- #1
spaghetti3451
- 1,344
- 33
Consider the radial differential equation
##\bigg( - \frac{d^2}{dr^2} + \frac{(l+\frac{d-3}{2})(l+\frac{d-1}{2})}{r^2} + V(r) + m^2 \bigg) \phi_l (r) = \lambda\ \phi_l (r)##,
which I've obtained by solving the Schrodinger equation in ##d## dimensions using the method of separation of variables.
Now, the boundary condition that I have been given is ##\phi_l (r) \sim r^{l+\frac{d-1}{2}}, r \rightarrow 0##.
However, I was expecting the boundary conditions ##\phi_l (0) = 0, \phi^{'}_{l} (0) = 1##.
Does anybody have an idea if there is a relation of some sort between the two sets of boundary conditions in the context of the given differential operator.
##\bigg( - \frac{d^2}{dr^2} + \frac{(l+\frac{d-3}{2})(l+\frac{d-1}{2})}{r^2} + V(r) + m^2 \bigg) \phi_l (r) = \lambda\ \phi_l (r)##,
which I've obtained by solving the Schrodinger equation in ##d## dimensions using the method of separation of variables.
Now, the boundary condition that I have been given is ##\phi_l (r) \sim r^{l+\frac{d-1}{2}}, r \rightarrow 0##.
However, I was expecting the boundary conditions ##\phi_l (0) = 0, \phi^{'}_{l} (0) = 1##.
Does anybody have an idea if there is a relation of some sort between the two sets of boundary conditions in the context of the given differential operator.