Question

C9M.6 Consider two quarks with masses mi and m2. The potential energy of the strong nuclear interaction between two quarks is roughly V(r) = br, where b is a constant. (a) Draw a qualitative sketch of V(r) for this system assuming that the quarks move in one dimension. (b) Describe the possible types of motion for this system. (c) Can the quarks ever have enough energy to be free of each other (that is, to move to infinite separation)?

Answer 1

VT) vs

b) Suppose the energy of the quarks is E as shown in the figure

The energy is constant and the quark starts from r = 0. At r = r1, the energy of the quark is equal to the potential energy.

$E=br_1,\Rightarrow r_1=E/b$

This is the point where the quark has zero kinetic energy and hence its velocity becomes zero. Thus r = r₁ is the turning point. The quark turns around and moves back to r = 0. Thus the quark with energy E is bound in the potential well from r = 0 to r = r₁

The motion of the quark in the potential well is to and from between r = 0 to r = r₁ and hence the motion is oscillatory

c) As described above, the quark's turning point is

$r_1=E/b$

For the quarks to be free, r₁ has to be infinity. Since the energy of the quarks is finite, the separation can never be infinite and hence they can never be free from each other.