Protons can be accelerated to speeds near that of light in...

Protons can be accelerated to speeds near that of light in particle accelerators. Estimate the wavelength (in $\mathrm{nm}$ ) of such a proton moving at $2.90 \times 10^{8} \mathrm{m} / \mathrm{s}$ (mass of a proton $=1.673 \times 10^{-27} \mathrm{kg}$ ).

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Key Concepts

de Broglie Hypothesis

This concept posits that every particle exhibits wave-like behavior and that its wavelength is inversely proportional to its momentum. It forms a cornerstone of quantum mechanics by linking the moving particle’s properties with the characteristics of matter waves.

Momentum

Momentum, defined as the product of an object's mass and velocity, is critical in determining the de Broglie wavelength. In the context of particles moving at speeds close to that of light, adjustments from classical momentum to relativistic momentum may be required.

Relativistic Effects

Relativistic effects become significant when particles travel at speeds near the speed of light. These effects, such as the Lorentz factor, modify the momentum and energy computations, ensuring that calculations remain consistent with the theory of relativity.

Planck's Constant

Planck's constant is a fundamental quantity in quantum mechanics that bridges the gap between the particle and wave descriptions of matter. It appears in formulas like the de Broglie wavelength and represents the scale at which quantum effects become significant.

Transcript

00:02 So in this question, we are looking for the wavelength in nanometer of a proton.

00:11 We're given the velocity of the proton is 2 .90 times tens of the eighth meters per second.

00:22 And the mass of the proton is 1 .673 times 10 .9273.

00:30 Now anytime we're looking for the wavelength of a particle that has mass, unlike a photon which doesn't have mass, right? here we have a proton which has mass and so we're going to be using the bigogli equation, which is lambda equals flanks constant divided by mass and velocity.

00:54 So thanks constant is still the 6 .63 times centered on negative 34, dual seconds.

01:05 And this is the wavelength of that particle and it's in meters.

01:13 The mass has to be in kilograms.

01:21 And the reason for that is that a jewel is one kilogram meter square per second square.

01:32 So when the units cancel that, the kilogram will cancel that with the kilograms down here.

01:37 And the meters per second, at least one of them, right, with left over with another meter per second.

01:43 I'll cancel that with this.

01:47 And then this is the velocity, which is in meters per second.

01:59 So if we use that equation, we plug in everything, we get 6 .63 times 10 in negative 34.

02:10 And i'm going to write kilgram meters square per second squared in here...

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