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Quarks carry spin 1/2. Three quarks bind together to make a baryon (such as the proton or neutron); two quarks (or more precisely a quark and an antiquark) bind together to make a meson (such as the pion or the kaon). Assume the quarks are in the ground state (so the orbital angular momentum is zero).

(a) What spins are possible for baryons?

(b) What spins are possible for mesons?

Short Answer

Expert verified

(a) Spin 12or 32

(b) Spin 1 or 0

Step by step solution

01

Definition of baryon and meson

The subatomic particles having large mass and are the combination of three quarks are known as baryons.

The existence quark and an antiquark in a subatomic particle are termed Meson

02

(a) Determination of possible spins for baryon

Combine two quarks,12and12.

Spin 1 (If they are parallel),

12+12or-12+-12

Zero (If they are antiparallel),

role="math" localid="1658124052031" 12+-12or12+-12

Combine the result with the third quark for the case to get spin1, 12 and1.

32(if they are parallel),

12+1or-12-1

12(if they are antiparallel),

role="math" localid="1658123797200" -12+1or12-1

For the case to get zero, only 12 is there only.

Thus, the possible spins for baryons are 12and 32.

03

(b) Determination of possible spins for mesons

Combine two quarks, 12 and 12.

Spin 1 (if they are parallel),

12+12or-12+-12

Zero (if they are antiparallel),

12+-12or12+-12

Thus, the possible spins for mesons are 1 and 0.

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Most popular questions from this chapter

Deduce the condition for minimum uncertainty inSx andSy(that is, equality in the expression role="math" localid="1658378301742" SxSy(/2)|<Sz>|, for a particle of spin 1/2 in the generic state (Equation 4.139). Answer: With no loss of generality we can pick to be real; then the condition for minimum uncertainty is that bis either pure real or else pure imaginary.

(a) Use the recursion formula (Equation 4.76) to confirm that whenI=n-1 the radial wave function takes the form

Rn(n-1)=Nnrn-1e-r/na and determine the normalization constant by direct integration.

(b) Calculate 200a and <r2> for states of the form n(n-1)m

(c) Show that the "uncertainty" in r(r) is<r>/2n+1for such states. Note that the fractional spread in decreases, with increasing (in this sense the system "begins to look classical," with identifiable circular "orbits," for large ). Sketch the radial wave functions for several values of, to illustrate this point.

[Refer to Problem 4.59 for background.] In classical electrodynamics the potentials Aandare not uniquely determined; 47 the physical quantities are the fields, E and B.

(a) Show that the potentials

'-t,A'A+

(whereis an arbitrary real function of position and time). yield the same fields asand A. Equation 4.210 is called a gauge transformation, and the theory is said to be gauge invariant.

(b) In quantum mechanics the potentials play a more direct role, and it is of interest to know whether the theory remains gauge invariant. Show that

'eiq/

satisfies the Schr枚dinger equation (4.205) with the gauge-transformed potentials'andA', Since'differs fromonly by a phase factor, it represents the same physical state, 48and the theory is gauge invariant (see Section 10.2.3for further discussion).

Suppose two spin -1/2particles are known to be in the singlet configuration (Equation Let Sa(1)be the component of the spin angular momentum of particle number 1 in the direction defined by the unit vectora^ Similarly, letSb(2) be the component of 2鈥檚 angular momentum in the directionb^ Show that

Sa(1)Sb(2)=-24肠辞蝉胃

where is the angle between a^ andb^

(a) For a functionf()that can be expanded in a Taylor series, show that f(+)=eiLz/f() (where is an arbitrary angle). For this reason, Lz/ is called the generator of rotations about the Z-axis. Hint: Use Equation 4.129 , and refer Problem 3.39.More generally, Ln^/ is the generator of rotations about the direction n^, in the sense that exp(iLn^/)effects a rotation through angle (in the right-hand sense) about the axis n^ . In the case of spin, the generator of rotations is Sn^/. In particular, for spin 1/2 '=ei(n^)/2tells us how spinors rotate.

(b) Construct the (22)matrix representing rotation by 180about the X-axis, and show that it converts "spin up" +into "spin down"- , as you would expect.

(c) Construct the matrix representing rotation by 90about the Y-axis, and check what it does to

+

(d) Construct the matrix representing rotation by 360about the -Zaxis, If the answer is not quite what you expected, discuss its implications.

(e) Show thatei(n^)/2=cos(/2)+i(n^)sin(/2)

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