A particle moves in a central potential U(r) = −k/re^−r/ξ (screened potential, with k > 0 and ξ > 0). For what values of the angular momentum l_0 a bounded motion of the particle (i.e. when the particle does not escape to the infinity) is possible?
Hint: A bounded motion is possible if the effective potential has a minimum – you can admit that if the effective potential has an extremum, then this is a minimum. You might find it useful to introduce the dimensionless variable
u = r/ξ, and to study the function f(u) = u(1 + u)e^−u.
To determine the values of angular momentum l_0 for which bounded motion of the particle is possible, we need to analyze the behavior of the effective potential. The effective potential is given by V_eff(r) = U(r) + l^2/(2mr^2), where l is the angular momentum and m is the mass of the particle.
Since the potential U(r) is central, the angular momentum is a conserved quantity. We can write l^2 = l_0^2, where l_0 is a constant.
Now, let's substitute the dimensionless variable u = r/ξ and rewrite the effective potential in terms of u:
V_eff(u) = -k/u * e^(-u/ξ) + l_0^2/(2mu^2)
To determine whether bounded motion is possible, we need to find the critical points of the effective potential, where its derivative with respect to u is zero:
dV_eff(u)/du = -k/u^2 * e^(-u/ξ) - 1/ξ * (-k/u * e^(-u/ξ)) + l_0^2/(mu^3) = 0
Simplifying this equation, we get:
-k/u^2 * e^(-u/ξ) + k/ξu * e^(-u/ξ) + l_0^2/(mu^3) = 0
Dividing through by k e^(-u/ξ) and multiplying through by ξu^3, we obtain:
-ξ/u + 1 + l_0^2/(mu^3) = 0
Multiplying through by mu^3, we get:
-ξu^2 + mu^3 + l_0^2 = 0
Now, we can define the function f(u) = u(1 + u)e^(-u) as given in the hint. The critical points of the effective potential correspond to the solutions of f(u) = 0.
By analyzing the behavior of the function f(u) = u(1 + u)e^(-u), we can determine the values of u for which bounded motion is possible.
It would be helpful to plot the function f(u) and observe the points where it intersects the x-axis. By observing the intersections, we can find the values of u for which f(u) = 0.
Once we have the values of u, we can relate them back to r using the equation u = r/ξ. So, the values of r for which bounded motion is possible can be found by multiplying the values of u by ξ.
To summarize, to find the values of the angular momentum l_0 for which bounded motion of the particle is possible, follow these steps:
1. Rewrite the effective potential in terms of the dimensionless variable u = r/ξ.
2. Find the critical points of the effective potential by solving the equation dV_eff(u)/du = 0.
3. Rewrite the equation in terms of the function f(u) = u(1 + u)e^(-u).
4. Plot the function f(u) and determine the values of u for which f(u) = 0.
5. Relate the values of u back to r using the equation u = r/ξ.
6. The values of r for which bounded motion is possible are given by multiplying the values of u by ξ.
Note: This explanation assumes that you are familiar with plotting functions and solving equations.