Nuclear Force

Question 1

What is the nuclear spin of a deuteron?

Answer 1

I = 1

Question 2

Total angular momentum of deuteron, I eqn

Answer 2

I = s_n + s_p + L

Question 3

Expectation values for L^2 and L_z (and s^2 and s_z)

Answer 3

<L^2> = L(L+1) hbar^2

<L_z> - hbar m_L
</L_z>

Question 4

Magnetic quantum numbers m_L (and m_s)

Answer 4

m_L = -L, … , 0 , … , L

Question 5

Total angular momentum J

Answer 5

J = L + s

Question 6

What are the allowed values of J

Answer 6

integer steps from |L-s| to L+s

Question 7

What are allowed values of s for deuteron

Answer 7

s = 1 and s = 0

Question 8

What are the allowed values of I,
what do we want for deuteron?

Answer 8

integer steps from |s-L| to s+L

we want I = 1 for deuteron

Question 9

If s = 0 what are values of L that allow I = 1

Answer 9

L = 1

Question 10

If s = 1 what are the allowed values of L for I = 1

Answer 10

L = 0, L = 1, L = 2

Question 11

What are the 4 states that satisfy I = 1

Answer 11

s = 0 , L = 1
s = 1 , L = 0
s = 1 , L = 1
s = 1 , L = 2

Question 12

Parity of deuteron, proton and neutron are all even so L must be

Answer 12

even

Question 13

What are the 2 states that satisfy I = 1and parity

Answer 13

s = 1 , L = 0 (96%)
s = 1 , L = 2

Question 14

What are the key properties of the strong nuclear force (5)?

Answer 14

Strong (~100× electromagnetic), short range (~2 fm), attractive with repulsive core, charge independent, spin dependent.

Question 15

What does spin dependence of the nuclear force mean?

Answer 15

The strength of the force depends on the relative spin orientation of nucleons.

Question 16

What are the 2 states that satisfy I = 1 and parity

Answer 16

s = 1 , L = 0 (96%)
s = 1 , L = 2

Question 17

How is the deuteron potential modeled in the square well model?

Answer 17

As a 3D square well with radius R and depth V₀.

Question 18

What state do we assume the deuteron is in?

Answer 18

L = 0

Question 19

What does r represent in the deuteron square well model?

Answer 19

The separation distance between proton and neutron.

Question 20

Why is there no Coulomb barrier in the deuteron square well model?

Answer 20

Because the deuteron consists of a proton and neutron, and the neutron has no charge.

Question 21

How can a two-body quantum problem be simplified?

Answer 21

By using the reduced mass to convert it into an effective one-body problem.

Question 22

What happens to the wavefunction inside the square well?

Answer 22

It oscillates.

Question 23

What happens to the wavefunction outside the square well?

Answer 23

It decays exponentially.

Question 24

Why must the wavefunction remain finite?

Answer 24

Because physical wavefunctions cannot diverge to infinity.

Question 25

What is the approximate radius of the nuclear potential well for the deuteron?

Answer 25

About 2.1 fm.

Question 26

What is the binding energy of the deuteron?

Answer 26

About 2.224 MeV.

Question 27

What is the approximate depth of the nuclear potential well?

Answer 27

About 35 MeV.

Question 28

What does it mean that the deuteron is weakly bound?

Answer 28

Its binding energy is small and close to the top of the potential well.

Question 29

What would happen if the nuclear force were slightly weaker?

Answer 29

The deuteron would not exist.

Question 30

What spin configuration is dominant in the deuteron?

Answer 30

L = 0 , s = 1, about 96%.

Question 31

What is the triplet state spin configuration?

Answer 31

Parallel spins with total spin s = 1.

Question 32

Why is it called a triplet state?

Answer 32

It has three possible spin projections: −1, 0, +1. s = 1 m_s = 1, 0, -1

Question 33

Why is it called a singlet state?

Answer 33

It has only one possible spin projection. s = 0

Question 34

Why do di-proton and di-neutron bound states, and deuteron anti-parallel s=0 not exist?

Answer 34

The nuclear force is weaker for anti-parallel spins due to spin dependence. PEP need anti parallel protons/neutrons

Question 35

Overall cross section of deuteron

Answer 35

sigma total = 3/4 sigma triplet + 1/4 sigma singlet

Question 36

What did Yukawa propose about the origin of the nuclear force?

Answer 36

It is caused by exchange of virtual particles.

Question 37

What particles mediate the nuclear force?

Answer 37

Pions.

Question 38

What spin must the exchange particle have?

Answer 38

Integer spin (boson).

Question 39

What charges can exchange particles have?

Answer 39

−1, 0, or +1 e.

Question 40

What are virtual particles?

Answer 40

Short-lived particles that temporarily exist during interactions.

Question 41

What principle allows virtual particles to exist?

Answer 41

The Heisenberg uncertainty principle.

Question 42

How is the range of a force related to exchange particle mass?

Answer 42

Shorter range corresponds to larger mass. m_ex c^2 ~ (hbar)c/R

Question 43

What diagram represents particle interactions visually?

Answer 43

Feynman diagram.

Question 44

What do the lines in Feynman diagrams represent?

Answer 44

Particles and exchanged mediators.

Question 45

What do vertices in Feynman diagrams represent?

Answer 45

Interaction points where charge and momentum are conserved.

Question 46

How does time flow in Feynman diagrams?

Answer 46

From left to right.

Question 47

What do dotted or curvy lines represent in nuclear force diagrams?

Answer 47

Exchange particles such as pions.

Question 48

What conservation laws apply at each vertex?

Answer 48

Conservation of momentum and charge.