Consider the following purely attractive potentials:
|
Square Well |
-V0*Theta (R-r) |
|
Delta Shell |
-V0*delta (r-R). |
|
Exponential |
-V0*exp(-r/R) |
|
Yukawa |
-V0*exp(-r/R)/r |
|
Gaussian |
-V0*exp(-r^2/R^2) |
|
Poeschl-Teller |
-V0*sech(r/R)^2 |
For each of these potentials, calculate the critical strength V0 necessary to give zero energy binding for a two body system. Suppose the masses are equal. Express the well strength in units of hbar^2/m.
Suppose that the NN int. gives a semibound state, and, for Nc colors, also (Nc-1)/2 deep bound states. (These must be removed by some kind of repulsion.)
This model leads a intermediate range Yukawa attraction: V(r) = - g_s**2/4pi * exp(-mu*r)/r
where g_s = 2*pi*sqrt(Nc), i.e. g_s**2/4pi = pi * Nc = 9.45 (OBEP = 8.1) mu = 2*m_q = 2M/Nc (in units hbar = c = 1)
It is convenient to discuss the Born approximation scattering length. a_B = (M/4pi) * int (V(r) d3r)/4pi For the NJL Yukawa potential, we obtain: a_B = (pi/4) * Nc**3 / M
For Nc = 1, and range mu = 2M, this potential has just about the right strength to give a semibound state. For Nc = 3, it leads, of course, to a deeply bound state, but the first excited state is not far from semibound.
For the PT potential: V(r) = -(1/M) * be**2 * N(N+1) * sech(be*r)**2 (N = odd integer) the Schroedinger equation can be solved analytically. We get a bound state at zero energy, and (N-1)/2 deeply bound states in addition. (To get rid of the latter, we need a repulsion, but that is another story!) For large r, the asymptotic form of V(r) is exp (-2*be*r). Thus, let us identify be with mu/2 = m_q = M/Nc.
A simple calculation gives: a_B = N(N+1)*Nc * (pi**2/12) /M We use here: int (s**2 * sech(s)**2)ds = pi**2/12 Identifying N with Nc, we get: a_B = 0.822 * Nc**2 * (Nc+1) /M which is close to the Yukawa value: a_B = 0.785 * Nc**3 /M
This is an approximation to the above potentials for which the zero energy solutions can be obtained analytically: For V(r) = -(1/M) * be**2 * j(0,s)**2 * exp(-2*be*r) the zero energy wavefunction is: u(r) = J0 (j(0,s) * exp(-be*r)), which approaches 1 for r 1. Here j(0,s) is the s'th zero of j0. We obtain: a_B = j(0,s)**2/(4*be)
Now a very good approximation to the square of the zeros is: j(0,s)**2 = ((N+0.5)*pi/2)**2 + 0.25 (For s = 1, we get 2.4087 (exact = 2.4048), and it is even better for s1. Using be = M/Nc, we find: a_B = (0.0625*pi**2*Nc*(Nc+0.5)**2 + 0.125)/M For Nc = 1, a_B = 1.513/M (1.645/M for PT), For Nc 1, a_B = 0.616*Nc**3/M (0.822*Nc**3/M for PT)
.. For an Introduction to Nuclear Theory of the 1930's
..For an introduction to quark chemistry
..For an introduction to the three body problem by Macek and Fedorov
Updated 9-3-98 Copyright Steven A. Moszkowski