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Chapter 33: Q2PE (page 1212)

Calculate the mass in \[{\rm{GeV/}}{{\rm{c}}^{\rm{2}}}\]of a virtual carrier particle that has a range limited to \[{\rm{1}}{{\rm{0}}^{{\rm{ - 30}}}}{\rm{\;m}}\]by the Heisenberg uncertainty principle. Such a particle might be involved in the unification of the strong and electroweak forces.

Short Answer

Expert verified

The mass of the virtual carrier particle is \[9.9 \times {10^{13}}\;{\rm{GeV}}/{{\rm{c}}^2}\].

Step by step solution

01

Definition of Heisenberg’s uncertainty principle

The Grand Unified Theory states that for very high energy levels, the strong nuclear force can be unified with the electroweak forces, which include the electromagnetic and weak nuclear forces.

02

Given Data

Range of the particle is- \(d = {10^{ - 30}}\;{\rm{m}}\)

03

Finding the mass of the particle 

Using the uncertainty equation as a guide

\[{\rm{\Delta E\Delta t = }}\frac{{\rm{h}}}{{{\rm{4\pi }}}}\],

we can get the energy,\[{\rm{\Delta E}}\], which is equal to the mass in\[{\rm{GeV/}}{{\rm{c}}^{\rm{2}}}\].

Also, uncertainty in time is given as-

\[\Delta t = \frac{d}{c}\]

Using the above-mentioned equations-

\[\begin{array}{c}\Delta E = \frac{{hc}}{{4\pi c}}\\ = \frac{{\left( {4.14 \times {{10}^{ - 24}}\;{\rm{GeV}}{\rm{.s}}} \right) \times \left( {{{10}^3} \times {{10}^8}\;{\rm{m/s}}} \right)}}{{4\pi \times {{10}^{ - 30}}\;{\rm{m}}}}\\ = 9.9 \times {10^{13}}\;{\rm{GeV}}\end{array}\]

Hence, the mass of the particle is \[{\rm{9}}{\rm{.9 \times 1}}{{\rm{0}}^{{\rm{13}}}}\;{\rm{GeV/}}{{\rm{c}}^{\rm{2}}}{\rm{.}}\]

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Most popular questions from this chapter

The \[{{\rm{\pi }}^{\rm{0}}}\]is its own antiparticle and decays in the following manner: \[{\pi ^0} \to \gamma + \gamma \]. What is the energy of each \[\gamma \]ray if the \[{\pi ^0}\] is at rest when it decays?

Accelerators such as the Triangle Universities Meson Facility (TRIUMF) in British Columbia produce secondary beams of pions by having an intense primary proton beam strike a target. Such "meson factories" have been used for many years to study the interaction of pions with nuclei and, hence, the strong nuclear force. One reaction that occurs is\({{\rm{\pi }}^{\rm{ + }}}{\rm{ + p}} \to {{\rm{\Delta }}^{{\rm{ + + }}}} \to {{\rm{\pi }}^{\rm{ + }}}{\rm{ + p}}\), where the \({{\rm{\Delta }}^{{\rm{ + + }}}}\)is a very short-lived particle. The graph in Figure \({\rm{33}}{\rm{.26}}\)shows the probability of this reaction as a function of energy. The width of the bump is the uncertainty in energy due to the short lifetime of the\({{\rm{\Delta }}^{{\rm{ + + }}}}\).

(a) Find this lifetime.

(b) Verify from the quark composition of the particles that this reaction annihilates and then re-creates a d quark and a \({\rm{\bar d}}\)antiquark by writing the reaction and decay in terms of quarks.

(c) Draw a Feynman diagram of the production and decay of the \({{\rm{\Delta }}^{{\rm{ + + }}}}\)showing the individual quarks involved.

A proton and an antiproton collide head-on, with each having a kinetic energy of 7.00TeV (such as in the LHC at CERN). How much collision energy is available, taking into account the annihilation of the two masses? (Note that this is not significantly greater than the extremely relativistic kinetic energy.)

The decay mode of the positive tau is\({{\bf{\tau }}^ + } \to {\rm{ }}{{\bf{\mu }}^ + }{\rm{ }} + {\rm{ }}{{\bf{\nu }}_{\bf{\mu }}}{\rm{ }} + {\rm{ }}{{\bf{\bar \nu }}_{\bf{\tau }}}\).

(a) What energy is released?

(b) Verify that charge and lepton family numbers are conserved.

(c) The \({\tau ^ + }\)is the antiparticle of the \({\tau ^ - }\). Verify that all the decay products of the \({\tau ^ + }\)are the antiparticles of those in the decay of the \({\tau ^ - }\) given in the text.

Discuss how we know that \[{\rm{\pi }}\]-mesons\[\left( {{{\rm{\pi }}^{\rm{ + }}}{\rm{,\pi ,}}{{\rm{\pi }}^{\rm{0}}}} \right)\]) are not fundamental particles and are not the basic carriers of the strong force.
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