y***u
发帖数: 25 | 1 V=0 as ra
say, the wave function is w=R(r)O(o), o is the angle
By using the polar coordinate the equation becomes
R''O+R'O/r+RO''/r^2=(V-k^2)RO
where V is the potential,which is zero here. k^2=2mE/h^2
so
r^2*R''/R+r*R'/R+r^2*k^2=-O''/O=m^2
where m is an integer in order to keep O(o+2*Pi)=O(o)
so
O(o)=Exp[I*m*o]
The radial equation is
R''+R'/r+[k^2-m^2/r^2]R=0
set r=x/k
R''+R'/x+[1-m^2/x^2]R=0
which is exactly the Bessel equation. The two linearly independent solutions
are
J_ |
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