mass is 55 g (converted to kg as 0.055 kg) and its velocity is given as 35 m/s. By using the same formula, the momentum is calculated as \( p = 0.055 \, \text{kg} \times 35 \, \text{m/s} \). These calculations show how objects much larger than electrons retain classical properties, where momentum scales with mass.

Electron Velocity

Electron velocity plays a critical role in determining its properties at such small scales. Electrons, which are subatomic particles found in atoms, have minuscule mass, making their velocity significant in physics. When an electron travels at 10% of the speed of light, this high velocity profoundly impacts its behavior as described by quantum mechanics.
The speed of light is a universal constant, \( c = 3 \times 10^8 \, \text{m/s} \), and speeds are often compared in percentage terms for ease of calculation. In the given problem, the electron's velocity is 10% of \( c \), which translates to \( v = 0.1c = 0.1 \times 3 \times 10^8 \, \text{m/s} \).
This calculation allows us to see how such significant velocities affect an electron's quantum properties, such as wave-particle duality and makes it critical in understanding fine concepts like their de Broglie wavelength.

Wave-Particle Duality

Wave-particle duality is a fundamental concept in quantum mechanics that states matter exhibits both particle-like and wave-like properties. This duality is particularly evident in microscopic particles like electrons. The duality suggests that even if electrons are considered particles, they can also exhibit wave characteristics under certain conditions, such as when determining their de Broglie wavelength.
The de Broglie hypothesis states that every particle has a wavelength \( \lambda \) associated with it, given by \( \lambda = \frac{h}{p} \), where \( h \) is the Planck constant \( 6.626 \times 10^{-34} \, \text{Js} \) and \( p \) is momentum. This wavelength becomes significant for very small masses, like electrons, making their wave nature observable.
In contrast, for large objects such as the tennis ball in the exercise, the de Broglie wavelength is exceedingly small and thus practically non-observable. This showcases how wave-particle duality is a unique feature of quantum-scale phenomena, observable with minute, high-speed particles but negligible with larger, everyday objects.