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Free particle in spherical polar coords

  1. Jun 2, 2009 #1
    1. The problem statement, all variables and given/known data

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

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

    2. Relevant equations

    Radial Equation:

    [tex]u(r) = rR(r)[/tex]

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

    3. The attempt at a solution

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

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

    with solution,

    [tex]u = A\sin(kr) + B\cos(kr)[/tex],

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

    [tex]R(r) = \frac{A}{r}\sin(kr)[/tex]

    and to normalize,

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

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

    How do I use the dirac delta function?
     
    Last edited: Jun 3, 2009
  2. jcsd
  3. Jun 3, 2009 #2
    Should I keep [itex]u[/itex] as,

    [tex]u(r)=Ae^{ikr}[/tex]

    [tex]R(r)=\frac{A}{r}e^{ikr}[/tex]
     
  4. Aug 5, 2009 #3
    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

    [tex]\int sin\theta d\theta \int d\varphi[/tex]

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

    [tex]dv= r^2 sin\theta dr d\theta d\varphi[/tex]

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