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Chapter 5 Quark Model

Chapter 5 5.1 Introduction 5.2 Quark Model 5.3 Meson and Baryon wave function 5.4 Magnetic moment and masses of baryons 5.5 Interactive Exercise

5.5 Interactive Exercise

Quark Model

Particle Physics

Interactive Exercise Outline: I have designed an interactive exercise to write baryon wave function in Quark Model in Adobe Flash Professional CS5 in which user interacts with the on the exercise. Baryons Baryons have half integral spin, thus they are fermions. Most familiar baryons are protons and neutrons. By eightfold Way classification, baryons can be classified into groups of 1, 8 or 10 members which are called singlets, octets and decuplets respectively. Lightest baryons supermultiplets are octet J =1/2+ +


and decuplet, Jp=3/2+. The 1/2 octet includes the nucleons, the Λ and Σ particles, together with cascade particles Ξ. Wave Functions Baryon is a three body system, so two orbital angular momentum should be considered. We will consider the ground state for l=l'=0. Angular momentum of baryon comes entirely from combined spins of three quarks. The quarks spin can combine to give either a total 1/2 or 3/2. Spin -3/2 combinations are completely symmetric in the sense that interchanging any two particles leaves the state untouched.

Dayalbagh Educational Institute


Quark Model

Particle Physics

The spin-1/2 combinations are partially antisymmetric, means interchange of any two particles reverses the sign. Then wave function for system is ψ (1,2) = ψα(1) ψβ(2), if particle 1 is in ψα and other is in state ψβ. Or ψ (1, 2) = ψβ(1) ψα(2), if particle 1 is in ψβ and other is in state ψα . If particles are identical bosons, the wavefunction is the symmetric combination ψ(1,2) =1/√2(ψα(1) ψβ(2) + ψβ(1) ψα(2)) And if they are identical fermions, the wave function is the antisymmetric combination ψ(1,2) =1/√2(ψα(1) ψβ(2) – ψβ(1) ψα(2)) If we put two fermions in same state ψα = ψβ, ψ (1,2) =0.

Exercise By using the antisymmetric combination of identical fermions, we can write the wavefunction of outer elements






First in








interchanging 1st and 2nd quark reverses the sign of wavefunction. Then wavefunction antisymmetric in 2 and 3 and finally wavefunction antisymmetric in 1 and 3. Dayalbagh Educational Institute


Quark Model

Particle Physics

The quark content of neutron is (udd), proton (udu), ∑ (dsd), ∑+(usu), Ξ-(dss), Ξ0(uss). Quark content of each particle is given. The quarks should be arranged so that it gives nd

wavefunction antisymmetric in 1st and 2nd quark, 2 and rd



3 quark, and 1 and 3 quark. Let us take the example of proton (udu). If we will put two








wavefunction formula, wavefunction will be zero. So one u quark should be kept out of the bracket and u and d quarks should be arranged inside bracket. If positions of u and d quark at 1st and 2nd position is interchanged the resultant wavefunction differs the previous one by a negative sign. Proton wavefunction is: ψp (1,2) = 1/√2 (ud–du) u If we interchange 1


and 2


quark position, means u

and d quark, the resultant wavefunction differs this wavefunction by negative sign. Similarly wavefunction of all baryons can be written and checked for antisymmetrization.

Dayalbagh Educational Institute


Quark Model

Particle Physics

Front page of the exercise

In this excercise the outer octet baryons are represented as hexagonal structure.

Dayalbagh Educational Institute


Quark Model

Particle Physics

Exercise on baryon spin wavefunction. Spin are aligned in two directions, upward direction, up spin ‘↑’ or in downward direction, down spin ‘↓’. Up and down Spins are arranged in boxes and checked by clicking check button. Symmetric wave functions are: |

〉 = (↑↑↑)


½ 〉 = (↑↑↓ + ↑↑↓ + ↓↑↓)


½ 〉 = (↓↓↑ + ↓↑↓ + ↑↓↓) |

〉 = (↓↓↓)

Antisymmetric wave function formula, where subscripts represents in which particles wave function is antisymmetric, in 1and 2 or 2 and 3 or 1 and 3. |½

½ 〉12 = (↑↓ - ↓↑) ↑ /√2

½ 〉12 = (↑↓ - ↓↑) ↓ /√2

½〉23 = ↑ (↑↓ - ↓↑) /√2

½ 〉23 = ↓ (↑↓ - ↓↑) /√2

½〉 13 = (↑↑↓ - ↓↑↑) /√2

½ 〉13 = (↑↓↓ - ↓↓↑) /√2

Dayalbagh Educational Institute


Quark Model

Particle Physics

Front page of exercise.

The stars on the page are two buttons, on clicking the symmetric and antisymmetric wave function formula page appears.

Dayalbagh Educational Institute


Quark Model

Particle Physics

This is the end of sub section of the chapter 5

Click to go on first slide

Dayalbagh Educational Institute


Chapter 5.5