(a) An electron moves with a speed of 4.70x10^6 m/s. What is its de Broglie wavelength? (b) A proton moves with the same speed. Determine its de Broglie wavelength.

Verified step by step guidance

1. Identify the formula for the de Broglie wavelength, which is \(\lambda = \frac{h}{p}\), where \(h\) is the Planck constant (\(6.626 \times 10^{-34} \, \text{Js}\)) and \(p\) is the momentum of the particle.
2. Calculate the momentum of the electron using the formula \(p = mv\), where \(m\) is the mass of the electron (\(9.11 \times 10^{-31} \, \text{kg}\)) and \(v\) is the given speed of the electron (\(4.70 \times 10^6 \, \text{m/s}\)).
3. Substitute the calculated momentum of the electron into the de Broglie wavelength formula to find the wavelength of the electron.
4. Calculate the momentum of the proton using the same formula \(p = mv\), but use the mass of the proton (\(1.67 \times 10^{-27} \, \text{kg}\)) and the same speed (\(4.70 \times 10^6 \, \text{m/s}\)).
5. Substitute the calculated momentum of the proton into the de Broglie wavelength formula to find the wavelength of the proton.

Key Concepts

de Broglie Wavelength

The de Broglie wavelength is a fundamental concept in quantum mechanics that describes the wave-like behavior of particles. According to de Broglie's hypothesis, every moving particle has an associated wavelength, which can be calculated using the formula λ = h/p, where λ is the wavelength, h is Planck's constant, and p is the momentum of the particle.

Momentum

Momentum is a vector quantity defined as the product of an object's mass and its velocity (p = mv). In the context of de Broglie wavelength, momentum is crucial because it directly influences the wavelength of a particle. For electrons and protons, their respective masses will significantly affect their momentum and, consequently, their de Broglie wavelengths when moving at the same speed.

Planck's Constant

Planck's constant (h) is a fundamental physical constant that relates the energy of a photon to its frequency. It plays a critical role in quantum mechanics, particularly in the calculation of the de Broglie wavelength. The value of Planck's constant is approximately 6.626 x 10^-34 Js, and it is essential for determining the wave properties of particles like electrons and protons.