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

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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.

Rearrange the formula to solve for velocity \( v \): \( v = \frac{h}{m\lambda} \).

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.

Calculate the velocity \( v \) using the rearranged formula and the substituted values.

Ensure the units are consistent and check the calculation for any possible errors.

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주요 개념

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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.