b) Assuming thatR(r) is given, write down the energy eigenfunctions for the ground state
and the first excited state.
c) Assuming that both masses are about 1000 MeV, how does the excitation energy of the
first excited state compare to thermal energies at 300◦K.15 The Radial Equation and Constant Potentials*
15.1 The Radial Equation*.
After separation of variables, the radial equation depends onℓ.
− ̄h^2
2 μ[
1
r^2(
r∂
∂r) 2
+
1
r∂
∂r−
ℓ(ℓ+ 1)
r^2]
REℓ(r) +V(r)REℓ(r) =EREℓ(r)It can be simplified a bit.
− ̄h^2
2 μ[
∂^2
∂r^2+
2
r∂
∂r−
ℓ(ℓ+ 1)
r^2]
REℓ(r) +V(r)REℓ(r) =EREℓ(r)Theterm due to angular momentumis often included with the potential.
- − ̄h^2
2 μ
[
∂^2
∂r^2+
2
r∂
∂r]
Rnℓ(r) +(
V(r) +
ℓ(ℓ+ 1) ̄h^2
2 μr^2)
Rnℓ(r) =ERnℓ(r)This pseudo-potential repels the particle from the origin.
15.2 Behavior at the Origin*.
Thepseudo-potential dominates the behavior of the wavefunction at the originif the
potential is less singular thanr^12.
− ̄h^2
2 μ[
∂^2
∂r^2+
2
r∂
∂r]
Rnℓ(r) +(
V(r) +
ℓ(ℓ+ 1) ̄h^2
2 μr^2)
Rnℓ(r) =ERnℓ(r)For smallr, the equation becomes
[
∂^2
∂r^2+
2
r∂
∂r]
Rnℓ(r)−ℓ(ℓ+ 1)
r^2
Rnℓ(r) = 0The dominant term at the origin will be given by some power ofr
R(r) =rs.