Quarks and gluons are fundamental particles that will be...

Quarks and gluons are fundamental particles that will be discussed in Chapter $44 .$ A proton, which is a bound state of two up quarks and a down quark, has a rest mass of $m_{\mathrm{p}}=1.67 \times 10^{-27} \mathrm{~kg}$. This is significantly greater than the sum of the rest mass of the up quarks, which is $m_{\mathrm{u}}=4.12 \times 10^{-30} \mathrm{~kg}$ each, and the rest mass of the down quark, which is $m_{\mathrm{d}}=8.59 \times 10^{-30} \mathrm{~kg} .$ Suppose we (incorrectly) model the rest energy of the proton $m_{\mathrm{p}} c^{2}$ as derived from the kinetic energy of the three quarks, and we split that energy equally among them. (a) Estimate the Lorentz factor $\gamma=\left(1-v^{2} / c^{2}\right)^{-1 / 2}$ for each of the up quarks using Eq. $(37.36) .$ (b) Similarly estimate the Lorentz factor $\gamma$ for the down quark. (c) Are the corresponding speeds $v_{\mathrm{u}}$ and $v_{\mathrm{d}}$ greater than $99 \%$ of the speed of light? (d) More realistically, the quarks are held together by massless gluons, which mediate the strong nuclear interaction. Suppose we model the proton as the three quarks, each with a speed of $0.90 c,$ with the remainder of the proton rest energy supplied by gluons. In this case, estimate the percentage of the proton rest energy associated with gluons. (e) Model a quark as oscillating with an average speed of $0.90 c$ across the diameter of a proton, $1.7 \times 10^{-15} \mathrm{~m}$. Estimate the frequency of that motion.

Quarks and gluons are fundamental particles that will be discussed in Chapter $44 .$ A proton, which is a bound state of two up quarks and a down quark, has a rest mass of $m_{\mathrm{p}}=1.67 \times 10^{-27} \mathrm{~kg}$. This is significantly greater than the sum of the rest mass of the up quarks, which is $m_{\mathrm{u}}=4.12 \times 10^{-30} \mathrm{~kg}$ each, and the rest mass of the down quark, which is $m_{\mathrm{d}}=8.59 \times 10^{-30} \mathrm{~kg} .$ Suppose we (incorrectly) model the rest energy of the proton $m_{\mathrm{p}} c^{2}$ as derived from the kinetic energy of the three quarks, and we split that energy equally among them.
(a) Estimate the Lorentz factor $\gamma=\left(1-v^{2} / c^{2}\right)^{-1 / 2}$ for each of the up quarks using Eq. $(37.36) .$ (b) Similarly estimate the Lorentz factor $\gamma$ for the down quark.
(c) Are the corresponding speeds $v_{\mathrm{u}}$ and $v_{\mathrm{d}}$ greater than $99 \%$ of the speed of light? (d) More realistically, the quarks are held together by massless gluons, which mediate the strong nuclear interaction. Suppose we model the proton as the three quarks, each with a speed of $0.90 c,$ with the remainder of the proton rest energy supplied by gluons. In this case, estimate the percentage of the proton rest energy associated with gluons.
(e) Model a quark as oscillating with an average speed of $0.90 c$ across the diameter of a proton, $1.7 \times 10^{-15} \mathrm{~m}$. Estimate the frequency of that motion.

Key Concepts

Relativistic Kinematics

This concept involves the use of the Lorentz factor, ?, which is central to the theory of special relativity. The Lorentz factor quantifies how time, length, and energy transform for an object moving at a high fraction of the speed of light. It is used to relate the rest energy of particles to their kinetic energy when they approach relativistic speeds, and it helps in calculating how close a particle’s speed is to the speed of light.

Mass–Energy Equivalence

Rooted in Einstein’s famous equation, E = mc², mass–energy equivalence provides the framework for understanding that the mass of a composite system is not simply the sum of the individual rest masses. This principle is applied in particle physics to explain the additional mass arising from binding energy and the kinetic energy of the constituents in systems like the proton.

Quark Model of Hadrons

The quark model describes hadrons (such as protons and neutrons) as composite particles made up of quarks bound together by strong interactions. In this model, the properties of hadrons emerge not only from the individual quarks and their rest masses but also from the dynamic energies associated with their interactions and motions.

Strong Nuclear Force and Gluons

This concept is essential for understanding how quarks are held together within hadrons. The strong nuclear force, mediated by massless gluons, is far more powerful than electromagnetic forces at nuclear distances. Gluons contribute significantly to the overall energy and mass of particles like the proton by providing the binding energy necessary to confine quarks within a small volume.

Oscillatory Motion and Frequency Estimation in Quantum Systems

When considering quarks confined within a proton, one can model their motion as oscillatory across a finite spatial region. Estimating the frequency of such motion requires associating the average speed and the confinement size of the particle. This concept connects the ideas of quantized motion and energy levels with spatial confinement inherent in subatomic systems.

Transcript

00:01 For this problem on the topic of relativity, we want to model the rest energy of a proton as derived from the kinetic energy of the three quarks, and we split that energy equally among them.

00:11 We then want to estimate the lorentz factor for each of the up quarks and estimate the lorentz factor for the down quark.

00:18 And we then want to know if the corresponding speeds for the up quark and down quark are greater than 99 % the speed of light, and estimate the percentage of the proton rest energy that are associated with the gluons.

00:30 And lastly, you want to model the quark as an oscillating object with an average speed of 0 .9c across the diameter of a proton and estimate the frequency of that motion.

00:44 Now we use relativistic energy to investigate the quark model of the proton.

00:49 And we first want the tolerance factor for the up quark gamma u.

00:54 And so we can use the equation m -p -c -squared is equal to two times the kinetic energy of the up quark, since there are two up -quarks plus the kinetic energy of the down quark, k -d, and this is equal to three times the kinetic energy of the up -quark.

01:20 And so this kinetic energy, k -u, is equal to one -third m -c -c -squared.

01:34 And this is equal to the mass of the up quark times c squared into gamma u minus one so if we rearrange and solve for gamma u we get gamma u to be 136 given the values in the problem now for b we want gamma d so using the same procedure as above we find the larynx factor for the down quark gamma d to be 65 .8.

02:24 For part c, we want the speed vu over c, which is equal to the square root of 1 minus 1 over gamma u squared, which gives us 0 .9997.

02:50 And we can see that this is greater than 99 % the speed of light.

02:56 So the answer here is yes...

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