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 nm) of such a proton moving at $2.90 \times 10^{8} \mathrm{m} / \mathrm{s}$ (Mass of a proton $\left.=1.673 \times 10^{-27} \mathrm{kg} .\right)$

Key Concepts

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.

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.

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.

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.

Transcript

00:01 Okay, this question is asking us to estimate the wavelength of a particle moving very fast.

00:07 So, using a altered version of debrogly's equation, we can say that wavelength is equal to plank's constant divided by square root two times the mass of the particle times the kinetic energy of the particle.

00:29 Okay? and we can find the kinetic energy, which is equal to one half mass of the particle times velocity squared.

00:41 We know that the velocity is in the mass, so we can find the kinetic energy, which is one half, 1 .673 times 10 to the negative 27 kilograms.

01:10 Applied by the speed, which is 2 .9 times 10 to the 8 meters per second.

01:19 And that is going to be squared.

01:29 Just that.

01:30 So kinetic energy comes out to be, once we plug it into the calculator, negative energy is 7 .03 times 10 to the negative 11th joules.

01:44 And since we got joules as our variable we know that that's at least we know we got the we got the units correct because jules is a units of energy and when we go ahead and cancel out when we go ahead and do this kilogram times meters per second times meter squared over second squared is also known as jules so when we go ahead and plug the kinetic energy mass and planks constant into the equation we get that plank's constant is 6 .63 times 10 to the negative 34th joules per second we're going to go ahead and set the square root of two times the mass of the particle which is 1 .673 times 10 to the negative 27th kilograms times the kinetic energy this is still under the square root which is 7 .03 times 10 to the negative 11th joules so when we plug all of that into the calculator we get that the wavelength for this particle which is traveling almost as fast as the speed of light is going to be 1 .36 times 10 to the negative 15 meters now this question asked for the wavelength in nanometers...

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