A proton in a linear accelerator has a de Broglie wavelength of 122 pm. What is the speed of the proton?

Verified step by step guidance

1. Identify the de Broglie wavelength formula: \( \lambda = \frac{h}{mv} \), where \( \lambda \) is the wavelength, \( h \) is Planck's constant, \( m \) is the mass of the particle, and \( v \) is the velocity.
2. Rearrange the formula to solve for velocity \( v \): \( v = \frac{h}{m\lambda} \).
3. Substitute the known values into the equation: Planck's constant \( h = 6.626 \times 10^{-34} \) Js, the mass of a proton \( m = 1.673 \times 10^{-27} \) kg, and the wavelength \( \lambda = 122 \times 10^{-12} \) m.
4. Calculate the velocity \( v \) using the rearranged formula and the substituted values.
5. Ensure the units are consistent and check the calculation for any possible errors.

Key Concepts

de Broglie Wavelength

The de Broglie wavelength is a fundamental concept in quantum mechanics that relates the wavelength of a particle to its momentum. It is given by the formula λ = h/p, where λ is the wavelength, h is Planck's constant, and p is the momentum of the particle. This concept illustrates the wave-particle duality of matter, indicating that particles like protons exhibit wave-like properties.

Momentum

Momentum is a vector quantity defined as the product of an object's mass and its velocity (p = mv). In the context of particles, especially in quantum mechanics, momentum plays a crucial role in determining the behavior of particles. For a proton, knowing its mass allows us to calculate its momentum when its speed is known, which is essential for finding its de Broglie wavelength.

Planck's Constant

Planck's constant (h) is a fundamental constant in quantum mechanics that relates the energy of a photon to its frequency. It has a value of approximately 6.626 x 10^-34 Js. In the context of the de Broglie wavelength, Planck's constant is used to connect the wave properties of particles to their momentum, enabling calculations that reveal the wave-like behavior of matter at the quantum level.