Problem 40 - College Physics

Question

\(\bullet\) (a) An electron moves with a speed of \(4.70 \times 10^{6} \mathrm{m} / \mathrm{s}\) . What is its de Broglie wavelength? (b) A proton moves with the same speed. Determine its de Broglie wavelength.

Step-by-Step Solution

Answer

Electron: \(1.55 \times 10^{-10} \text{ m}\), Proton: \(8.40 \times 10^{-14} \text{ m}\).

Step 1: Calculate the de Broglie Wavelength for Electron

The de Broglie wavelength is given by the formula \( \lambda = \frac{h}{mv} \), where \( h \) is Planck's constant \( 6.626 \times 10^{-34} \text{ J}\cdot\text{s} \), \( m \) is the mass of the particle, and \( v \) is the velocity. For an electron, \( m = 9.11 \times 10^{-31} \text{ kg} \). The velocity \( v \) is given as \( 4.70 \times 10^{6} \text{ m/s} \). Plug these values into the formula:\[\lambda_{\text{electron}} = \frac{6.626 \times 10^{-34}}{9.11 \times 10^{-31} \times 4.70 \times 10^{6}}\]Calculate \( \lambda_{\text{electron}} \).

Step 2: Plug in Values and Solve for Electron

Substitute the numerical values into the equation and solve for \( \lambda_{\text{electron}} \):\[\lambda_{\text{electron}} = \frac{6.626 \times 10^{-34}}{42.817 \times 10^{-25}}\]Calculate the above to find \( \lambda_{\text{electron}} = 1.55 \times 10^{-10} \text{ m} \).

Step 3: Calculate de Broglie Wavelength for Proton

Using the same de Broglie wavelength formula \( \lambda = \frac{h}{mv} \), and knowing that a proton has a mass \( m = 1.67 \times 10^{-27} \text{ kg} \), use the same velocity \( v = 4.70 \times 10^{6} \text{ m/s} \). Insert these values into the formula:\[\lambda_{\text{proton}} = \frac{6.626 \times 10^{-34}}{1.67 \times 10^{-27} \times 4.70 \times 10^{6}}\]Calculate \( \lambda_{\text{proton}} \).

Step 4: Plug in Values and Solve for Proton

Substitute the numerical values into the equation for \( \lambda_{\text{proton}} \):\[\lambda_{\text{proton}} = \frac{6.626 \times 10^{-34}}{7.849 \times 10^{-21}}\]Calculate the above to find \( \lambda_{\text{proton}} = 8.40 \times 10^{-14} \text{ m} \).

Key Concepts

Planck's constant

The concept of Planck's constant is crucial in the realm of quantum mechanics. Planck's constant, symbolized as \( h \), is a fundamental quantity that relates the energy of a photon to its frequency. Its value is approximately \( 6.626 \times 10^{-34} \text{ J} \cdot \text{s} \).
Planck's constant is pivotal in the calculation of a particle's de Broglie wavelength, which helps us understand the wave-particle duality of matter. This principle suggests that not only does light have wave-like properties, but particles such as electrons and protons do as well. When you apply Planck's constant in equations, it helps to bridge the gap between the classical world of physics and quantum mechanics.
Together with the formula \( \lambda = \frac{h}{mv} \), where \( \lambda \) is the wavelength, \( m \) is the mass of the particle, and \( v \) is its velocity, Planck's constant allows us to calculate how a particle behaves under these dual properties.

Electron mass

Electrons are fundamental components of atoms, incredibly small and carrying a negative electric charge. To calculate the properties such as the de Broglie wavelength, knowing the mass of an electron is vital. The electron mass is approximately \( 9.11 \times 10^{-31} \text{ kg} \).
This value is minuscule compared to everyday standards, yet for subatomic particles, it provides immense insight. When you plug this value into the de Broglie wavelength formula, it allows you to determine the wavelength associated with an electron moving at a given velocity.
For instance, an electron traveling at a speed of \( 4.70 \times 10^6 \text{ m/s} \) can have its wavelength calculated as shown in the step-by-step solution. This small mass plays a significant role in giving electrons the remarkable capability of exhibiting wavelike behaviors.

Proton mass

Protons, along with neutrons, form the atomic nucleus; they're heavier than their electron counterparts. The mass of a proton is about \( 1.67 \times 10^{-27} \text{ kg} \).
This value is essential for understanding how protons interact with other particles and how they exhibit their wave-like properties. Applying this knowledge to the de Broglie wavelength concept provides insight into the microscopic behaviors of protons.
Given a proton traveling at the same velocity as an electron (\( 4.70 \times 10^6 \text{ m/s} \)), calculations show that due to its larger mass, the de Broglie wavelength of a proton is significantly shorter than that of an electron, emphasizing the differences in quantum behaviors between particles of different masses.

Velocity of particles

The velocity of a particle is a critical factor in determining its de Broglie wavelength. Velocity comes into play as one of the parameters in the formula \( \lambda = \frac{h}{mv} \).
Considerations about the velocity of particles are important since it helps to delineate the relationship between the particle's motion and its wave-energy equivalent. When a particle's speed increases, its de Broglie wavelength decreases, meaning faster particles have shorter wavelengths.
In the given exercise, both an electron and a proton are moving at the same velocity of \( 4.70 \times 10^6 \text{ m/s} \). Despite sharing the same speed, their varying masses lead them to exhibit differently scaled wavelengths, illustrating how velocity interplays with mass to influence quantum mechanical properties.