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Homework Statement
How much of the time are the proton and neutron in a deuteron outside the range of the strong force? Suppose the strong force can be described by a spherical potential with parameters
##V_0 = 35 MeV##, ##R = 2.1fm##. The binding energy for deuteron is ##E_b = 2.22 MeV## and the ground state is mostly an ##s##-state.
Homework Equations
If the radial part of the wave equation is ##\Psi(r) = f(r)/r## then the radial equation is
##\frac{d^2f}{dr^2}+\frac{2m}{\hbar^2}\left[E-V(r)-\frac{\hbar^2}{2m}\frac{l(l+1)}{r^2}\right]f = 0##.
The Attempt at a Solution
The well is
##V(r) = \begin{cases}
-V_0, \; \; \; r<R\\
0, \; \; \; \; \; \; r>R
\end{cases}##
and we have a bound state so ##E = -E_b##. As I understand the question to find how much the time the proton and neutron is outside the strong force I should find the probability of finding the nucleus outside the well.
For an ##s##-state the radial equation becomes
##\frac{d^2f}{dr^2}+\frac{2m}{\hbar^2} \left(E-V(r)\right) = 0##.
The solutions become
##f_1(r) = A\sin k_1 r + B\cos k_1 r## for ##r<R## and
##f_ 2(r) = Ce^{-k_2r}+De^{k_2r}## for ##r>R##.
##f(r)/r## should be finite for all ##r##. It follows that ##B=D=0## so we the solutions are
##f_1(r) = A\sin k_1 r## for ##r<R## and
##f_ 2(r) = Ce^{-k_2r}## for ##r>R##
with ##k_1 = \sqrt{\frac{2m(E+V_0)}{\hbar^2}}## and ##k_2 = \sqrt{-2mE/\hbar^2}##.
The boundary conditions at ##r=R## for ##f## and ##f'_r## give us
##\begin{cases}A\sin k_1 R = Ce^{-k_2 R}\\
Ak_1 \cos k_1 R = -Ck_2 e^{-k_2 R}\end{cases} \Longrightarrow k_1 \cot k_1 R = -k_2.##
The above values for ##E## and ##V_0## satisfies this condition.
From the first boundary equation it's possible to calculate ##\frac{C}{A}=\frac{\sin k_2 R}{e^{-k_2R}} \approx 1.4645## (I used the mass of the deuteron ##m=2.014u## and converted everything to SI units). If I could norm the function it would be possible to find the probability. The norming condition becomes
##1 = 4\pi A^2 \left[ \int_0^R r^2 \sin^2k_1 r dr + 1.4645^2 \int_R^\infty r^2e^{-2k_2 r} dr \right]## however the values seem to be so small at this point that numerical integration give zero for the second integral. Am I going at this the right way?
Perhaps it would be simpler to write this is terms of normed eigenfunctions first but while this is simple for ##\sin k_1 r## I still get the same problem for the exponential function.