Free particle in spherical polar coords

[cscott]

Homework Statement

Consider the time-independent Schrodinger equation in spherical polar coordinates for a free particle, in the case where we have an azimuthal quantum number l=0.

(a) Solve the radial equation to find the (unnormalized) radial wavefunction R(r).
(b) Normalize R(r), using the definition of the dirac delta function \delta(k'-k).

Homework Equations

Radial Equation:

u(r) = rR(r)

-\frac{\hbar^2}{2m}\frac{d^2u}{dr^2} + \left[V + \frac{\hbar^2}{2m}\frac{l(l+1)}{r^2}\right]u = Eu

The Attempt at a Solution

For a free particle, V=0, and with l = 0 the radial equation reduces to,

\frac{d^2u}{dr^2}=-k^2u,~~k=\frac{\sqrt{2mE}}{\hbar},

with solution,

u = A\sin(kr) + B\cos(kr),

but u(r)=rR(r), so B=0 for a normalizable wavefunction (considering r->0). Therefore,

R(r) = \frac{A}{r}\sin(kr)

and to normalize,

\int_{0}^{\inf} r^2|R(r)|^2~dr = 1

|A|^2 \int_{0}^{\inf} \sin^2(kr)~dr=1

How do I use the dirac delta function?

 

[cscott]

Should I keep u as,

u(r)=Ae^{ikr}

R(r)=\frac{A}{r}e^{ikr}

 

[esorolla]

Hi

Sorry, but I don't see why do you need to use the delta of k'-k. I would integrate the sine square but the problem is that you forgot the integration for the angular variables. Thus there is a "4 times pi" factor missed which should be at RHS of the last equation as denominator of 1/(4*pi) since the 4*pi is the result of the integration of

\int sin\theta d\theta \int d\varphi

This comes from the fact that to normalize you have to integrate in a volume whose differential element is

dv= r^2 sin\theta dr d\theta d\varphi

But I don't see anything else to be added.